Characterization of Non-Smooth Pseudodifferential Operators

Abstract

Smooth pseudodifferential operators on $\mathbb {R}^{n}$ can be characterized by their mapping properties between $L^p-$Sobolev spaces due to Beals and Ueberberg. In applications such a characterization would also be useful in the non-smooth case, for example to show the regularity of solutions of a partial differential equation. Therefore, we will show that every linear operator P, which satisfies some specific continuity assumptions, is a non-smooth pseudodifferential operator of the symbol-class $C^{\tau } S^m_{1,0}(\mathbb {R}^n \times \mathbb {R}^n)$. The main new difficulties are the limited mapping properties of pseudodifferential operators with non-smooth symbols.

1 Introduction

In the smooth case some characterizations of pseudodifferential operators are already proved e.g. for the Hörmander class $S^{m}_{\rho ,\delta }(\mathbb {R}^{n}\times \mathbb {R}^{n})$. Here a smooth function p is an element of the symbol-class $S^{m }_{\rho ,\delta }(\mathbb {R}^{n}\times \mathbb {R}^{n})$ with $m \in \mathbb {R}$ and $0 \le \delta \le \rho \le 1$ if and only if

$$\begin{aligned} |p|^{(m)}_k := \max _{|\alpha |,|\beta |\le k} \sup _{x, \xi \in \mathbb {R}^n}\bigg |\partial ^{\alpha }_{\xi }\partial ^\beta _x p(x,\xi )\bigg |\langle \xi \rangle ^{-(m-\rho |\alpha |+\delta |\beta |)} < \infty \end{aligned}$$

holds for all $k \in \mathbb {N}_0$. During the whole paper $\mathbb {N}$ denotes the set o f all natural numbers without zero and $\mathbb {N}_0$ is the set of all natural numbers, where zero is included. The associated pseudodifferential operator is defined by

Here $\mathcal {S}(\mathbb {R}^n)$ denotes the Schwartz space, the space of all rapidly decreasing smooth functions and $\hat{u}$ is the Fourier transformation of u. We also denote $\hat{u}$ via $\mathscr {F}[u]$. The set of all pseudodifferential operators with symbols in $S^{m}_{\rho ,\delta }(\mathbb {R}^{n}\times \mathbb {R}^{n})$ is denoted via $OPS^{m}_{\rho ,\delta }(\mathbb {R}^{n}\times \mathbb {R}^{n})$.

In 1977 Beals [4] proved a characterization of smooth pseudodifferential operators, for example of the Hörmander class $S^{m}_{\rho ,\delta }(\mathbb {R}^{n}\times \mathbb {R}^{n})$ with $0 \le \delta \le \rho \le 1$ and $\delta <1$. Later Ueberberg [22] generalized this characterization for $L ^p-$Sobolev spaces. In the literature there are some other characterizations in the smooth case, e.g. [10, 13] or [19]. But the most important one for this paper is that one of Ueberberg, cf. [22]. It is based on the method for characterizing algebras of pseudodifferential operators developed by Beals [4, 5], Coifman, Meyer [6] and Cordes [7, 8]. Since non-smooth pseudodifferential operators are used in the regularity theory for partial differential equations, such a characterization is also useful in the non-smooth case. We use the main ideas of the characterization of Ueberberg in the smooth case, cf. [22], in order to derive a characterization for non-smooth pseudodifferential operators of the symbol- class $p \in C^{\tau }_{*} S^{m}_{\rho ,0}(\mathbb {R}^{n}\times \mathbb {R}^{n}; M)$ with $\rho \in \{ 0,1 \}$. Here the Hölder-Zygmund space $C^{\tau }_{*}(\mathbb {R}^{n})$, $\tau >0$, is defined by

$$\begin{aligned} C^{\tau }_{*}(\mathbb {R}^{n}):=\left\{ f \in \mathcal {S'}(\mathbb {R}^n): \Vert f\Vert _{C^{\tau }_{*} }:= \sup _{j \in \mathbb {N}_0} 2^{j\tau } \bigg \Vert \mathscr {F}^{-1}[\varphi _j \hat{f}] \bigg \Vert _{L^ {\infty }} < \infty \right\} , \end{aligned}$$

where $\mathscr {F}^{-1}[u]$ is the inverse Fourier transformation of $u \in \mathcal {S'}(\mathbb {R}^n)$, the dual space of $\mathcal {S}(\mathbb {R}^n)$ and $(\varphi _j)_{j\in \mathbb {N}_0}$ is a dyadic partition of unity on $\mathbb {R}^n$. Moreover a function $p: \mathbb {R}^n \times \mathbb {R}^n\rightarrow \mathbb {C}$ is an element of the symbol-class $C^{\tau }_{*} S^{m}_{\rho ,0 }(\mathbb {R}^{n}\times \mathbb {R}^{n}; M)$ with $m \in \mathbb {R}$, $\tau >0$, $0 \le \rho \le 1$ and $M\in \mathbb {N}_0 \cup \{ \infty \}$ if and only if

• (i) $\partial _x^{\beta } p(x, .) \in C^M(\mathbb {R}^{n})$ for all $x \in \mathbb {R}^{n}$,

• (ii) $\partial _x^{\beta } \partial _{\xi }^{\alpha } p \in C^{0}(\mathbb {R}^n_x \times \mathbb {R}^n_{\xi })$,

• (iii) $\Vert \partial _{\xi }^{\alpha } p(.,\xi ) \Vert _{C^{\tau }_{*}(\mathbb {R}^n)} \le C _{\alpha }\langle \xi \rangle ^{m-\rho |\alpha |}$ for all $\xi \in \mathbb {R}^n$

holds for all $\alpha , \beta \in \mathbb {N}_0^n$ with $|\alpha | \le M$ and $|\beta | \le \tau$. The associated pseudodifferential operator $p(x,D_x)$, also denoted by $OP(p)$, to such a symbol p is defined in the same way as in the smooth case. The set of all pseudodifferential operators with symbols in the symbol-class $C^{\tau }_{*} S^{m}_{\rho ,0}(\mathbb {R}^{n}\times \mathbb {R}^{n}; M)$ is denoted via $OPC^{\tau }_{*} S^{m}_{\rho ,0}(\mathbb {R}^{n}\times \mathbb {R}^{n}; M)$.

Similary to the smooth case, see [22] and [4] we define the following set of operators:

Definition 1.1

Let $m\in \mathbb {R}$, $M \in \mathbb {N}_0 \cup \{ \infty \}$ and $0\le \rho \le 1$. Additionally let $\tilde{m} \in \mathbb {N}_0 \cup \{ \infty \}$ and $1< q < \infty$. Then we define $\mathcal {A}^{m,M}_{\rho ,0}(\tilde{m},q)$ as the set of all linear and bounded operators $P: H^m_q(\mathbb {R}^{n}) \rightarrow L^q(\mathbb {R}^{n})$, such that for all $l\in \mathbb {N}$, $\alpha _1, \ldots , \alpha _l \in \mathbb {N}_0^n$ and $\beta _1, \ldots , \beta _l \in \mathbb {N}_0^n$ with $|\alpha _1| + |\beta _1|= \cdots = |\alpha _l| + |\beta _l|= 1$, $|\alpha | \le M$ and $| \beta |\le \tilde{m}$ the iterated commutator of P

$$\begin{aligned} {{\mathrm{ad}}}(-ix)^{\alpha _1}{{\mathrm{ad}}}(D_x)^{\beta _1}\ldots {{\mathrm{ad}}}(-ix)^{\alpha _l}{{\mathrm{ad}}}(D_x)^{\beta _l} P: H^{m-\rho |\alpha |}_q(\mathbb {R}^{n}) \rightarrow L^q(\mathbb {R}^{n}) \end{aligned}$$

is continuous. Here $\alpha := \alpha _1 + \cdots + \alpha _l$ and $\beta :=\beta _1 + \cdots + \beta _l$.

In the case $M = \infty$ we write $\mathcal {A}^{m}_{\rho ,0}(\tilde{m},q )$ instead of $\mathcal {A}^{m,\infty }_{\rho ,0}(\tilde{m},q)$. For the definition of the iterated commutators, see Definition 2.3 below. All the other notations and basics needed in this paper, also in the Introduction, are given in Sect. 2.

Choosing $M=\tilde{m}=\infty$ the proof of the characterization in the smooth case, cf. [22, Chapter 3] provides that each $T \in \mathcal {A}^{m,\infty }_{\rho ,0}(\infty ,q)$ is a smooth pseudodifferential operator of the class $S^{m}_{\rho ,0}(\mathbb {R}^{n}\times \mathbb {R}^{n})$. But we even get more: Smooth pseudodifferential operators of the symbol-class $S^{m}_{\rho ,0}(\mathbb {R}^{n}\times \mathbb {R}^{n})$ are elements of $\mathcal {A}^{m}_{\rho ,0}(\infty ,q)$ due to Remark 2.5 and Theorem 3.6. Thus we have $\mathcal {A}^{m}_{\rho ,0}(\infty ,q) = OPS^{m}_{\rho ,0}(\mathbb {R}^{n}\times \mathbb {R}^{n})$. In the case $\tilde{m} \ne \infty$ we obtain a similar result: Non-smooth pseudodifferential operators of the class $C^{\tau }_{*} S^m_{\rho , 0}( \mathbb {R}^n \times \mathbb {R}^n)$ with $\rho \in \{ 0,1 \}$ are elements of such sets.

Example 1.2

Let $\tau > 0$, $\tau \notin \mathbb {N}$, $m \in \mathbb {R}$ and $\rho \in \{ 0, 1 \}$. Considering a non-smooth symbol $p \in C^{\tau }_{*} S^{m}_{\rho ,0}(\mathbb {R}^{n}\times \mathbb {R}^{n})$ we get for $\tilde{m}:= \max \{ k \in \mathbb {N}_0: \tau -k>n/2 \}$ and $1< q < \infty$:

• (i) $p(x, D_x) \in \mathcal {A}^{m+n/2}_{0,0}(\lfloor \tau \rfloor ,2)$ if $\rho = 0$,

• (ii) $p(x, D_x) \in \mathcal {A}^{m}_{0,0}(\tilde{m},2)$ if $\rho = 0$,

• (iii) $p(x, D_x) \in \mathcal {A}^m_{1,0}(\lfloor \tau \rfloor ,q)$ if $\rho = 1$.

The previous example motivates to take the set $\mathcal {A}^{m,M}_{\rho ,0 }(\tilde{m},q)$ as characterization set for non-smooth pseudodifferential operators. Hence we want to show, that elements of $\mathcal {A}^{m,M}_{\rho ,0}(\tilde{m},q)$ are non-smooth pseudodifferential operators. As in the smooth case the characterization of non-smooth pseudodifferential operators of the symbol-class $C^{\tau }_{*} S^{m}_{1,0}(\mathbb {R}^{n}\times \mathbb {R}^{n}; M)$ is reduced to the characterization of those ones of the symbol-class $C^{\tau }_{*} S^{m}_{0,0}(\mathbb {R}^{n}\times \mathbb {R}^{n}; M)$. To this end the following property of the set $\mathcal {A}^{m,M}_{\rho ,0}(\tilde{m},q)$ is needed:

Lemma 1.3

Let $m\in \mathbb {R}$, $M \in \mathbb {N}_0 \cup \{ \infty \}$ and $0\le \rho _1 < \rho _2 \le 1$. Furthermore, let $\tilde{m} \in \mathbb {N}_0$ and $1< q < \infty$. Then

$$\begin{aligned} \mathcal {A}^{m, M}_{\rho _2,0}(\tilde{m},q) \subseteq \mathcal {A}^{m, M }_{\rho _1,0}(\tilde{m},q). \end{aligned}$$

Proof

On account of the continuous embedding $H^{m-\rho _1|\alpha |}_q(\mathbb {R}^{n}) \hookrightarrow H^{m-\rho _2|\alpha |}_q(\mathbb {R}^{n})$ the claim holds. $\square$

The main goal of this paper is to show that each element of $\mathcal {A}^ {m,M}_{1,0}(\tilde{m},q)$ is a non-smooth pseudodifferential operator with coefficients in a Hölder space. This is the topic of Sect. 4.5. We will see that M has to be sufficiently large. In analogy to the proof of J. Ueberberg in the smooth case one reduces this statement to the following: each element of the set $\mathcal {A}^{m,M}_{0,0}(\tilde{m},q)$ is a non-smooth pseudodifferential operator with coefficients in a Hölder space. Making use of order reducing pseudodifferential operators we obtain the characterization of non-smooth pseudodifferential operators of arbitrary order m from that. Details for deriving this result are explained in Sect. 4.4.

The first three subsections of Sect. 4 serve to develop some auxiliary tools needed for the proof of the case $m= 0$. In Sect. 4.1 we start by showing that a bounded sequence in $C^{\tilde{m}, s} S^{0}_{0,0}(\mathbb {R}^{n}\times \mathbb {R}^{n}; M)$ has a subsequence which converges in the symbol-class $C^{\tilde{m}, s} S^{0}_{0, 0}(\mathbb {R}^{n}\times \mathbb {R}^{n}; M-1)$. Section 4.2 is devoted to the symbol reduction of non-smooth double symbols to non-smooth single symbols. Details for the third tool are proved in Sect. 4.3. There a family $( T_{\varepsilon } )_{\varepsilon \in (0,1]}$ fulfilling the following property is constructed: $T_{\varepsilon }: \mathcal {S'}(\mathbb {R}^n)\rightarrow \mathcal {S}(\mathbb {R}^n)$ is continuous for all $\varepsilon \in (0,1]$ and converges pointwise if $\varepsilon \rightarrow 0$. Moreover, all iterated commutators of $T_{\varepsilon }$ are uniformly bounded with respect to $\varepsilon$ as maps from $L^q(\mathbb {R}^{n})$ to $L^q(\mathbb {R}^{n})$.

In Sect. 5 we illustrate the usefulness of such a characterization: We show that the composition PQ of two non-smooth pseudodifferential operators P and Q is a non-smooth pseudodifferential operator again if Q is smooth enough. This is done by means of the characterization of non-smooth pseudodifferential operators.

Section 3 is devoted to some properties of pseudodifferential operators with single symbols, cf. Sect. 3.1, and pseudodifferential operators with double symbols, cf. Sect. 3.2.

This paper is based on a part of the PhD-thesis of the second author, cf.  [17] advised by the first author. In this thesis some proofs are given in more details.

2 Preliminaries

During the whole paper, we consider $n \in \mathbb {N}$ except when stated otherwise. In particular $n \ne 0$. Considering $x \in \mathbb {R}$ we define

$$\begin{aligned} x^+:=\max \{0;x \} \quad \text {and} \; \lfloor x \rfloor := \max \{k \in \mathbb {Z}: k \le x \}. \end{aligned}$$

Moreover

Additionally we scale partial derivatives with respect to a variable $x\in \mathbb {R}^{n}$ with the factor $-i$ and denote it by

$$\begin{aligned} D_x^{\alpha }:= (-i)^{|\alpha |} \partial ^{\alpha }_x := (-i)^{|\alpha |} \partial ^ {\alpha _1}_{x_1} \ldots \partial ^{\alpha _n}_{x_n} . \end{aligned}$$

Here $\alpha =(\alpha _1, \ldots , \alpha _n) \in \mathbb {N}_0^n$ is a multi -index. For arbitrary $j\in \{1,\ldots , n\}$ we define the j-th canonical unit vector $e_j \in \mathbb {N}^n_0$ as $(e_j)_k =1$ if $j=k$ and $(e_j)_k=0$ else.

In view of two Banach spaces X, Y the set $\mathscr {L}(X,Y)$ consists of all linear and bounded operators $A:X \rightarrow Y$. We also write $\mathscr {L}(X)$ instead of $\mathscr {L}(X,X)$.

We finally note that the dual space of a topological vector space V is denoted by $V'$. If V is even a Banach space the duality product V is denoted by ${\langle .,. \rangle }_{V; V'}$.

2.1 Functions on $\mathbb {R}^{n}$ and Function Spaces

In this subsection we are going to introduce some function spaces which play a central role during this paper. To begin with, the Hölder space of the order $m \in \mathbb {N}_0$ with Hölder continuity $s \in (0,1]$ is denoted by

$$\begin{aligned} C^{m,s}(\mathbb {R}^{n}):= \bigg \{ f \in C^m({\mathbb {R}^{n}}): \sup _{x \ne y} \frac{|\partial _x^\alpha f(x)-\partial _x^\alpha f(y)| }{|x-y|^{s}} < \infty \bigg \} \ \ \forall |\alpha |\le m \end{aligned}$$

and also by $C^{m+s}(\mathbb {R}^{n})$ if $s \ne 1$. Note, that $C^{s}(\mathbb {R}^{n}) = C^s_ {*}(\mathbb {R}^{n})$ if $s \notin \mathbb {N}_0$. For arbitrary $s \in \mathbb {R}$ and $1<p<\infty$ the Bessel Potential Space $H^s_p(\mathbb {R}^{n})$ is defined by

$$\begin{aligned} H^s_p(\mathbb {R}^{n}):= \bigg \{ f \in \mathcal {S'}(\mathbb {R}^n): \langle D_x \rangle ^s f \in L^p(\mathbb {R}^{n}) < \infty \bigg \} \end{aligned}$$

where $\langle D_x \rangle ^s:=OP(\langle \xi \rangle ^s)$.

Let us mention a characterization of functions in a Bessel potential space needed later on:

Lemma 2.1

Let $1<p<\infty$, $s < 0$ and $m := -\lfloor s \rfloor$. Then for each $f \in H^s_p(\mathbb {R}^{n})$ there are functions $g_{\alpha } \in H^{s-\lfloor s \rfloor }_p(\mathbb {R}^{n})$, where $\alpha \in \mathbb {N}_0^n$ with $|\alpha | \le m$, such that

• $f= \sum \limits _{|\alpha | \le m} \partial ^{\alpha }_x g_{\alpha }$,

• $\sum \limits _{|\alpha | \le m} \Vert g_{\alpha } \Vert _{H^{s-\lfloor s \rfloor }_p} \le C \Vert f \Vert _{H^s_p}$,

where C is independent of f and $g_{\alpha }$.

Proof

We define the operator $T: H^{-s}_q(\mathbb {R}^{n}) \rightarrow (H^{\lfloor s \rfloor -s}_q(\mathbb {R}^{n}) )^N$ with $1/p + 1/q =1$ and $N:= \sharp \{ \alpha \in \mathbb {N}_0^n: |\alpha | \le m \}$ in the following way:

$$\begin{aligned} T(\varphi ) = \big (\partial _x^{\alpha } \varphi \big )_{|\alpha | \le m}. \end{aligned}$$

The norm $\Vert . \Vert _{X_q}$ of $X_q:=( H^{\lfloor s \rfloor -s}_q(\mathbb {R}^{n}))^N$ is defined by

$$\begin{aligned} \Vert f \Vert _{ X_q }&: = \sum _{i=1}^N \Vert f_i\Vert _{H^{\lfloor s \rfloor - s}_q} \quad \text {for all }f \in \left( H^{\lfloor s \rfloor -s}_q (\mathbb {R}^{n}) \right) ^N . \end{aligned}$$

For an arbitrary sequence $(f_l)_{l \in \mathbb {N}}$ in $Y:=T(H^{-s}_q(\mathbb {R}^{n}))$ , which converges in $X_q$, we get an weakly convergent sequence $(\varphi _l)_{l \in \mathbb {N}}$ in $H^{-s}_q(\mathbb {R}^{n})$ with $f_l=T\varphi _l$ for all $l \in \mathbb {N}$. Due to the weak convergence of $(\varphi _l)_{l \in \mathbb {N}}$, we can verify that the limit of $(f_l)_{l \in \mathbb {N}}$ lies in Y. Hence Y is closed. By means of an application of the bounded inverse theorem we can show the existence of the inverse of $T: H^{-s}_q(\mathbb {R}^{n}) \rightarrow T(H^{-s}_q(\mathbb {R}^{n}))=:Y$ and that $T^{-1} \in \mathscr {L}(Y, H^{-s}_q(\mathbb {R}^{n}))$. For more details see [17, Proposition 2.50]. In view of $T^{-1} \in \mathscr {L}(Y,H^{-s}_q(\mathbb {R}^{n}))$ we get

$$\begin{aligned} |f \circ T^{-1} (g)| = \bigg |{\langle f,T^{-1} g \rangle }_{ H^s_p; H^{-s}_q }\bigg | \le \Vert f \Vert _{H^s_p} \Vert T^{-1} g \Vert _{H^{-s}_q} \le C \Vert f \Vert _{H^s_p} \Vert g \Vert _{ X_q} \end{aligned}$$

for all $f \in H^s_p(\mathbb {R}^{n})$ and $g \in Y$. An application of the Theorem of Hahn Banach to $\tilde{f}:=f \circ T ^{-1} \in Y'$ provides the existence of a linear functional $F \in \left( X_q \right) ' = \left( H_p^{s-\lfloor s \rfloor } (\mathbb {R}^{n})\right) ^N$ such that

• (i) $F|_{Y} = \tilde{f}$,

• (ii) $|F(g)| \le C \Vert f \Vert _{H^s_p} \Vert g \Vert _{ X_q}$ for all $g \in X_q$.

For arbitrary $\varphi \in \mathcal {S}(\mathbb {R}^n)\subseteq H^{-s}_q(\mathbb {R}^{n})$ we can apply property i) on account of $T \varphi \in Y$ and get:

$$\begin{aligned} \bigg \langle \sum _{|\alpha | \le m} (-1)^{|\alpha |} \partial ^{\alpha }_x F^{\alpha } , \varphi \bigg \rangle _{H^s_p; H^{-s}_q}&= \bigg \langle F, \left( \partial ^{\alpha }_x \varphi \right) _{|\alpha | \le m} \bigg \rangle _{ \left( X_q \right) '; X_q} = \langle F, T \varphi \rangle _{ \left( X_q \right) '; X_q} \\&= \langle \tilde{f} , T \varphi \rangle _{ \left( X_q \right) '; X _q} = f \circ T^{-1} (T \varphi ) = {\langle f,\varphi \rangle }_{H^s_p; H^{-s}_q}. \end{aligned}$$

Due to the density of $\mathcal {S}(\mathbb {R}^n)$ in $H^{-s}_q(\mathbb {R}^{n})$ we obtain

$$\begin{aligned} \sum _{|\alpha | \le m} (-1)^{|\alpha |} \partial ^{\alpha }_x F^{\alpha } = f \quad \text {in } H^s_p(\mathbb {R}^{n}). \end{aligned}$$

Additionally we get the claim due to ii):

$$\begin{aligned} \sum _{|\alpha | \le m} \Vert (-1)^{|\alpha |} F^{\alpha } \Vert _{H^{s-\lfloor s \rfloor }_p} = \Vert F \Vert _{\left( X_q \right) '} = \sup _{ \Vert g\Vert _{X_q} \le 1 } | {\langle F,g \rangle }_{ \left( X_q \right) '; X_q} | \le C \Vert f \Vert _{H^s_p}. \end{aligned}$$

$\square$

In this paper the translation function $\tau _y(g): \mathbb {R}^{n}\rightarrow \mathbb {C}$, $y \in \mathbb {R}^{n}$ of $g \in L^1(\mathbb {R}^{n})$, is defined for all $x\in \mathbb {R}^{n}$ as $\tau _y(g)(x):=g(x-y).$

2.2 Kernel Theorem

An important ingredient of the characterization is the next kernel theorem:

Theorem 2.2

Every continuous linear operator $T: \mathcal {S'}(\mathbb {R}^n)\rightarrow \mathcal {S}(\mathbb {R}^n)$ has a Schwartz kernel $t(x,y) \in \mathcal {S}(\mathbb {R}^n \times \mathbb {R}^n)$. Thus for every $u \in \mathcal {S}(\mathbb {R}^n)$ we have

$$\begin{aligned} Tu(x) = \int \limits _{\mathbb {R}^{n}}t(x,y) u(y) dy \quad \text {for all } \; x \in \mathbb {R}^{n}. \end{aligned}$$

Proof

This claim is a consequence of [21, Theorem 51.6] and [3, Theorem 1.48] if one uses that $\mathcal {S}(\mathbb {R}^n)$ and $\mathcal {S'}(\mathbb {R}^n)$ are nuclear and conuclear, see e.g. [21, p. 530]. For more details we refer to [17, Theorem 2.62]. $\square$

We can apply the previous kernel theorem for the iterated commutators of linear and bounded operators $P: \mathcal {S'}(\mathbb {R}^n)\rightarrow \mathcal {S}(\mathbb {R}^n)$. These operators are defined in the following way:

Definition 2.3

Let $X,Y \in \{ \mathcal {S}(\mathbb {R}^n), \mathcal {S'}(\mathbb {R}^n)\}$ and $T: X \rightarrow Y$ be linear. We define the linear operators ${{\mathrm{ad}}}(-ix_j) T: X \rightarrow Y$ and ${{\mathrm{ad}}}(D_{x_j}) T: X \rightarrow Y$ for all $j\in \{ 1, \ldots , n\}$ and $u \in X$ by

$$\begin{aligned} {{\mathrm{ad}}}(-ix_j) T u:= -ix_j Tu + T \left( ix_j u \right) \quad \text {and} \quad {{\mathrm{ad}}}(D_{x_j}) T u:= D_{x_j} \left( T u \right) - T \left( D_{x _j} u \right) . \end{aligned}$$

For arbitrary multi-indices $\alpha , \beta \in \mathbb {N}_0^n$ we denote the iterated commutator of T as

$$\begin{aligned} {{\mathrm{ad}}}(-ix)^{\alpha }{{\mathrm{ad}}}(D_{x})^{\beta } T := [{{\mathrm{ad}}}(-ix_1)]^{\alpha _1} \ldots [{{\mathrm{ad}}}(-ix_n)]^{\alpha _n} [{{\mathrm{ad}}}(D_{x_1})]^{\beta _1} \ldots [{{\mathrm{ad}}}(D_{x_n} )]^{\beta _n} T. \end{aligned}$$

On account of the previous definition all iterated commutators of $P: \mathcal {S'}(\mathbb {R}^n)\rightarrow \mathcal {S}(\mathbb {R}^n)$ map $\mathcal {S'}(\mathbb {R}^n)$ to $\mathcal {S}(\mathbb {R}^n)$. Consequently an application of Theorem 2.2 provides:

Corollary 2.4

Let $\alpha , \beta \in \mathbb {N}_0^n$ and $P: \mathcal {S'}(\mathbb {R}^n)\rightarrow \mathcal {S}(\mathbb {R}^n)$ be a linear operator. Then the operator ${{\mathrm{ad}}}(-ix)^{\alpha } {{\mathrm{ad}}}(D_x)^{\beta } P: \mathcal {S'}(\mathbb {R}^n)\rightarrow \mathcal {S}(\mathbb {R}^n)$ has a Schwartz kernel $f^{\alpha ,\beta } \in \mathcal {S}(\mathbb {R}^n \times \mathbb {R}^n)$, i.e., for all $u \in \mathcal {S}(\mathbb {R}^n)$

$$\begin{aligned} {{\mathrm{ad}}}(-ix)^{\alpha } {{\mathrm{ad}}}(D_x)^{\beta } P u(x) = \int \limits _{\mathbb {R}^{n}}f^{\alpha , \beta }(x,y) u(y) dy \quad \text {for all } x \in \mathbb {R}^{n}. \end{aligned}$$

(1)

Iterated commutators of pseudodifferential operators are pseudodifferential operators again due to the properties of the Fourier transformation.

Remark 2.5

Let $\tilde{m}\in \mathbb {N}_0$, $M \in \mathbb {N}_0 \cup \{ \infty \}$, $0< \tau \le 1$, $m\in \mathbb {R}$ and $0 \le \rho \le 1$. We assume that $p \in C^{\tilde{ m}, \tau } S^m_{\rho ,0} (\mathbb {R}^n \times \mathbb {R}^n; M)$. We define $P:=p(x,D_x)$. Using integration by parts and some properties of the Fourier transformation, one can calculate for each $u \in \mathcal {S}(\mathbb {R}^n)$ at once that

$$\begin{aligned} {{\mathrm{ad}}}(-ix_j) P u (x)&= -ix_j P u (x) + P [ ix_j u (x) ] = ( \partial _{\xi _j} p) (x,D_x) u (x),\\ {{\mathrm{ad}}}(D_{x_j}) P u (x)&= D_{x_j} \{ P u (x)\} - P [ D_{x_j} u (x) ] = ( D_{x_j} p) (x,D_x) u (x) \end{aligned}$$

for all $x \in \mathbb {R}^{n}$. Applying $p(x,\xi ) \in C^{\tilde{m}, \tau } S^{m}_{ \rho ,0}(\mathbb {R}^n_{x} \times \mathbb {R}^n_{\xi };M)$, we obtain

$$\begin{aligned} ( \partial _{\xi _j} p) (x,\xi )&\in C^{\tilde{m}, \tau } S^{m-\rho }_{\rho ,0}\big (\mathbb {R}^n_{x} \times \mathbb {R}^n_{\xi };M- 1 \big ) \quad \text {and} \\ (D_{x_j} \tilde{p}) (x,\xi )&\in C^{\tilde{m}-1, \tau } S^{m }_{\rho ,0}\big (\mathbb {R}^n_{x} \times \mathbb {R}^n_{\xi };M\big ). \end{aligned}$$

Now let $l \in \mathbb {N}$, $\alpha _1, \ldots , \alpha _l \in \mathbb {N}_0^n$ and $\beta _1, \ldots , \beta _l \in \mathbb {N}_0^n$ with $|\alpha _j + \beta _j| = 1$ for all $j \in \{ 1, \ldots , l\}$, $|\alpha | \le M$ and $|\beta | \le \tilde{m}$. Here $\alpha$ and $\beta$ are defined by $\alpha :=\alpha _1 + \cdots + \alpha _l$ and $\beta := \beta _1 + \cdots +\beta _l$. By induction with respect to l we can prove, that the operator

$$\begin{aligned} {{\mathrm{ad}}}(-ix)^{\alpha _1} {{\mathrm{ad}}}(D_x)^{\beta _1} \ldots {{\mathrm{ad}}}(-ix)^{\alpha _l} {{\mathrm{ad}}}(D_x)^{\beta _l} p(x,D_x) \end{aligned}$$

is a pseudodifferential operator with the symbol

$$\begin{aligned} \partial _{\xi }^{\alpha } D^{\beta }_x p(x,\xi ) \in C^{\tilde{m}- |\beta |, \tau } S^ {m-\rho |\alpha |}_{\rho ,0} \big (\mathbb {R}^n_{x} \times \mathbb {R}^n_{\xi }; M-|\alpha |\big ). \end{aligned}$$

If we even have $p \in S^m_{\rho ,0} (\mathbb {R}^n \times \mathbb {R}^n)$, then $\partial _{\xi }^{\alpha } D^{\beta }_x p(x,\xi ) \in S^{m-\rho |\alpha | }_{\rho ,0} (\mathbb {R}^n \times \mathbb {R}^n)$.

An application of the kernel theorem provides:

Lemma 2.6

Let $g \in \mathcal {S}(\mathbb {R}^n)$. For all $y \in \mathbb {R}^{n}$ we denote $g_y:=\tau _y(g):= g(. -y)$. Moreover, let $P:\mathcal {S'}(\mathbb {R}^n)\rightarrow \mathcal {S}(\mathbb {R}^n)$ be linear and continuous. We define $p:\mathbb {R}^n \times \mathbb {R}^n\times \mathbb {R}^n\rightarrow \mathbb {C}$ by

$$\begin{aligned} p (x,\xi ,y):= e^{-ix \cdot \xi } P \left( e_{\xi } g_y \right) (x) \quad \text {for all } x,\xi , y \in \mathbb {R}^{n}. \end{aligned}$$

Here $e_{\xi }(x):= e^{ix\xi }$ for all $x \in \mathbb {R}^{n}$. Then we have for all $\alpha , \beta , \gamma \in \mathbb {N}_0^n$:

$$\begin{aligned}&\partial _{\xi }^{\alpha } D_x^{\beta } D_y^{\gamma } p(x,\xi ,y) \\&\quad = (-1)^{\gamma } \sum _{\beta _1 + \beta _2 = \beta } \left( {\begin{array}{c}\beta \\ \beta _1\end{array}}\right) e^{-ix \cdot \xi } \left( {{\mathrm{ad}}}(-ix)^{\alpha } {{\mathrm{ad}}}(D_x)^{\beta _1} P \right) \left( e_{\xi } D_x^{\beta _2 + \gamma } g_y \right) (x). \end{aligned}$$

Proof

Theorem 2.2 provides the existence of a Schwartz kernel $f \in \mathcal {S}(\mathbb {R}^n \times \mathbb {R}^n)$ of P. Due to $g \in \mathcal {S}(\mathbb {R}^n)$ and $f \in \mathcal {S}(\mathbb {R}^n \times \mathbb {R}^n)$ we get for all $x \in \mathbb {R}^{n}$:

$$\begin{aligned} D_y^{\gamma } \left\{ e^{-ix \cdot \xi } P (e_{\xi } g_y)(x) \right\}&= e^{-ix \cdot \xi } D_y^{\gamma } \int f(x,z) e^{iz \cdot \xi } g_y(z) dz \\&= e^{-ix \cdot \xi } \int f(x,z) e^{iz \cdot \xi } D^{\gamma }_y g_y(z) dz\\&= (-1)^{|\gamma |} e^{-ix \cdot \xi } P \big (e_{\xi } D^{\gamma }_x g_y \big )(x ). \end{aligned}$$

Inductively with respect to $|\beta |$ we can show for all $\beta ,\gamma \in \mathbb {N}_0^n$ and each $x \in \mathbb {R}^{n}$:

$$\begin{aligned}&D_x^{\beta } D_y^{\gamma } \left\{ e^{-ix \cdot \xi } P (e_{\xi } g_y)( x) \right\} \nonumber \\&\quad = (-1)^{|\gamma |} D_x^{\beta } \left\{ e^{-ix \cdot \xi } P \big (e_{\xi } D^{\gamma }_x g_y \big )(x) \right\} \nonumber \\&\quad = (-1)^{|\gamma |} \sum _{\beta _1 + \beta _2 = \beta } \left( {\begin{array}{c}\beta \\ \beta _1\end{array}}\right) e^{-ix \cdot \xi } {{\mathrm{ad}}}(D_x)^{\beta _1} P \left( e_{\xi } D_x^{\beta _2 + \gamma } g_y \right) (x). \end{aligned}$$

(2)

With Corollary 2.4 at hand, the iterated commutator ${{\mathrm{ad}}}(D_x)^{\beta _1} P$ has a Schwartz kernel $f^{\beta _1} \in \mathcal {S}(\mathbb {R}^n \times \mathbb {R}^n)$. Due to $e_{\xi } D_x^{\beta _2 + \gamma } g_y \in \mathcal {S}(\mathbb {R}^n)$ and $(ix)^{\alpha _2} e_{\xi } D_x^{\beta _2 + \gamma } g_y(x) \in \mathcal {S}(\mathbb {R}^n_{x})$ an application of the Leibniz rule and interchanging the derivatives with the integral yields for all $x \in \mathbb {R}^{n}$:

$$\begin{aligned}&\partial _{\xi }^{\alpha } \left\{ e^{-ix \cdot \xi } {{\mathrm{ad}}}(D_x)^{\beta _1} P \left( e_{\xi } D_x^{\beta _2 + \gamma } g_y \right) (x) \right\} \nonumber \\&\qquad = \sum _{\alpha _1\,+\,\alpha _2 = \alpha } \left( {\begin{array}{c}\alpha \\ \alpha _1\end{array}}\right) (-ix)^{\alpha _1} e^{-ix \cdot \xi } \int f^{\beta _1} (x,z) (iz)^{\alpha _2} e^{iz \cdot \xi } D_z^{\beta _2 + \gamma } g_y(z) dz \nonumber \\&\qquad = e^{-ix \cdot \xi } \big ({{\mathrm{ad}}}(-ix)^{\alpha } {{\mathrm{ad}}}(D_x)^{\beta _1} P \big ) \left( e_{\xi } D_x^{\beta _2 + \gamma } g_y\right) (x). \end{aligned}$$

(3)

Finally, the combination of (2) and (3) finishes the proof:

$$\begin{aligned}&\partial _{\xi }^{\alpha } D_x^{\beta } D_y^{\gamma } p(x,\xi ,y) \\&\quad = \partial _{\xi }^{ \alpha } D_x^{\beta } D_y^{\gamma } \left\{ e^{-ix \cdot \xi } P \left( e_{\xi } g_y \right) (x) \right\} \\&\quad = (-1)^{|\gamma |} \sum _{\beta _1 + \beta _2 = \beta } \left( {\begin{array}{c}\beta \\ \beta _1\end{array}}\right) e^{-ix \cdot \xi } \big ({{\mathrm{ad}}}(-ix)^{\alpha } {{\mathrm{ad}}}(D_x)^{\beta _1 } P \big ) \left( e_{\xi } D_x^{\beta _2 + \gamma } g_y\right) (x) \end{aligned}$$

for all $x, \xi , y \in \mathbb {R}^{n}$. $\square$

2.3 Extension of the Space of Amplitudes

An important technique for working with smooth pseudodifferential operators are the oscillatory integrals, defined by

with $\chi \in \mathcal {S}(\mathbb {R}^n \times \mathbb {R}^n)$ and $\chi (0,0) = 1$ for all elements a of the space of amplitudes $\mathscr {A}^m_{\tau }(\mathbb {R}^n \times \mathbb {R}^n)$ $(m,\tau \in \mathbb {R})$, the set of all smooth functions $a:\mathbb {R}^{n}\times \mathbb {R}^{n}\rightarrow \mathbb {C}$ such that

$$\begin{aligned} \bigg |\partial ^{\alpha }_{\eta } \partial ^{\beta }_y a(y,\eta ) \bigg | \le C_{\alpha , \beta } (1 +|\eta |)^m (1+|y|)^{\tau } \end{aligned}$$

uniformly in $y,\eta \in \mathbb {R}^{n}$ for all $\alpha , \beta \in \mathbb {N}_0^n$. In order to use the oscillatory integral in the non-smooth case we extend the space of amplitudes in the following way:

Definition 2.7

Let $m, \tau \in \mathbb {R}$ and $N \in \mathbb {N}_0 \cup \{ \infty \}$. We define $\mathscr {A}^{m,N}_{\tau }(\mathbb {R}^n \times \mathbb {R}^n)$ as the set of all functions $a: \mathbb {R}^n \times \mathbb {R}^n\rightarrow \mathbb {C}$ with the following properties: For all $\alpha , \beta \in \mathbb {N}_0^n$ with $|\alpha | \le N$ we have

• (i) $\partial ^{\alpha }_{\eta } \partial ^{\beta }_{y} a(y,\eta ) \in C^0(\mathbb {R}^n_{y} \times \mathbb {R}^n_{\eta })$,

• (ii) $\left| \partial ^{\alpha }_{\eta } \partial ^{\beta }_{y} a(y, \eta ) \right| \le C_{\alpha , \beta } (1 + |\eta |)^m (1 + |y|)^{\tau }$ for all $y, \eta \in \mathbb {R}^{n}$.

Note that $\mathscr {A}^{m,\infty }_{\tau }(\mathbb {R}^n \times \mathbb {R}^n) = \mathscr {A}^{m}_{\tau }(\mathbb {R}^n \times \mathbb {R}^n)$.

Remark 2.8

For $m=2k$, $k \in \mathbb {N}$ we can show $e^{ix\cdot \xi } = \langle \xi \rangle ^{-m} \langle D_x \rangle ^{m} e^{ix\cdot \xi }$, if we write $\langle D_x \rangle ^{m}=\sum _{|\gamma | \le k} a_{\gamma , m} D_x^{2\gamma }$. Now let $m=2k+1$, $k \in \mathbb {N}_0$. Using

$$\begin{aligned} \langle \xi \rangle ^{m}= \langle \xi \rangle ^{2k} \frac{\langle \xi \rangle ^2}{\langle \xi \rangle } = \langle \xi \rangle ^{2 k} \left\{ \frac{1}{\langle \xi \rangle } + \sum _{j=1}^n \frac{\xi _j^2}{\langle \xi \rangle } \right\} \end{aligned}$$

we get:

$$\begin{aligned} e^{ix\cdot \xi } = \langle \xi \rangle ^{-m} \langle \xi \rangle ^{m} e^{ix\cdot \xi } =\langle \xi \rangle ^{-m-1} \langle D_x \rangle ^{m-1}e^{ix\cdot \xi } + \sum _{j=1}^n \langle \xi \rangle ^{-m} \frac{\xi _j}{\langle \xi \rangle } \langle D_x \rangle ^{m-1} D_{x_j} e^{ix\cdot \xi }. \end{aligned}$$

On account of the previous remark, we define for all $m \in \mathbb {N}$

$$\begin{aligned} \begin{array}{lll} &{}\displaystyle A^m(D_{x},\xi ) := \langle \xi \rangle ^{-m} \langle D_x \rangle ^{m} &{}\quad \text {if } m \text { is even},\\ &{}\displaystyle A^m(D_{x},\xi ) := \langle \xi \rangle ^{-m-1} \langle D_x \rangle ^{m-1} -\sum _{j=1}^n \langle \xi \rangle ^{-m} \frac{\xi _j}{\langle \xi \rangle } \langle D_x \rangle ^{m-1} D_{x_j} &{}\quad \text {else}. \end{array} \end{aligned}$$

Remark 2.9

Let $m, \tau \in \mathbb {R}$ and $N \in \mathbb {N}_0 \cup \{ \infty \}$. Moreover let $\mathscr {B} \subseteq \mathscr {A}^{m,N}_{\tau }(\mathbb {R}^n \times \mathbb {R}^n)$ be bounded, i.e., for all $\alpha ,\beta \in \mathbb {N}_0^n$ with $|\beta | \le N$ we have

$$\begin{aligned} \bigg |\partial ^{\alpha }_x \partial ^{\beta }_{\xi } a(x, \xi ) \bigg | \le C_{\alpha , \beta } \langle \xi \rangle ^m \langle x \rangle ^{\tau } \quad \text {for all } a \in \mathscr {B}. \end{aligned}$$

(4)

If we use the Leibniz rule and write $\langle D_x \rangle ^{l-1}=\sum _{|\gamma | \le (l-1)/2} a_{\gamma , l} D_x^{2\gamma }$ we obtain due to (4), $\xi _j \langle \xi \rangle ^{-1} \in S^0_{1,0}(\mathbb {R}^n \times \mathbb {R}^n)$ and $\langle \xi \rangle ^s \in S^s_{1,0}(\mathbb {R}^n \times \mathbb {R}^n)$ for all $\alpha ,\beta \in \mathbb {N}_0^n$ with $|\beta | \le N$ and $l\in \mathbb {N}$ odd:

$$\begin{aligned}&\left| \partial ^{\alpha }_x \partial ^{\beta }_{\xi } \left\{ \langle \xi \rangle ^{-l-1} \langle D_x \rangle ^ {l-1} a(x,\xi )-\sum _{j=1}^n \langle \xi \rangle ^{-l} \frac{\xi _j}{\langle \xi \rangle } \langle D_x \rangle ^{l-1} D_{x_j} a(x,\xi ) \right\} \right| \\&\quad \le C_{\alpha }\langle \xi \rangle ^{-l+m} \langle x \rangle ^{\tau } \end{aligned}$$

for all $a \in \mathscr {B}$ .

Definition 2.7 enables us to extend the definition of the oscillatory integral for functions in the set $\mathscr {A}^{m,N}_{\tau }(\mathbb {R}^n \times \mathbb {R}^n)$. It can be proved similarly to e.g. Theorem 6.4 in [11] while using Remarks 2.8 and 2.9.

Theorem 2.10

Let $m, \tau \in \mathbb {R}$ and $N \in \mathbb {N}_0 \cup \{ \infty \}$ with $N > n+ \tau$. Moreover, let $\chi \in \mathcal {S}(\mathbb {R}^n \times \mathbb {R}^n)$ with $\chi (0,0)=1$ be arbitrary. Then the oscillatory integral

exists for each $a \in \mathscr {A}^{m,N}_{\tau }(\mathbb {R}^n \times \mathbb {R}^n)$. Additionally for all $l,l' \in \mathbb {N}_0$ with $l > n+m$ and $N \ge l' > n + \tau$ we have

Therefore the definition does not depend on the choice of $\chi$.

Next, we want to convince ourselves that the properties of the oscillatory integral even hold for all functions of the set $\mathscr {A}^{m,N}_{\tau }(\mathbb {R}^n \times \mathbb {R}^n)$.

Theorem 2.11

Let $m,\tau \in \mathbb {R}$ and $k \in \mathbb {N}$. We define $\tilde{\tau }:= \tau$ if $\tau \ge -k$, $\tilde{\tau }:= -k-0.5$ if $\tau \in \mathbb {Z}$ and $\tau < -k$ and $\tilde{\tau }:= -k-(|\tau | - \lfloor -\tau \rfloor )/2$ else. Moreover, we define $\hat{\tau }:= \tau _+$ if $\tau \ge -k$ and $\hat{\tau }:= \tau - \tilde{\tau }$ else. Additionally let $N \in \mathbb {N}_0 \cup \{ \infty \}$ and $M:= \max \{ m \in \mathbb {N}_0: N-m \ge l > k+ \tilde{\tau } \text { for one } l \in \mathbb {N}_0 \}$. Assuming an $a \in \mathscr {A}^{m,N}_{\tau }(\mathbb {R}^{n+k} \times \mathbb {R}^{n+k})$ we define $b: \mathbb {R}^n \times \mathbb {R}^n\rightarrow \mathbb {C}$ via

If there is an $\tilde{l} \in \mathbb {N}_0$ with $M \ge \tilde{l} > n + \hat{\tau }$ we obtain

(5)

If there is an $\tilde{l} \in \mathbb {N}_0$ with $N \ge \tilde{l} > k + \tau$ we have $b \in \mathscr {A}^{m_+, M}_{\hat{\tau }}(\mathbb {R}^{n} \times \mathbb {R}^{n})$ and

(6)

for each $\alpha , \beta \in \mathbb {N}_0^n$ with $|\beta | \le M$.

Proof

We can show (5) in the same manner as Theorem 3.13 in [2] while using Remark 2.9. Now let $\tilde{l} \in \mathbb {N}_0$ with $N \ge \tilde{l} > k + \tau$. On account of $\langle (\eta , \eta ') \rangle ^2 \ge \langle \eta \rangle ^2$ for all $\eta , \eta ' \in \mathbb {R}^{n}$ and of Peetre’s inequality, cf. [2, Lemma 3.7], we get

$$\begin{aligned} \langle (\eta , \eta ') \rangle ^m \langle (y,y') \rangle ^{\tau } \le C \langle \eta \rangle ^{m_+} \langle \eta ' \rangle ^m \langle y \rangle ^{ \hat{\tau } } \langle y' \rangle ^{ \tilde{\tau } } \quad \text {for all } \eta , \eta ', y,y' \in \mathbb {R}^{n}. \end{aligned}$$

Hence we obtain for fixed $y,\eta \in \mathbb {R}^{n}$ and for all $\alpha , \beta \in \mathbb {N}_0^n$, $\tilde{\alpha }, \tilde{\beta } \in \mathbb {N}^k_0$ with $|\beta | \le M$ and $|\tilde{\beta }| \le N-|\beta |$:

$$\begin{aligned} \bigg |\partial _y^{\tilde{\alpha }} \partial _{\eta }^{\tilde{\beta }} \partial ^{\alpha }_y \partial ^{\beta }_{\eta } a(y,y',\eta , \eta ') \bigg | \le C_{y,\eta } \langle \eta ' \rangle ^m \langle y' \rangle ^{ \tilde{\tau } } \quad \text {for all } \; \eta ',y' \in \mathbb {R}^{n}. \end{aligned}$$

Using the previous inequality (6) can be verified in the same way as [2, Theorem 3.13] while using Remark 2.9. $\square$

In the same way as Theorem 6.6 and Theorem 6.8 in [11] we can verify the following statements while using Remarks 2.8 and 2.9:

Theorem 2.12

Let $m,\tau \in \mathbb {R}$ and $N \in \mathbb {N}_0 \cup \{ \infty \}$ with $N > n + \tau$. Moreover, let $l_0, \tilde{l}_0 \in \mathbb {N}_0$ with $\tilde{l}_0 \le N$. Then

for every $a \in \mathscr {A}^{m, N}_{\tau }(\mathbb {R}^{n} \times \mathbb {R}^{n})$.

Corollary 2.13

Let $m,\tau \in \mathbb {R}$ and $N \in \mathbb {N}_0 \cup \{ \infty \}$ with $N > n + \tau$. Additionally let $(a_j)_{j \in \mathbb {N}} \subseteq \mathscr {A}^{m, N }_{\tau }(\mathbb {R}^{n} \times \mathbb {R}^{n})$ be bounded, i.e., for all $\alpha , \beta \in \mathbb {N}_0^n$ with $|\alpha | \le N$:

$$\begin{aligned} \left| \partial ^{\beta }_y \partial ^{\alpha }_{\eta } a_j(y, \eta ) \right| \le C_{\alpha , \beta } \langle \eta \rangle ^m \langle y \rangle ^{\tau } \quad \text {for all} \; y, \eta \in \mathbb {R}^{n}\text { and } j \in \mathbb {N}. \end{aligned}$$

Moreover, there is an $a \in \mathscr {A}^{m, N}_{\tau }(\mathbb {R}^{n} \times \mathbb {R}^{n})$ such that

$$\begin{aligned} \lim _{j \rightarrow \infty } \partial ^{\alpha }_{\eta } \partial ^{\beta }_y a_j(y,\eta ) = \partial ^{\alpha }_{\eta } \partial ^{\beta }_y a(y,\eta ) \quad \text { for all} \; y, \eta \in \mathbb {R}^{n}\end{aligned}$$

for each $\alpha , \beta \in \mathbb {N}_0^n$ with $|\alpha | \le N$. Then

Theorem 2.14

Let $m,\tau \in \mathbb {R}$ and $N \in \mathbb {N}_0 \cup \{ \infty \}$ with $N > n + \tau$. For $a \in \mathscr {A}^{m, N}_{\tau }(\mathbb {R}^{n} \times \mathbb {R}^{n})$ we have:

3 Pseudodifferential Operators

3.1 Properties of Pseudodifferential Operators

For derivatives of non-smooth symbols we are able to verify the next statement:

Lemma 3.1

Let $m\in \mathbb {N}_0$, $M \in \mathbb {N}_0 \cup \{ \infty \}$ and $0<s\le 1$. Additionally let $\mathscr {B} \subseteq C^{m,s}S^0_{0,0}(\mathbb {R}^n \times \mathbb {R}^n; M)$ be a bounded subset. Considering $\alpha , \gamma \in \mathbb {N}_0^n$ with $|\gamma | \le M-1$ and $|\alpha | \le m$, the set $\{ \partial ^{\alpha }_x \partial _{\xi }^{\gamma } a : a \in \mathscr {B} \} \subseteq C^{0,s}(\mathbb {R}^n \times \mathbb {R}^n)$ is bounded.

Proof

First of all we choose arbitrary $\alpha , \gamma \in \mathbb {N}_0^n$ with $|\gamma | \le M-1$ and $|\alpha | \le m$. Since $\mathscr {B} \subseteq C^{m, s}S^0_{0,0}(\mathbb {R}^n \times \mathbb {R}^n; M)$ is a bounded subset, we get that

$$\begin{aligned}&\{ \partial ^{\alpha }_x \partial _{\xi }^{ \gamma } a : a \in \mathscr {B} \} \subseteq C^0_b(\mathbb {R}^n \times \mathbb {R}^n) \text { is bounded and}, \end{aligned}$$

(7)

$$\begin{aligned}&\sup _{(x,\xi )\ne (y,\eta )} \frac{| \partial ^{\alpha }_x \partial _{\xi }^{ \gamma } a(x,\eta )- \partial ^{\alpha }_x \partial _{\xi }^{ \gamma } a(y,\eta )|}{|( x,\xi ) - (y,\eta )|^{s}} \le \sup _{\eta } \Vert \partial _{\xi }^{ \gamma } a(.,\eta ) \Vert _{C^{m,s}(\mathbb {R}^{n})} < C_{\gamma } \forall a \in \mathscr {B}. \end{aligned}$$

(8)

On account of the fundamental theorem of calculus in the case $|\xi -\eta |<1$ with $\xi \ne \eta$ and because of (7) for $|\xi -\eta |\ge 1$ we can show

$$\begin{aligned} \sup _{(x,\xi )\ne (y,\eta )} \frac{| \partial ^{\alpha }_x \partial _{\xi }^{ \gamma } a(x,\xi )- \partial ^{\alpha }_x \partial _{\xi }^{\gamma } a(x,\eta )|}{|(x,\xi ) - (y,\eta )|^{s}} \le C_{\gamma } \quad \text {for all } a \in \mathscr {B}. \end{aligned}$$

(9)

Collecting the estimates (7), (8) and (9) we finally obtain the claim. $\square$

Now let us mention some boundedness results for pseudodifferential operators needed later on.

Theorem 3.2

Let $p \in S^{m}_{1,0}(\mathbb {R}^{n}\times \mathbb {R}^{n})$ with $m \in \mathbb {R}$. Then

$$\begin{aligned} p(x,D_x): \mathcal {S}(\mathbb {R}^n)\rightarrow \mathcal {S}(\mathbb {R}^n)\end{aligned}$$

is a bounded mapping. More precisely, for every $k \in \mathbb {N}_0$ we can show

$$\begin{aligned} |p(x,D_x)f|_{k,\mathcal {S}} \le C_k |p|^{(m)}_k |f|_{\tilde{m}, \mathcal {S}} \quad \text {for all } f \in \mathcal {S}(\mathbb {R}^n), \end{aligned}$$

where $\tilde{m}:= \max \{ 0, m + 2(n+1) + k\}$ if $m \in \mathbb {Z}$ and $\tilde{m}:= \max \{ 0, \lfloor m \rfloor + 2n+3 + k\}$ else.

We refer to e. g. [2, Theorem 3.6] for the proof. Non-smooth pseudodifferential operators with coefficients in a Banach space X with $C^{\infty }_c(\mathbb {R}^{n}) \subseteq X \subseteq C^0(\mathbb {R}^{n})$, see e. g. [20] for the definition, have similar properties if the next estimate holds for some $N \in \mathbb {N}$ and $C_{\tilde{m},\tau }>0$:

$$\begin{aligned} \Vert e_{\xi } \cdot a(.,\xi ) \Vert _{X} \le C_{\tilde{m},\tau } \langle \xi \rangle ^{ N } \Vert a(.,\xi ) \Vert _{X} \quad \text {for all } \xi \in \mathbb {R}^{n}, \end{aligned}$$

(10)

where a denotes the symbol of the pseudodifferential operator. In the next remark we mention some spaces, where the previous estimate is fulfilled:

Remark 3.3

Let $X \in \{ C^{\tilde{m},\tau }, C^{\tilde{m} + \tau }_{*}, H^{\tilde{m}}_q, W^{\tilde{m}, q}_{uloc} \}$ with $\tilde{m} \in \mathbb {N}_0$, $0< \tau \le 1$ and $1<q < \infty$. Here

$$\begin{aligned} W^{\tilde{m}, q}_{uloc}(\mathbb {R}^{n}):= \bigg \{ f:\mathbb {R}^{n}\rightarrow \mathbb {C}: \sup _{x \in \mathbb {R}^{n}} \Vert f\Vert _{W^{\tilde{m}, q}(B_1(x))} < \infty \bigg \} \end{aligned}$$

Additionally we assume $\delta =0$ in the case $X \notin \{ C^{\tilde{m },\tau }, C^{\tilde{m} + \tau }_{*} \}$ and $\tilde{m}>n/q$ if $X \in \{ H^{\tilde{m}}_q, W^{\tilde{m}, q}_{uloc} \}$. For $0 \le \rho , \delta \le 1$ and $M \in \mathbb {N}_0 \cup \{ \infty \}$ we choose an arbitrary $a \in X S^{m}_{\rho ,\delta }(\mathbb {R}^{n}\times \mathbb {R}^{n}; M)$. Then inequality (10) holds for $N = \tilde{m} +2$ in the case $X=C^{\tilde{m} + \tau }_{*}$, for $N= \tilde{m}+1$ in the case $X=C^{\tilde{m}, \tau }$ and for $N= \tilde{m}$ else. For $X \in \{ H^{\tilde{m}}_q, W^{\tilde{m}, q}_{uloc} \}$ the claim can be verified by using the definition of these spaces and the Leibniz rule. With the multiplication property

$$\begin{aligned} \Vert fg \Vert _{C^s_{*}} \le C \Vert f\Vert _{C^s_{*}} \Vert g\Vert _{C^s_{*}} \quad \text {for all } \; f,g \in C^s_{*}(\mathbb {R}^{n}), \end{aligned}$$

and the embedding $C_b^{\tilde{m} +\lfloor \tau \rfloor +1 }(\mathbb {R}^{n}) \hookrightarrow C_{*}^{\tilde{m} +\tau }(\mathbb {R}^{n})$ at hand, we are in the position to prove the remark for $X=C^{\tilde{m},\tau }_{*}$:

$$\begin{aligned} \Vert e_{\xi } \cdot a(.,\xi ) \Vert _{C^{\tilde{m},\tau }_{*}} \le C_{\tilde{m},\tau } \Vert e_{\xi } \Vert _{ C_b^{\tilde{m} +\lfloor \tau \rfloor +1 } } \Vert a(.,\xi ) \Vert _{C^{\tilde{m},\tau }_{*}} \le C_{\tilde{m},\tau } \langle \xi \rangle ^{ \tilde{m}+2 } \Vert a(.,\xi ) \Vert _{C^{\tilde{m},\tau }_{*}} \end{aligned}$$

for all $\xi \in \mathbb {R}^{n}$. It remains to prove the case $X=C^{\tilde{m},\tau }$. Using the mean value theorem in the case $|x_1-x_2| \le 1$, $x_1 \ne x_2$ we obtain

$$\begin{aligned} \sup _{x_1 \ne x_2} \frac{|e^{ix_1 \cdot \xi } - e^{ix_2 \cdot \xi }|}{ |x_1-x_2|^\tau } \le 2 \langle \xi \rangle \quad \text {for all } \xi \in \mathbb {R}^{n}. \end{aligned}$$

On account of the previous inequality we are able to verify the next estimate:

$$\begin{aligned} \max _{|\alpha | \le \tilde{m}} \sup _{x_1 \ne x_2} \frac{|e^{ix_1 \cdot \xi } \partial _x^{\alpha } a(x_1, \xi ) - e^{ix_2 \cdot \xi } \partial _x^{\alpha } a (x_2, \xi ) |}{|x_1-x_2|^\tau } \le C_{\tilde{m},\tau } \langle \xi \rangle \Vert a(.,\xi )\Vert _{C^{\tilde{m},\tau }_{*}} \end{aligned}$$

(11)

for all $\xi \in \mathbb {R}^{n}$. Moreover we are able to show

$$\begin{aligned} \Vert e_{\xi } \cdot a(.,\xi ) \Vert _{C^{\tilde{m}}_{b}} \le C_{\tilde{m}} \langle \xi \rangle ^{ \tilde{m} } \Vert a(.,\xi ) \Vert _{C^{\tilde{m},\tau } } \quad \text {for all } \xi \in \mathbb {R}^{n}. \end{aligned}$$

(12)

A combination of the inequalities (11) and (12) yields

$$\begin{aligned} \Vert e_{\xi } \cdot a(.,\xi ) \Vert _{C^{\tilde{m},\tau }} \le C_{\tilde{m},\tau } \langle \xi \rangle ^{ \tilde{m}+1 } \Vert a(.,\xi ) \Vert _{C^{\tilde{m},\tau }} \quad \text {for all } \xi \in \mathbb {R}^{n}. \end{aligned}$$

The previous remark enables us to prove the next boundedness result:

Lemma 3.4

Let X be a Banach space such that $C^{\infty }_c(\mathbb {R}^{n}) \subseteq X \subseteq C^0(\mathbb {R}^{n})$ and that inequality (10) holds. Moreover let $m\in \mathbb {R}$, $M \in \mathbb {N}_0 \cup \{ \infty \}$. Additionally let $\delta =0$ if $X \notin \{ C^{\tilde{m},\tau }, C^{\tilde{m} + \tau }_{*} \}$ and $0 \le \rho ,\delta \le 1$ else. Assuming $p \in XS^{m}_{\rho ,\delta }(\mathbb {R}^{n}\times \mathbb {R}^{n}; M)$, we obtain the continuity of $p(x, D_x): \mathcal {S}(\mathbb {R}^n)\rightarrow X$.

Proof

Let $u \in \mathcal {S}(\mathbb {R}^n)$ be arbitrary. An application of $p \in XS^{m}_{\rho ,\delta }(\mathbb {R}^{n}\times \mathbb {R}^{n}; M)$, $u \in \mathcal {S}(\mathbb {R}^n)$ and Remark 3.3 yields

$\square$

In the case $X=C^{\tilde{m},\tau }$ this statement was already proven in [9, Theorem 3.6]. For a bounded subset of $\mathscr {B} \subseteq XS^{m}_{\rho ,\delta }(\mathbb {R}^{n}\times \mathbb {R}^{n}; M)$, where $X, m, \rho ,\delta$ and M are defined as in the previous lemma, we are even able to improve the statement of Lemma 3.4. Verifying the proof of Lemma 3.4 yields the boundedness of

$$\begin{aligned} \left\{ p(x, D_x) : p \in \mathscr {B} \right\} \subseteq \mathscr {L}(\mathcal {S}(\mathbb {R}^n); X). \end{aligned}$$

(13)

In the literature such problems are mostly not investigated. Usually just boundedness results are shown in different cases. Verifying these proofs in order to get similar results as (13) is often very complex. With the next lemma at hand, such problems are much easier to prove.

Lemma 3.5

Let $N \in \mathbb {N}_0 \cup \{ \infty \}$ and X be a Banach space with $C^{\infty }_c(\mathbb {R}^{n}) \subseteq X \subseteq C^0(\mathbb {R}^{n})$. Additionally let m, $\rho$ and $\delta$ be as in the last lemma. We consider that $\mathscr {B}$ is the topological vector space $S^m_{\rho , \delta } (\mathbb {R}^n \times \mathbb {R}^n)$ or $X S^m_{\rho , \delta } (\mathbb {R}^n \times \mathbb {R}^n;N)$. In the case $\mathscr {B}=S^m_{\rho , \delta } (\mathbb {R}^n \times \mathbb {R}^n)$ we set $N:=\infty$. Moreover, let $X_1,X_2$ be two Banach spaces with the following properties:

• (i) $\mathcal {S}(\mathbb {R}^n)\subseteq X_1, X_2 \subseteq \mathcal {S'}(\mathbb {R}^n)$,

• (ii) $\mathcal {S}(\mathbb {R}^n)$ is dense in $X_1$ and in $X'_2$,

• (iii) $a(x,D_x) \in \mathscr {L}(X_1, X_2)$ for all $a \in \mathscr {B}$.

Then there is a $k \in \mathbb {N}$ with $k \le N$ such that

$$\begin{aligned} \Vert a(x, D_x)f \Vert _{\mathscr {L}(X_1;X_2)} \le C |a|^{(m)}_{k} \quad \text {for all} \; a \in \mathscr {B}. \end{aligned}$$

Proof

First of all we define for $f,g \in \mathcal {S}(\mathbb {R}^n)$ with $\Vert f\Vert _{X_1} \le 1$ and $\Vert g\Vert _{X'_2} \le 1$ the operator $OP_{f,g} : \mathscr {B} \rightarrow \mathbb {C}$ by $OP_{f,g}(a):= {\langle a(x, D_x)f,g \rangle }_{X_2, X'_2}$. Using iii) we get the existence of a constant C, independent of $f,g \in \mathcal {S}(\mathbb {R}^n)$ with $\Vert f \Vert _{X_1} \le 1$ and $\Vert g\Vert _{X'_2} \le 1$, such that

$$\begin{aligned} |{\langle a(x, D_x)f,g \rangle }_{X_2, X'_2}|&\le C \left\| a(x, D_x)\right\| _{\mathscr {L}(X_1; X_2) } \left\| f \right\| _{X_1} \left\| g \right\| _{X_2'} \\&\le C \left\| a(x, D_x)\right\| _{\mathscr {L}(X_1; X_2) }. \end{aligned}$$

Consequently the set

$$\begin{aligned} \left\{ OP_{f,g(a)} : f,g \in \mathcal {S}(\mathbb {R}^n)\text { with } \Vert f\Vert _{X_1} \le 1 \text { and } \Vert g\Vert _{X'_2} \le 1 \right\} \subseteq \mathbb {C}\end{aligned}$$

is bounded for each $a \in \mathscr {B}$. An application of the theorem of Banach-Steinhaus, cf. e.g. [18, Theorem 2.5] provides that

$$\begin{aligned} \left\{ OP_{f,g} : f,g \in \mathcal {S}(\mathbb {R}^n)\text { with } \Vert f\Vert _{X_1} \le 1 \text{ and } \Vert g\Vert _{X'_2} \le 1 \right\} \end{aligned}$$

is equicontinuous. With the equicontinuity of the previous set at hand, we get the existence of a $k \in \mathbb {N}$ with $k \le N$ and a constant $C>0$ such that

$$\begin{aligned} |OP_{f,g}(a)| \le C | a |^{(m)}_{k} \quad \text {for all } a \in \mathscr {B}, f,g \in \mathcal {S}(\mathbb {R}^n)\text { with } \Vert f\Vert _{X_1} \le 1, \Vert g\Vert _{X'_2} \le 1. \end{aligned}$$

Since $\mathcal {S}(\mathbb {R}^n)$ is dense in $X_1$ and in $X'_2$, the previous inequality even holds for all $f \in X_1$ and $g \in X_2'$ with $\Vert f\Vert _{X_1} \le 1$ and $\Vert g\Vert _{X'_2} \le 1$. This implies the claim:

$$\begin{aligned} \Vert a(x, D_x) \Vert _{\mathscr {L}(X_1;X_2)}&= \sup _{\Vert f\Vert _{X_1} \le 1} \Vert a(x, D_x)f \Vert _{X_2}\\&= \sup _{\Vert f\Vert _{X_1} \le 1} \sup _{\Vert g\Vert _{X_2'} \le 1} |OP_{f,g}(a )| \le C |a|^{(m)}_{k} \end{aligned}$$

for all $a \in \mathscr {B}$. $\square$

Next we summarize boundedness results for pseudodifferential operators as maps between two Bessel potential spaces. In the smooth case we refer e.g. to [2, Theorem 5.20].

Theorem 3.6

Let $m\in \mathbb {R}$, $p \in S^{m}_{1,0}(\mathbb {R}^{n}\times \mathbb {R}^{n})$ and $1<q<\infty$. Then $p(x,D_x)$ extends to a bounded linear operator

$$\begin{aligned} p(x,D_x): H_q^{s+m}(\mathbb {R}^{n}) \rightarrow H^s_q(\mathbb {R}^{n}) \quad \text {for all}\,s \in \mathbb {R}. \end{aligned}$$

Theorem 3.7

Let $m \in \mathbb {R}$, $0 \le \delta \le \rho \le 1$ with $\rho > 0$ and $1< p < \infty$. Additionally let $\tau > \frac{1- \rho }{1-\delta } \cdot \frac{n}{2}$ if $\rho <1$ and $\tau >0$ if $\rho =1$ respectively. Moreover, let $N \in \mathbb {N}\cup \{ \infty \}$ with $N> n/2$ for $2 \le p < \infty$ and $N>n/p$ else. Denoting $k_p := (1- \rho )n \left| 1/2 - 1/p \right|$, let $\mathscr {B} \subseteq C^{\tau }_{*} S^{m- k_p}_{\rho ,\delta }(\mathbb {R}^{n}\times \mathbb {R}^{n}; N)$ be a bounded subset. Then for each real number s with the property

$$\begin{aligned} (1-\rho )\frac{n}{p}-(1-\delta )\tau< s < \tau \end{aligned}$$

there is a constant $C_s>0$, independent of $a \in \mathscr {B}$, such that

$$\begin{aligned} \Vert a(x, D_x)f \Vert _{ H_p^{s} } \le C_s \Vert f \Vert _{ H_p^{s+m} } \quad \text {for all } f \in H_p^{s+m}(\mathbb {R}^{n}) \text { and } a \in \mathscr {B}. \end{aligned}$$

Proof

In the case $2 \le p <\infty$ the theorem was shown in [15, Theorem 2.7] for $\sharp \mathscr {B} = 1$. The case $1<p<2$ has been proved in [15, Theorem 4.2] for $\sharp \mathscr {B} = 1$. Thus it remains to verify whether the constant $C_s$ is independent of $a \in \mathscr {B}$. We define $p'$ by $1/p + 1/p' = 1$. Since $\mathcal {S}(\mathbb {R}^n)$ is dense in $H^{s+m}_p(\mathbb {R}^{n})$ and $H^{-s}_{p'}(\mathbb {R}^{n})$, the theorem holds because of Lemma 3.5. $\square$

In the case $\sharp \mathscr {B} = 1$, the previous theorem also holds for $p = 1$ or $p=\infty$, cf. [15, Theorem 2.7 and Theorem 4.2 ].

On account of Theorem 2.1 in [15] and Lemma 2.9 in [15] the next boundedness results hold:

Theorem 3.8

Let $m \in \mathbb {R}$ and $\tau > \frac{n}{2}$. Moreover, let $N \in \mathbb {N}\cup \{ \infty \}$ with $N> n/2$. Additionally let $a \in C^{\tau }_{*} S^{ m}_{0,0}(\mathbb {R}^{n}\times \mathbb {R}^{n}; N)$. Then for each real number $s \in \left( \frac{n}{2}-\tau , \tau \right)$ there is a constant $C_s>0$ such that

$$\begin{aligned} \Vert a(x, D_x)f \Vert _{ H_2^{s} } \le C_s \Vert f \Vert _{ H_2^{s+m} } \quad \text {for all } f \in H_2^{s+m}(\mathbb {R}^{n}). \end{aligned}$$

Theorem 3.9

Let $m \in \mathbb {R}$, $N> n/2$ and $\tau > 0$. Moreover let P be an element of $OPC^{\tau }_{*} S^{m-n/2}_{0,0}(\mathbb {R}^{n}\times \mathbb {R}^{n}; N)$. Then the operator

$$\begin{aligned} P: H_2^{s+m}(\mathbb {R}^{n}) \rightarrow H_2^s(\mathbb {R}^{n}) \quad \text {is continuous for all } -\tau< s < \tau . \end{aligned}$$

Lemma 3.10

Let $s \in \mathbb {R}^+$ with $s \notin \mathbb {N}$, $m \in \mathbb {R}$ and $0 \le \rho , \delta \le 1$. Additionally let $M \in \mathbb {N}_0 \cup \{ \infty \}$. Moreover, $\mathscr {B} \subseteq C^s S^{m}_{\rho ,\delta }(\mathbb {R}^{n}\times \mathbb {R}^{n}; M)$ should be a bounded subset and $u \in \mathcal {S}(\mathbb {R}^n)$. For every $N \in \mathbb {N}_0$ with $2N \le M$ we have

$$\begin{aligned} |a(x,D_x)u(x)| \le C_{N,n} \langle x \rangle ^{-2N} \quad \text {for all } x \in \mathbb {R}^{n}\text { and } a \in \mathscr {B}. \end{aligned}$$

Note that $C_{N,n}$ is dependent on $u \in \mathcal {S}(\mathbb {R}^n)$.

Proof

Let $N \in \mathbb {N}_0$ with $2N \le M$. Choosing $M_{m,n} \in \mathbb {N}$ with $-M_{m,n} < -n -|m|$, we get for all $a \in \mathscr {B}$ and all $x \in \mathbb {R}^{n}$ by means of $u \in \mathcal {S}(\mathbb {R}^n)$ and the boundedness of $\mathscr {B} \subseteq C^s S^{m}_{\rho ,\delta }(\mathbb {R}^{n}\times \mathbb {R}^{n}; M)$:

$$\begin{aligned} \left| \langle D_{\xi } \rangle ^{2N} \left[ a(x,\xi ) \hat{u}(\xi ) \right] \right| \le C_{N,n} \langle \xi \rangle ^{m -M_{m,n}} \in L^1(\mathbb {R}^{n}_{\xi }). \end{aligned}$$

(14)

Here $C_{N,n}$ is independent of $x,\xi \in \mathbb {R}^{n}$ and $a \in \mathscr {B}$. On account of (14) and integration by parts with respect to $\xi$ we conclude the claim:

for all $a \in \mathscr {B}$ and $x \in \mathbb {R}^{n}$. $\square$

3.2 Double Symbols

Definition 3.11

Let $0<s \le 1$, $m \in \mathbb {N}_0$ and $\tilde{m},m'\in \mathbb {R}$. Furthermore, let $N \in \mathbb {N}_0 \cup \{ \infty \}$ and $0 \le \rho \le 1$. Then the space of non-smooth double (pseudodifferential) symbols $C^{m,s} S^{\tilde{m},m'}_{\rho , 0}(\mathbb {R}^n \times \mathbb {R}^n\times \mathbb {R}^n \times \mathbb {R}^n;N)$ is the set of all functions $p:\mathbb {R}^n_x \times \mathbb {R}^n_{\xi } \times \mathbb {R}^n_{x'} \times \mathbb {R}^n_{\xi '} \rightarrow \mathbb {C}$ such that

• (i) $\partial _{\xi }^{\alpha } \partial ^{\beta '}_{x'} \partial _{\xi '}^{\alpha '} p \in C^s(\ R^n_x)$ and $\partial _x^{\beta } \partial _{\xi }^{\alpha } \partial ^{\beta '}_{x'} \partial _{\xi '}^{\alpha ' } p \in C^{0}(\mathbb {R}^n_x \times \mathbb {R}^n_{\xi } \times \mathbb {R}^n_{x'} \times \mathbb {R}^n_{\xi '} )$,

• (ii) $\Vert \partial _{\xi }^{\alpha } \partial ^{\beta '}_{x'} \partial _{\xi '}^{\alpha '} p(.,\xi , x', \xi ') \Vert _{C^{m,s}(\mathbb {R}^n)} \le C_{\alpha , \beta ', \alpha '} \langle \xi \rangle ^{\tilde{m}-\rho |\alpha |} \langle \xi ' \rangle ^{m'-\rho |\alpha '|}$

for all $\xi , x', \xi ' \in \mathbb {R}^{n}$ and arbitrary $\beta , \alpha , \beta ', \alpha ' \in \mathbb {N}_0^n$ with $|\beta | \le m$ and $|\alpha | \le N$. Here the constant $C_{\alpha , \beta ', \alpha '}$ is independent of $\xi , x', \xi ' \in \mathbb {R}^{n}$. In the case $N= \infty$ we write $C^{m,s} S^{\tilde{m},m'}_{\rho , 0}( \mathbb {R}^n \times \mathbb {R}^n\times \mathbb {R}^n \times \mathbb {R}^n)$ instead of $C^{m,s} S^{\tilde{m},m'}_{\rho , 0}(\mathbb {R}^n \times \mathbb {R}^n\times \mathbb {R}^n \times \mathbb {R}^n;\infty )$. Furthermore, we define the set of semi-norms $\{|.|^{\tilde{m},m'}_k : k \in \mathbb {N}_0 \}$ by

$$\begin{aligned} |p|^{\tilde{m},m'}_k&:= \max _{\begin{array}{c} |\alpha | + |\beta '| + |\alpha '| \le k \\ |\alpha | \le N \end{array} } \sup _{\xi , x', \xi ' \in \mathbb {R}^{n}} \bigg \Vert \partial _{\xi }^{\alpha } \partial ^{\beta '}_{x'} \partial _{ \xi '}^{\alpha '} p(.,\xi , x', \xi ') \bigg \Vert _{C^{m,s}(\mathbb {R}^n)} \\&\qquad \langle \xi \rangle ^{-(\tilde{ m}-\rho |\alpha |)} \langle \xi ' \rangle ^{-(m'-\rho |\alpha '|)}. \end{aligned}$$

Due to the previous definition, $p \in C^{m,s}S^{\tilde{m}}_{\rho ,\delta }(\mathbb {R}^{n}\times \mathbb {R}^{N})$ is often called a non-smooth single symbol.

The associated operator of a non-smooth double symbol is defined in the following way:

Definition 3.12

Let $0<s \le 1$, $m \in \mathbb {N}_0$, $0 \le \rho \le 1$ and $\tilde{m},m'\in \mathbb {R}$. Additionally let $N \in \mathbb {N}_0 \cup \{ \infty \}$. Assuming $p \in C^{m,s} S^{\tilde{m},m'}_{\rho , 0}(\mathbb {R}^n \times \mathbb {R}^n\times \mathbb {R}^n \times \mathbb {R}^n; N)$, we define the pseudodifferential operator $P = p(x,D_x, x', D_{x'})$ such that for all $u \in \mathcal {S}(\mathbb {R}^n)$ and $x \in \mathbb {R}^{n}$

Note, that we can verify the existence of the previous oscillatory integral by using the properties of such integrals. For more details, see [17, Lemma 4.64].

The set of all non-smooth pseudodifferential operators whose double symbols are in the symbol-class $C^{m,s} S^{\tilde{m},m'}_{\rho , 0}(\mathbb {R}^n \times \mathbb {R}^n\times \mathbb {R}^n \times \mathbb {R}^n; N)$ is denoted by

$$\begin{aligned} OPC^{m,s} S^{\tilde{m},m'}_{\rho , 0}(\mathbb {R}^n \times \mathbb {R}^n\times \mathbb {R}^n \times \mathbb {R}^n; N). \end{aligned}$$

For later purposes we will need a special subset of the non-smooth double symbols $C^{m,s} S^{\tilde{m},0}_{\rho , 0}(\mathbb {R}^n \times \mathbb {R}^n\times \mathbb {R}^n \times \mathbb {R}^n;N)$: For $0<s \le 1$, $m \in \mathbb {N}_0$, $N \in \mathbb {N}_0 \cup \{ \infty \}$ and $\tilde{m} \in \mathbb {R}$ we denote the space $C^{m,s} S^{\tilde{m}}_{\rho , 0}(\mathbb {R}^n \times \mathbb {R}^n\times \mathbb {R}^{n};N)$ as the set of all non-smooth symbols $p \in C^{m,s} S^{\tilde{m},0}_{\rho , 0}(\mathbb {R}^n \times \mathbb {R}^n\times \mathbb {R}^n \times \mathbb {R}^n;N)$ which are independent of $\xi '$. Then we define the pseudodifferential operator $p(x,D_x,x')$ by

$$\begin{aligned} p(x,D_x,x') := p(x,D_x, x', D_{x'}). \end{aligned}$$

The set of all non-smooth pseudodifferential operators whose double symbols are in $C^{m,s} S^{\tilde{m}}_{\rho , 0}(\mathbb {R}^n \times \mathbb {R}^n\times \mathbb {R}^{n};N)$ is denoted by $OPC^{m,s} S^{\tilde{m}}_{\rho , 0}(\mathbb {R}^n \times \mathbb {R}^n\times \mathbb {R}^{n};N)$.

Pseudodifferential operators of the symbol-class $C^{m,s} S^{\tilde{m}}_{ \rho , 0}(\mathbb {R}^n \times \mathbb {R}^n\times \mathbb {R}^{n};N)$ applied on a Schwartz function can be presented in the following way:

Lemma 3.13

Let $0<s<1$, $\tilde{m} \in \mathbb {N}_0$, $0 \le \rho \le 1$, $m\in \mathbb {R}$ and $N \in \mathbb {N}_0 \cup \{ \infty \}$. Considering $a \in C^{\tilde{m},s} S^{m}_ {\rho , 0}(\mathbb {R}^n \times \mathbb {R}^n\times \mathbb {R}^{n};N)$, we obtain for all $u \in \mathcal {S}(\mathbb {R}^n)$:

$$\begin{aligned} a(x,D_x,x') u(x) = \text {Os-}\iint e^{i(x-y)\cdot \xi } a(x,\xi , y) u(y) dy {\textit{">}}\xi \quad \text {for all } x\in \mathbb {R}^{n}. \end{aligned}$$

Proof

Let $u \in \mathcal {S}(\mathbb {R}^n)$ and $x \in \mathbb {R}^{n}$ be arbitrary. Then

$$\begin{aligned} \langle y' \rangle ^{-2l} \langle D_{\xi '} \rangle ^{2l} a(x,\xi , z) u(z+y') \in \mathscr {A}_{- 2n-2}^{m_+, N} \bigg (\mathbb {R}^{2n}_{(z,y')} \times \mathbb {R}^{2n}_{(\xi ,\xi ')}\bigg ). \end{aligned}$$

With Theorem 2.10, Corollary 2.13, Theorems 2.14 and 2.11 at hand, we get

By means of

$$\begin{aligned} \langle y' \rangle ^{-2l} \langle D_{\xi '} \rangle ^{2l} u(z+y') \in \mathcal {S}(\mathbb {R}^n_{y'}) \subseteq \mathscr {A}^0_{-k}(\mathbb {R}^n_{y'} \times \mathbb {R}^n_{\xi '}) \end{aligned}$$

we are able to apply Theorems 2.12 and 2.14 and get

For the proof of the last equality we refer to [2, Example 3.11] . Combining all these results we conclude the proof. $\square$

Remark 3.14

Let $0<s<1$, $m \in \mathbb {R}$, $\tilde{m} \in \mathbb {N}_0$ and $N \in \mathbb {N}_0 \cup \{ \infty \}$. The boundedness of the subset $\mathscr {B} \subseteq C^{\tilde{m},s} S^{m}_{\rho , 0}(\mathbb {R}^n \times \mathbb {R}^n\times \mathbb {R}^{n}; N)$, $0\le \rho \le 1$, implies the boundedness of

$$\begin{aligned} \left\{ \partial ^{\delta }_x \partial _{\xi }^{\gamma } a : a \in \mathscr {B} \right\} \subseteq C^{\tilde{m}- |\delta |,s} S^{m-\rho |\gamma |}_{\rho , 0}(\mathbb {R}^n \times \mathbb {R}^n\times \mathbb {R}^{n}; N-|\gamma |) \end{aligned}$$

for each $\gamma , \delta \in \mathbb {N}_0^n$ with $|\delta | \le \tilde{m}$ and $|\gamma | \le N$.

Proof

The claim is a direct consequence of the definition of the double symbols. $\square$

4 Characterization of Non-Smooth Pseudodifferential Operators

Throughout the whole section $( \varphi _j)_{ j \in \mathbb {N}_0}$ is an arbitrary but fixed dyadic partition of unity on $\mathbb {R}^n$, that is a partition of unity with

$$\begin{aligned} \text {supp }\varphi _0 \subseteq \overline{B_2(0)} \quad \text {and} \quad \text {supp }\varphi _j \subseteq \{ \xi \in \mathbb {R}^{n}: 2^{j-1} \le |\xi | \le 2^{j+1}\} \end{aligned}$$

for all $j \in \mathbb {N}$.

4.1 Pointwise Convergence in $C^{m,s} S^0_{0,0}$

For a bounded sequence $(p_{\varepsilon })_{\varepsilon >0} \subseteq C^{m,s}S^0_{0,0}(\mathbb {R}^n \times \mathbb {R}^n;M)$, we show the existence of a subsequence of $(p_{\varepsilon })_{\varepsilon >0}$ which converges pointwise in $C^{m,s}S^0_{0,0}(\mathbb {R}^n \times \mathbb {R}^n;M-1)$. To reach this goal we need the next lemma:

Lemma 4.1

Let $m \in \mathbb {N}_0$, $0<s\le 1$ and $(p_{\varepsilon } )_{\varepsilon >0} \subseteq C^{m,s}(\mathbb {R}^{n})$ be a bounded sequence. Then there is a subsequence $(p_{\varepsilon _k} )_{k \in \mathbb {N}} \subseteq (p_{\varepsilon } )_{\varepsilon >0}$ with $\varepsilon _k \rightarrow 0$ for $k \rightarrow \infty$ and a $p \in C ^{m,s}(\mathbb {R}^{n})$ such that for all $\beta \in \mathbb {N}_0^n$ with $|\beta | \le m$

$$\begin{aligned} \partial ^{\beta }_x p_{\varepsilon _k} \xrightarrow []{k \rightarrow \infty } \partial ^{\beta }_x p \end{aligned}$$

converges uniformly on each compact set $K \subseteq \mathbb {R}^{n}$.

Proof

It is sufficient to prove the claim for each $\overline{B_j(0)}$, $j \in \mathbb {N}$. Due to the boundedness of $( p_{\varepsilon }|_{\overline{B_j(0)} })_{\varepsilon > 0} \subseteq C^{m,s}(\overline{B_j(0)})$ and the compactness of the embedding $C^{m,s}(\overline{B_j(0)}) \subseteq C^{m}(\overline{B_j(0)})$ we get by a diagonal sequence argument the existence of a subsequence $(p_{\varepsilon _k} )_{k \in \mathbb {N}} \subseteq (p_{\varepsilon } )_{\varepsilon >0}$ with $\varepsilon _k \rightarrow 0$ for $k \rightarrow \infty$ and of unique functions $p_{B_j(0)} \in C^m(\overline{B_j(0)})$ such that

$$\begin{aligned} p_{\varepsilon _k} \xrightarrow []{k \rightarrow \infty } p_{B_j(0)} \quad \text {in } C^m(\overline{B_j(0)}) \text { for all } j \in \mathbb {N}. \end{aligned}$$

We define $p:\mathbb {R}^{n}\rightarrow \mathbb {C}$ via $p(x) := p_{B_j(0)}(x)$ for all $x \in \overline{B_j(0)}$ and each $j \in \mathbb {N}$. This implies the uniform convergence of

$$\begin{aligned} \partial ^{\beta }_x p_{\varepsilon _k} \xrightarrow []{k \rightarrow \infty } \partial ^{\beta }_x p \quad \text {on } \overline{B_j(0)} \end{aligned}$$

for all $j \in \mathbb {N}$ and $\beta \in \mathbb {N}_0^n$ with $|\beta | \le m$. The definition of p provides $p \in C^m(\overline{B_j(0)})$. The boundedness of $(p_{\varepsilon } )_{\varepsilon >0} \subseteq C^{m,s}(\mathbb {R}^{n})$ and the pointwise convergence of $\partial _x^{\alpha } p_{\varepsilon } \rightarrow \partial _x^{\alpha }p$ if $\varepsilon \rightarrow 0$ for all $\alpha \in \mathbb {N}_0^n$ yields $p \in C^{m,s}(\mathbb {R}^{n})$. $\square$

The previous result enables us to show the next claim:

Lemma 4.2

Let $m \in \mathbb {N}_0$ and $0<s\le 1$. Furthermore, let $( \partial _x^{\beta } p_{\varepsilon } )_{\varepsilon >0} \subseteq C^{0,s}(\mathbb {R}^n_{x} \times \mathbb {R}^n_{\xi })$ be a bounded sequence for all $\beta \in \mathbb {N}_0^n$ with $|\beta | \le m$. Then there is a subsequence $(p_{\varepsilon _k} )_{k \in \mathbb {N}} \subseteq (p_{\varepsilon } )_{\varepsilon >0}$ with $\varepsilon _k \rightarrow 0$ for $k \rightarrow \infty$ and a $p \in C^{0,s}(\mathbb {R}^n \times \mathbb {R}^n)$ such that for all $\beta \in \mathbb {N}_0^n$ with $|\beta | \le m$ we have

• i) $\partial _x^{\beta } p \in C^{0,s}(\mathbb {R}^n_{x} \times \mathbb {R}^n_{\xi })$,

• ii) $\partial ^{\beta }_x p_{\varepsilon _k} \xrightarrow []{k \rightarrow \infty } \partial ^{\beta }_x p$ converges uniformly on each compact set $K \subseteq \mathbb {R}^n \times \mathbb {R}^n$.

Proof

It is sufficient to show the claim for all sets $\overline{B_j(0) \times B_i(0) }$, $i,j \in \mathbb {N}$. Since the subset $( \partial _x^{\beta } p_{\varepsilon } )_{\varepsilon >0}$ is bounded in $C^{0,s}(\mathbb {R}^n_{x} \times \mathbb {R}^n_{\xi })$, we iteratively conclude from Lemma 4.1 the existence of a subsequence $(p_{\varepsilon _k} )_{k \in \mathbb {N}}$ of $(p_{\varepsilon } )_{e>0}$ and of functions $q_{ \beta } \in C^{0,s}(\mathbb {R}^n_{x} \times \mathbb {R}^n_{\xi })$ such that

$$\begin{aligned} \partial ^{ \beta }_x p_{\varepsilon _k} \xrightarrow []{k \rightarrow \infty } q_{ \beta } \quad \text {uniformly in } \overline{ B_j(0) \times B_i(0) } \end{aligned}$$

(15)

for all $i,j \in \mathbb {N}$ and $\beta \in \mathbb {N}_0^n$ with $|\beta | \le m$. Choosing an arbitrary but fixed $\xi \in \mathbb {R}^{n}$, (15) implies the uniformly convergence of

$$\begin{aligned} \partial ^{ \beta }_x p_{\varepsilon _k}(., \xi ) \xrightarrow []{k \rightarrow \infty } q_{ \beta }(.,\xi ) \end{aligned}$$

(16)

in $\overline{B_j(0)}$ for all $\beta \in \mathbb {N}_0^n$ with $|\beta | \le m$ and all $j \in \mathbb {N}$. Hence $( p_{\varepsilon _k}(., \xi ) )_{k \in \mathbb {N}}$ is a Cauchy sequence in $C^m(\overline{B_j(0)})$. Due to the completeness of $C^m (\overline{B_j(0)})$ we have the convergence of $( p_{\varepsilon _k}(., \xi ) )_{k \in \mathbb {N}}$ to $\tilde{p}$ in $C^m (\overline{B_j(0)})$. Consequently we obtain for all $\beta \in \mathbb {N}_0^n$ with $|\beta | \le m$ and each $j \in \mathbb {N}$:

$$\begin{aligned} \partial ^{ \beta }_x p_{\varepsilon _k}(., \xi ) \xrightarrow []{k \rightarrow \infty } \partial ^{ \beta }_x \tilde{p} \end{aligned}$$

(17)

in $C^0(\overline{B_j(0)})$. Because of the uniqueness of the strong limit we get together with (16) that $\partial ^{ \beta }_x \tilde{p} = q _{\beta }(.,\xi )$ for each $\beta \in \mathbb {N}_0^n$ with $|\beta | \le m$. Thus with $p(x,\xi ):= q_0(x,\xi )$ for all $x,\xi \in \mathbb {R}^{n}$ the claim holds. $\square$

Finally we are able to show the main theorem of this subsection:

Theorem 4.3

Let $m \in \mathbb {N}_0$, $M \in \mathbb {N}\cup \{ \infty \}$ and $0<s\le 1$. Furthermore, let $(p_{\varepsilon } )_{\varepsilon >0} \subseteq C^{m,s}S^0_{0,0}(\mathbb {R}^n \times \mathbb {R}^n;M)$ be a bounded sequence. Then there is a subsequence $(p_{\varepsilon _l} )_{l \in \mathbb {N}} \subseteq (p_{\varepsilon } ) _{\varepsilon >0}$ with $\varepsilon _l \rightarrow 0$ for $l \rightarrow \infty$ and a function $p: \mathbb {R}^{n}_x \times \mathbb {R}^{n}_{\xi } \rightarrow \mathbb {C}$ such that for all $\alpha , \beta \in \mathbb {N}_0^n$ with $|\beta | \le m$ and $|\alpha | \le M-1$ we get

• i) $\partial _x^{\beta }\partial _{\xi }^{\alpha } p$ exists and $\partial _x^{\beta } \partial _{\xi }^{\alpha } p \in C^{0,s}(\mathbb {R}^n \times \mathbb {R}^n)$,

• ii) $\partial ^{\beta }_x \partial _{\xi }^{\alpha } p_{\varepsilon _l} \xrightarrow []{l \rightarrow \infty } \partial ^{\beta }_x \partial _{\xi }^{\alpha } p$ is uniformly convergent on each compact set $K \subseteq \mathbb {R}^n \times \mathbb {R}^n$.

In particular $p \in C^{m,s}S^0_{0,0}( \mathbb {R}^n \times \mathbb {R}^n; M-1)$.

Proof

It is sufficient to prove the claim for $\overline{B_j(0) \times B_{j}( 0) }$, $j \in \mathbb {N}$. Applying Lemma 3.1 we get for all $\beta , \gamma \in \mathbb {N}_0^n$ with $|\beta | \le m$ and $|\gamma | \le M-1$ the boundedness of the sequence $( \partial _x^{\beta } \partial _{\xi }^{\gamma } p_{\varepsilon } ) _{\varepsilon >0} \subseteq C^{0,s}(\mathbb {R}^n \times \mathbb {R}^n)$. Thus by Lemma 4.2 we inductively obtain the existence of a subsequence $(p_{\varepsilon _l} )_{l \in \mathbb {N}} \subseteq (p_{\varepsilon } )_{\varepsilon >0}$ and functions $q_{\alpha } \in C^{0,s}(\mathbb {R}^n \times \mathbb {R}^n)$ with the following properties: For all $j \in \mathbb {N}$ and $\alpha ,\beta \in \mathbb {N}_0^n$ with $|\beta | \le m$ and $|\alpha | \le M-1$ we have $\partial _ x^{\beta } q_{\alpha } \in C^{0,s}(\mathbb {R}^n_{x} \times \mathbb {R}^n_{\xi })$ and

$$\begin{aligned} \partial ^{\beta }_x \partial _{\xi }^{\alpha } p_{\varepsilon _l} \xrightarrow []{l \rightarrow \infty } \partial ^{\beta }_x q_{\alpha } \end{aligned}$$

(18)

converges uniformly on $\overline{B_j(0) \times B_{j}(0)}$. Now we choose an arbitrary but fixed $k \in \mathbb {N}_0$ with $k \le M-1$ and $x \in \mathbb {R}^{n}$. The boundedness of $( \partial _{\xi }^{\gamma } p_{\varepsilon _l})_{l \in \mathbb {N}} \subseteq C^{0,s}(\mathbb {R}^n \times \mathbb {R}^n)$ for all $\gamma \in \mathbb {N}_0^n$ with $|\gamma | \le k$ leads to

$$\begin{aligned} \Vert p_{\varepsilon _l}(x,.) \Vert _{ C^{k,s}(\mathbb {R}^{n}) } \le \max _{\begin{array}{c} \gamma \in \mathbb {N}_0^n \\ |\gamma | \le k \end{array} } \Vert \partial _{\xi }^{ \gamma } p_{\varepsilon _l} \Vert _{ C^{0,s}(\mathbb {R}^n \times \mathbb {R}^n) } \le C_{k} \end{aligned}$$

for all $x\in \mathbb {R}^{n}$ and $l \in \mathbb {N}$. By means of Lemma 4.1 we obtain via a diagonal sequence argument the existence of a subsequence of $( p_{\varepsilon _l} )_{l \in \mathbb {N}}$ denoted by $( p_{\varepsilon _{l_r}} )_{r \in \mathbb {N}}$ and of a function $\tilde{p} \in C^{M-1}(\mathbb {R}^{n})$ with the property

$$\begin{aligned} \partial _{\xi }^{\gamma } p_{\varepsilon _{l_r}}(x,\xi ) \xrightarrow []{r \rightarrow \infty } \partial _{\xi }^{\gamma } \tilde{p}(\xi ) \quad \text {pointwise for all } \xi \in \mathbb {R}^{n}\end{aligned}$$

(19)

and every $\gamma \in \mathbb {N}_0^n$ with $|\gamma | \le M-1$. On account of (18) and (19) the uniqueness of the limit gives us $q_{\alpha }(x ,.)= \partial _{\xi }^{\alpha } \tilde{p}$. This implies $p(x,.):= q_0(x,.) \in C^{M-1}(\mathbb {R}^{n})$ for all $x \in \mathbb {R}^{n}$, (i) and (ii). Note that (i) implies for all $\gamma \in \mathbb {N}_0^n$ with $|\gamma | \le M-1$

$$\begin{aligned} \Vert \partial _{\xi }^{\gamma }p(.,\xi ) \Vert _{ C^{m,s}(\mathbb {R}^{n}) } \le \max _{|\beta | \le m} \Vert \partial _x^{\beta } \partial _{\xi }^{\gamma } p \Vert _{ C^{0,s}(\mathbb {R}^n \times \mathbb {R}^n) } \le C_{\gamma } \quad \text { for all } \xi \in \mathbb {R}^{n}. \end{aligned}$$

Consequently $p \in C^{m,s}S^0_{0,0}( \mathbb {R}^n \times \mathbb {R}^n; M-1)$. $\square$

4.2 Reduction of Non-Smooth Pseudodifferential Operators with Double Symbol

In this subsection we derive a formula representing an operator with a non-smooth double symbol as an operator with a non-smooth single symbol. During the development of this work (however independent) Köppl generalized this result in his diploma thesis, cf. [9, Theorem 3.33], for non-smooth double symbols of the symbol-class $C^{\tilde{m}, \tau } S^{m,m'}_{\rho , \delta }(\mathbb {R}^n \times \mathbb {R}^n\times \mathbb {R}^n \times \mathbb {R}^n; N)$ where $N=\infty$. We can show the same result even for $N >n$. However, the smoothness of the reduced symbol with respect to $\xi$ is reduced by the order of n.

For the proof of the characterization of non-smooth pseudodifferential operators only the case $\rho =\delta =0$ is required. Thus we restrict the symbol reduction to this case. This significantly simplifies some proofs. The main idea of the symbol reduction is taken from that one of the smooth case, cf. e.g. [11, Theorem 2.5]. Since the symbols are non-smooth in both variables, the proof has to be adapted to this modified condition.

We begin with some auxiliary tools needed for the proof of the symbol reduction:

Lemma 4.4

Let $s > 0$, $s \notin \mathbb {N}_0$, $m,m' \in \mathbb {R}$ and $N \in \mathbb {N}_0 \cup \{ \infty \}$. Additionally we choose $l,l_0, l_0' \in \mathbb {N}_0$ such that

$$\begin{aligned} -2 l +m< -n, \quad - 2 l_0< -n \quad \text {and} \quad -2l_0' + 2 l + m' < -n. \end{aligned}$$

Furthermore, let $P:= p(x,D_x,x',D_{x'}) \in OPC^s_*S^{m,m'}_{0,0}( \mathbb {R}^n \times \mathbb {R}^n\times \mathbb {R}^n \times \mathbb {R}^n; N)$. For $u \in \mathcal {S}(\mathbb {R}^n)$ we define $\tilde{p}_u: \mathbb {R}^{5n} \rightarrow \mathbb {C}$ by

$$\begin{aligned} \tilde{p}_u(x,\xi ,x', \xi ',x''):= \langle -\xi + \xi ' \rangle ^{-2l} \langle D_{x'} \rangle ^{2l} p_u(x,\xi ,x', \xi ', x'') \end{aligned}$$

for all $x,\xi ,x',\xi ', x'' \in \mathbb {R}^{n}$, where

$$\begin{aligned} p_u(x,\xi ,x', \xi ', x''):= \langle x'-x'' \rangle ^{-2l_0} \langle D_{\xi '} \rangle ^{2l_0} \left[ \langle \xi ' \rangle ^{-2l_0'} \langle D_{x''} \rangle ^{2l_0'} p(x,\xi ,x', \xi ') u( x'') \right] \end{aligned}$$

for all $x,\xi ,x',\xi ', x'' \in \mathbb {R}^{n}$. Then we have for all $x \in \mathbb {R}^{n}$:

Proof

Let $x \in \mathbb {R}^{n}$ be arbitrary but fixed. Additionally let $u \in \mathcal {S}(\mathbb {R}^n)$ and $\chi \in \mathcal {S}(\mathbb {R}^{4n})$ with $\chi (0) = 1$. For each $0< \varepsilon < 1$ we denote $\chi _{\varepsilon }: \mathbb {R}^{4n} \rightarrow \mathbb {C}$ by

$$\begin{aligned} \chi _{\varepsilon }(\xi ,\xi ',y,y'):= \chi (\varepsilon \xi ,\varepsilon \xi ',\varepsilon y,\varepsilon y') \quad \text { for all } \xi ,\xi ',y,y' \in \mathbb {R}^{n}. \end{aligned}$$

We define $p_{\varepsilon ,u}: \mathbb {R}^n \times \mathbb {R}^n\times \mathbb {R}^n \times \mathbb {R}^n\times \mathbb {R}^{n}\rightarrow \mathbb {C}$ for every $0< \varepsilon < 1$ by

$$\begin{aligned} p_{\varepsilon ,u} (\tilde{x}, \xi , x', \xi ', x'') := \chi _{\varepsilon }(\xi ,\xi ', x'- x , x''-x') p(\tilde{x}, \xi , x', \xi ') u(x''). \end{aligned}$$

for all $\tilde{x}, \xi ,x',\xi ',x'' \in \mathbb {R}^{n}$. Using Leibniz’s rule, $p \in C^s_*S^{m,m'}_{0,0}(\mathbb {R}^n \times \mathbb {R}^n\times \mathbb {R}^n \times \mathbb {R}^n; N)$ and $\chi \in \mathcal {S}(\mathbb {R}^{4n})$ provides for all $\alpha , \beta , \gamma \in \mathbb {N}_0^n$:

$$\begin{aligned} \partial ^{\alpha }_{x''} \partial ^{\beta }_{\xi '} \partial ^{\gamma }_{x'} p_{\varepsilon ,u}(x,\xi , x',\xi ', x'') \in L^1(\mathbb {R}^n_{\xi } \times \mathbb {R}^n_{x'} \times \mathbb {R}^n_{\xi '} \times \mathbb {R}^n_{x''}) . \end{aligned}$$

(20)

Due to the definition of the oscillatory integral, the change of variables $x':=x+y$ and $x'':= x' + y'$ and Fubini’s theorem we obtain

(21)

Now we choose $l,l_0, l_0' \in \mathbb {N}_0$ as in the assumptions. Then we define for each $0< \varepsilon < 1$ the function $\tilde{p}_{\varepsilon ,u}: \mathbb {R}^{5n} \rightarrow \mathbb {C}$ by

$$\begin{aligned} \tilde{p}_{\varepsilon ,u}(\tilde{x},\xi ,x', \xi ',x''):= \langle -\xi + \xi ' \rangle ^{-2 l} \langle D_{x'} \rangle ^{2l} p^u_{\varepsilon }(\tilde{x},\xi ,x', \xi ', x'') \end{aligned}$$

for all $\tilde{x},\xi ,x', \xi ',x'' \in \mathbb {R}^{n}$, where the function $p^u_{ \varepsilon }: \mathbb {R}^{5n} \rightarrow \mathbb {C}$ is defined by

$$\begin{aligned} p^u_{\varepsilon }(\tilde{x},\xi ,x', \xi ', x''):= \langle x'-x'' \rangle ^{-2l_0} \langle D_{\xi '} \rangle ^{2l_0} \left[ \langle \xi ' \rangle ^{-2l_0'} \langle D_{x''} \rangle ^{2l_0'} p_{\varepsilon ,u}(\tilde{x},\xi ,x', \xi ',x'') \right] \end{aligned}$$

for each $\tilde{x},\xi ,x', \xi ',x'' \in \mathbb {R}^{n}$. Additionally we can integrate by parts in (21) due to (20) and get

(22)

An application of the Leibniz rule, $p \in C^s_*S^{m,m'}_{0,0}(\mathbb {R}^n \times \mathbb {R}^n\times \mathbb {R}^n \times \mathbb {R}^n; N)$ and $u \in \mathcal {S}(\mathbb {R}^n)$ yields the existence of a constant C, which is independent of $0< \varepsilon < 1$ , such that

$$\begin{aligned}&|e^{-i(x'-x) \cdot \xi -i (x''-x') \cdot \xi '} \tilde{p}_{\varepsilon ,u}(x,\xi ,x', \xi ',x'')| \nonumber \\&\quad \le C \langle \xi \rangle ^{m-2l} \langle \xi ' \rangle ^{-2l_0' +m'+2l} \langle x' \rangle ^{-2l_0} \langle x'' \rangle ^{2l_0 -M} \in L^1 \big (\mathbb {R}^n_{x''} \times \mathbb {R}^n_{\xi '} \times \mathbb {R}^n_{x'} \times \mathbb {R}^n_{\xi }\big ). \end{aligned}$$

(23)

Moreover, if we use the Leibniz rule, the pointwise convergence of $\chi _{\varepsilon }$ to 1 and the pointwise convergence of every derivative of $\chi _{\varepsilon }$ to 0, see e.g. [11, Lemma 6.3], we obtain

$$\begin{aligned} \tilde{p}_{\varepsilon ,u}(x,\xi ,x', \xi ',x'') \xrightarrow []{\varepsilon \rightarrow 0} \tilde{p}_u (x,\xi ,x', \xi ',x''). \end{aligned}$$

(24)

Hence applying Lebesgue’s theorem to (22) concludes the proof. $\square$

Making use of this integral representation we are able to show the following result:

Lemma 4.5

Let $s > 0$, $s \notin \mathbb {N}_0$ and $m,m' \in \mathbb {R}$. Additionally let $N \in \mathbb {N}_0 \cup \{ \infty \}$ and $l' \in \mathbb {N}_0$ with $l' \le N$. Furthermore, let $\mathscr {B} \subset C^s_*S^{m,m'}_{0,0}(\mathbb {R}^n \times \mathbb {R}^n\times \mathbb {R}^n \times \mathbb {R}^n; N)$ be bounded and $u \in \mathcal {S}(\mathbb {R}^n)$. Assuming $p \in C^s_*S^{m,m'}_{0,0 }(\mathbb {R}^n \times \mathbb {R}^n\times \mathbb {R}^n \times \mathbb {R}^n; N)$, we denote $P:= p(x,D_x, x', D_{x'})$. Then we obtain the existence of a constant C, independent of $x \in \mathbb {R}^{n}$ and $p \in \mathscr {B}$, such that

$$\begin{aligned} |P u(x)| \le C \langle x \rangle ^{-l'} \quad \text {for all } x \in \mathbb {R}^{n}. \end{aligned}$$

Proof

An application of Lemma 4.4, Remarks 2.8 and 2.9 and of integration by parts with respect to $\xi$ concludes the claim. $\square$

As in the smooth case, cf. e.g. [11], we define for all $a \in C^{\tilde{m},s}_*S^{m}_{0,0}(\mathbb {R}^n \times \mathbb {R}^n\times \mathbb {R}^n; N)$ the function . In order to verify that $a_L$ is a non-smooth single symbol, we need the next results:

Proposition 4.6

Let $m \in \mathbb {R}$ and X be a Banach space with $X \hookrightarrow L^{\infty }(\mathbb {R}^{n})$. Let $l_0 \in \mathbb {N}_0$ such that $l_0 > n$ and $\mathscr {B}$ be a set of functions $r: \mathbb {R}^n \times \mathbb {R}^n\times \mathbb {R}^n \times \mathbb {R}^n\rightarrow \mathbb {C}$ which are smooth with respect to the fourth variable such that the next inequality holds for all $l \in \mathbb {N}_ 0$:

$$\begin{aligned} \Vert \langle D_y \rangle ^{2l} r(.,\xi ,\eta ,y) \Vert _X \le C_{l} \langle y \rangle ^{-l_0} \langle \xi + \eta \rangle ^m \quad \text {for all } \xi , \eta , y \in \mathbb {R}^{n}, r \in \mathscr {B}. \end{aligned}$$

(25)

Then $\int e^{-iy\cdot \eta } r(x,\xi , \eta ,y) dy \in L^1(\mathbb {R}^{n}_{\eta })$ for all $x,\xi \in \mathbb {R}^{n}$. If we define

for arbitrary $x, \xi \in \mathbb {R}^{n}$ and $r \in \mathscr {B}$ we have

$$\begin{aligned} \left\| I(.,\xi ) \right\| _{X} \le C \langle \xi \rangle ^m \quad \text { for all } \xi \in \mathbb {R}^{n}\text { and } r \in \mathscr {B}. \end{aligned}$$

Proof

Let $\xi \in \mathbb {R}^{n}$ and $r \in \mathscr {B}$. Making use of the assumptions of the proposition we can show $\langle D_y \rangle ^{2\tilde{l}} r(x,\xi ,\eta ,y) \in L^1(\mathbb {R}^{n}_y)$ for each $x, \xi , \eta \in \mathbb {R}^{n}$ and $\tilde{l} \in \mathbb {N}_0$ due to (25). Consequently we can integrate by parts and we obtain for all $l \in \mathbb {N}_ 0$ and $x,\eta \in \mathbb {R}^{n}$:

$$\begin{aligned} \int e^{-iy \cdot \eta } r(x,\xi ,\eta ,y) dy = \langle \eta \rangle ^{-2l} \int e^ {-iy \cdot \eta } \langle D_y \rangle ^{2l} r(x,\xi ,\eta ,y) dy. \end{aligned}$$

(26)

Now we choose an $l \in \mathbb {N}_0$ with $|m| - 2l < -n$. Then we conclude

In particular this provides $\int e^{-iy \cdot \eta }r(x,\xi , \eta ,y) dy \in L^1(\mathbb {R}^{n}_{\eta })$ for all $x,\xi \in \mathbb {R}^{n}$. $\square$

Remark 4.7

In particular we can apply Proposition 4.6 on $X:= C^0_b(\mathbb {R}^{n})$ and on the function $r: \mathbb {R}^n \times \mathbb {R}^n\times \mathbb {R}^n \times \mathbb {R}^n\rightarrow \mathbb {C}$ defined by

$$\begin{aligned} r(x,\xi ,\eta ,y):= A^{l_0}(D_{\eta }, y) a(x,\xi +\eta ,x+y) \quad \text {for all } x, \xi , \eta , y \in \mathbb {R}^{n}. \end{aligned}$$

Proposition 4.8

Let $0< s < 1$, $\tilde{m} \in \mathbb {N}_0$ and $m \in \mathbb {R}$. Additionally let $N \in \mathbb {N}_0 \cup \{ \infty \}$ with $n < N$. Moreover, let $a \in C^{\tilde{m},s}_*S^{m}_{0,0}(\mathbb {R}^n \times \mathbb {R}^n\times \mathbb {R}^n; N)$. Considering an $l_0 \in \mathbb {N}_0$ with $n < l_0 \le N$, we define $r: \mathbb {R}^n \times \mathbb {R}^n\times \mathbb {R}^n \times \mathbb {R}^n\rightarrow \mathbb {C}$ as in Remark 4.7. Then $\int e^{-iy\cdot \eta } r(x,\xi , \eta ,y) dy \in L^1(\mathbb {R}^{n}_{\eta })$ for all $x,\xi \in \mathbb {R}^{n}$ and we obtain

Proof

On account of $a(x, \eta + \xi , x+y) \in \mathscr {A}^{m,N}_0(\mathbb {R}^n_{y} \times \mathbb {R}^n_{\eta })$ for all $x,\xi \in \mathbb {R}^{n}$ Theorem 2.12 yields the existence of the oscillatory integral. Assuming an arbitrary $\chi \in \mathcal {S}(\mathbb {R}^n)$ with $\chi (0)=1$, we get for fixed $x,\eta , \xi \in \mathbb {R}^{n}$:

$$\begin{aligned} e^{-iy\cdot \eta } \chi (\varepsilon y) r(x,\xi ,\eta ,y) \xrightarrow []{\varepsilon \rightarrow 0} e^{-iy\cdot \eta } r(x,\xi ,\eta ,y) \quad \text { for all } y \in \mathbb {R}^{n}. \end{aligned}$$

(27)

Now let $0 < \varepsilon \le 1$. We can prove the next two estimates if we use $\chi \in C^{\infty }_b(\mathbb {R}^{n})$ and Remark 2.9:

$$\begin{aligned} \bigg | \partial _y^{\alpha } r(x,\xi ,\eta ,y) \bigg |&\le C_{\alpha ,m} \langle y \rangle ^{-l_0} \langle \xi \rangle ^m \langle \eta \rangle ^{|m|} \quad \text {for all } \alpha \in \mathbb {N}_0^n,\end{aligned}$$

(28)

$$\begin{aligned} \bigg |\langle D_y \rangle ^{2l'} [\chi (\varepsilon y) r(x,\xi ,\eta ,y)] \bigg |&\le C_{l',m} \langle y \rangle ^{- l_0} \langle \xi \rangle ^m \langle \eta \rangle ^{|m|} \quad \text {for all } l' \in \mathbb {N}_0, \end{aligned}$$

(29)

uniformly in $x,\xi , \eta , y \in \mathbb {R}^{n}$ and in $0 < \varepsilon \le 1$. Let $x,\xi \in \mathbb {R}^{n}$ be fixed and $l \in \mathbb {N}_0$ with $|m|-2l < -n$ be arbitrary. Then we get by means of $\chi \in \mathcal {S}(\mathbb {R}^n)\subseteq C^{\infty }_b(\mathbb {R}^{n})$, integration by parts and (29):

$$\begin{aligned}&\left| \chi (\varepsilon \eta ) \int e^{-iy\cdot \eta } \chi (\varepsilon y) r(x,\xi ,\eta ,y) dy \right| \nonumber \\&\quad \quad \le C \int \left| e^{-iy\cdot \eta } \langle \eta \rangle ^{-2l} \langle D_y \rangle ^{2l} [\chi (\varepsilon y) r(x,\xi ,\eta ,y)] \right| dy \nonumber \\&\qquad \le C_{l,m,\xi } \langle \eta \rangle ^{-2l + |m|} \int \langle y \rangle ^{-l_0} dy \le C_{l,m,\xi } \langle \eta \rangle ^{-2l + |m|} \in L^1(\mathbb {R}^{n}_{\eta }). \end{aligned}$$

(30)

Here the constant $C_{l,m,\xi }$ is independent of $\varepsilon \in (0, 1]$. Setting $l'=0$, (29) provides for each fixed $x,\xi ,\eta \in \mathbb {R}^{n}$, that $\{ y \mapsto \chi (\varepsilon y) r(x,\xi ,\eta ,y): 0 < \varepsilon \le 1 \}$ has a $L^1(\mathbb {R}^{n}_y)$-majorant. Applying Lebesgue’s theorem we obtain

$$\begin{aligned} \chi (\varepsilon \eta ) \int e^{-iy\cdot \eta } \chi (\varepsilon y) r(x,\xi ,\eta ,y) dy \xrightarrow []{\varepsilon \rightarrow 0} \int e^{-iy\cdot \eta } r(x,\xi ,\eta ,y) dy \end{aligned}$$

(31)

for all $x,\xi ,\eta \in \mathbb {R}^{n}$. Applying Lebesgue’s theorem again we get for all $x,\xi \in \mathbb {R}^{n}$:

The assumptions of Lebesgue’s theorem are fulfilled because of (30) and (31). $\square$

The previous results enable us to show the following statement:

Lemma 4.9

Let $0< s < 1$, $\tilde{m} \in \mathbb {N}_0$ and $m \in \mathbb {R}$. Additionally let $\mathscr {B}$ be a bounded subset of $C^{\tilde{m},s}_*S^{m}_{0,0}(\mathbb {R}^n \times \mathbb {R}^n\times \mathbb {R}^n; N)$ and $N \in \mathbb {N}_0 \cup \{ \infty \}$ with $n < N$. We define for each $a \in \mathscr {B}$ the function $a_L: \mathbb {R}^n \times \mathbb {R}^n\rightarrow \mathbb {C}$ by

Then there is a constant C, independent of $x,\xi \in \mathbb {R}^{n}$ and $a \in \mathscr {B}$, such that

$$\begin{aligned} \big |\partial ^{\delta }_x a_L (x, \xi ) \big | \le C \langle \xi \rangle ^m \quad \text {for each } \delta \in \mathbb {N}_0^n\text { with } |\delta | \le \tilde{m}. \end{aligned}$$

Note that Theorem 2.10 yields the existence of $a_L(x, \xi )$ for all $x,\xi \in \mathbb {R}^{n}$ since $a(x, \eta + \xi , x+y) \in \mathscr {A}^{m, N}_0(\mathbb {R}^n_{y} \times \mathbb {R}^n_{\eta })$. For the proof of Lemma 4.9 we need:

Lemma 4.10

Let $N \in \mathbb {N}_0 \cup \{ \infty \}$ with $n < N$. Assuming $a \in C^{ \tilde{m} ,s} S^m_{\rho ,0}(\mathbb {R}^n \times \mathbb {R}^n\times \mathbb {R}^n;N)$ with $\tilde{m} \in \mathbb {N}_0$, $m \in \mathbb {R}$, $0 \le \rho \le 1$ and $0<s<1$, we define $a_L:\mathbb {R}^n \times \mathbb {R}^n\rightarrow \mathbb {C}$ as in Lemma 4.9. Then we get for each $\beta \in \mathbb {N}_0^n$ with $|\beta | \le \tilde{m}$:

Proof

The claim follows from Corollary 2.13 and from approximation of the function $\partial _{x_j} \{ a(x,\xi + \eta , x+y) \}$ by difference quotients. $\square$

Now we are able to prove Lemma 4.9:

Proof of Lemma 4.9 Using Theorem 2.12, Proposition 4.8 and Remark 4.7 we get for each $l_0 \in \mathbb {N}_0$ with $n< l_0 \le N$

for all $x,\xi \in \mathbb {R}^{n}$ and of $a \in \mathscr {B}$. Thus the claim holds for $\delta =0$. Now we assume $\delta \in \mathbb {N}_0^n$ with $|\delta | \le \tilde{m}$. Due to Remark 3.14 $\mathscr {B}^{\delta }: = \left\{ \partial ^{\delta }_x a : a \in \mathscr {B} \right\}$ is bounded in $C^{\tilde{m} - |\delta |,s}_*S^{m}_{0,0}(\mathbb {R}^n \times \mathbb {R}^n\times \mathbb {R}^n; N)$. On account of Lemma 4.10 the case $\delta =0$ applied on the set $\mathscr {B}^{\delta }$, gives us

$$\begin{aligned} \big |\partial ^{\delta }_x a_L (x, \xi ) \big | \le C \langle \xi \rangle ^m \quad \text {for all } x, \xi \in \mathbb {R}^{n}\text { and } a \in \mathscr {B}. \end{aligned}$$

$\square$

Having in mind the definition of the Hölder spaces, we need the next two statements to show that $a_L \in C^{\tilde{m},s}S^m_{0,0}(\mathbb {R}^n \times \mathbb {R}^n; \tilde{N})$ for some $\tilde{m}, \tilde{N} \in \mathbb {N}$.

Proposition 4.11

Let $0< s < 1$, $\tilde{m} \in \mathbb {N}_0$, $N \in \mathbb {N}_0 \cup \{ \infty \}$ and $m \in \mathbb {R}$. Moreover, let $\mathscr {B} \subseteq C^{\tilde{m},s}S^{m}_ {0,0}(\mathbb {R}^n \times \mathbb {R}^n\times \mathbb {R}^n; N)$ be bounded. Then we have for each $\gamma , \beta \in \mathbb {N}_0^n$ with $|\beta | \le N$

$$\begin{aligned}&\max _{|\alpha | \le \tilde{m}} \left\{ \frac{ | \partial _{x_1}^{\alpha } \partial _ y^{\gamma } \partial _{\eta }^{\beta } a(x_1, \xi + \eta , x_1 + y) - \partial _{x_2}^{\alpha } \partial _y^{\gamma } \partial _{\eta }^{\beta } a(x_2, \xi + \eta , x_2 + y) | }{|x_1 - x_2|^s} \right\} \\&\quad \le C \langle \xi + \eta \rangle ^m \end{aligned}$$

for all $x_1,x_2, y, \xi , \eta \in \mathbb {R}^{n}$ with $x_1 \ne x_2$ and $a \in \mathscr {B}$.

Proof

First of all we choose arbitrary $\alpha , \beta , \gamma \in \mathbb {N}_0^n$ with $|\alpha | \le \tilde{m}$ and $|\beta | \le N$ and let $x_1, x_2 \in \mathbb {R}^{n}$ . The boundedness of $\mathscr {B}$ in the set $C^{\tilde{m},s} S^{m}_{ 0,0}(\mathbb {R}^n \times \mathbb {R}^n\times \mathbb {R}^n; N)$ implies

$$\begin{aligned}&\sup _{\begin{array}{c} x,\tilde{x} \in \mathbb {R}^{n}\\ x \ne \tilde{x} \end{array} } \left\{ \frac{ | \partial _{x}^{\alpha } \partial _y^{\gamma } \partial _{\eta }^{\beta } a(x, \xi + \eta , x_1 + y) - \partial _{ \tilde{x} }^{\alpha } \partial _y^{\gamma } \partial _{\eta }^{\beta } a( \tilde{x}, \xi + \eta , x_1 + y) | }{|x - \tilde{x}|^s} \right\} \nonumber \\&\qquad \le \big \Vert \partial _y^{\gamma } \partial _{\eta }^{\beta } a(x, \xi + \eta , x_1 + y) \big \Vert _{ C^{\tilde{m}, s} (\mathbb {R}^{n}_{x}) } \le C_{\gamma ,\beta } \langle \xi + \eta \rangle ^m \end{aligned}$$

(32)

for all $\xi , \eta ,y \in \mathbb {R}^{n}$ and all $a \in \mathscr {B}$. By means of the fundamental theorem of calculus for $|x_1-x_2| <1$ and on account of $|x_1-x_2|^s \ge 1$ for $|x_1-x_2| \ge 1$ we obtain due to the boundedness of $\mathscr {B} \subseteq C^{\tilde{m},s} S^{m}_{0,0}( \mathbb {R}^n \times \mathbb {R}^n\times \mathbb {R}^n; N)$

$$\begin{aligned} \frac{ | \partial _{x}^{\alpha } \partial _y^{\gamma } \partial _{\eta }^{\beta } a(x, \xi + \eta , x_1 + y) - \partial _{x}^{\alpha } \partial _y^{\gamma } \partial _{\eta }^{\beta } a(x, \xi + \eta , x_2 + y) | }{|x_1 - x_2|^s} \le C_{\beta , \gamma } \langle \xi + \eta \rangle ^m\nonumber \\ \end{aligned}$$

(33)

for all $a \in \mathscr {B}$ and $x,\xi , \eta ,x_1, x_2,y \in \mathbb {R}^{n}$, $x_1 \ne x_2$. Finally, the proposition follows from (32) and (33) by means of the triangle inequality. $\square$

Lemma 4.12

Let $N \in \mathbb {N}_0 \cup \{ \infty \}$ with $N > n$. Moreover, we define $\tilde{N}:= N-(n+1)$. For $a \in C^{ \tilde{m} ,s} S^m_{\rho ,0}(\mathbb {R}^n \times \mathbb {R}^n\times \mathbb {R}^n; N)$ with $\tilde{m} \in \mathbb {N}_0$, $m \in \mathbb {R}$, $0 \le \rho \le 1$ and $0<s<1$, we define $a_L:\mathbb {R}^n \times \mathbb {R}^n\rightarrow \mathbb {C}$ as in Lemma 4.9. Then $\partial _x^{\delta } \partial _{\xi }^{\gamma } a_L \in C^{0}(\mathbb {R}^n \times \mathbb {R}^n)$ for every $\gamma , \delta \in \mathbb {N}_0^n$ with $|\delta | \le \tilde{m}$ and $|\gamma | \le \tilde{N}$.

Proof

Let $\alpha , \beta \in \mathbb {N}_0^n$ with $|\beta | \le \tilde{m}$ and $|\alpha | \le \tilde{N}$. On account of Remark 3.14 we know that $\partial _x^{\beta } \partial _{\xi }^{\alpha } a \in C^0(\mathbb {R}^n \times \mathbb {R}^n\times \mathbb {R}^n)$. With $a(x, \xi + \eta , x+y) \in \mathscr {A}_0^{m^+, N}(\mathbb {R}^{2n}_{(y,y') } \times \mathbb {R}^{2n}_{(\xi ,\eta )})$ at hand we are able to apply Theorem 2.11 and get together with Lemma 4.10 for all $x,\xi \in \mathbb {R}^{n}$:

(34)

In order to show the continuity of $\partial _x^{\beta } \partial _{\xi }^{\alpha } a_L$, we want to apply Corollary 2.13. To this end let $(x, \xi ) \in \mathbb {R}^n \times \mathbb {R}^n$ be arbitrary. Additionally let $(x ', \xi ') \in \mathbb {R}^n \times \mathbb {R}^n$ with $|x-x'|, |\xi - \xi '| < 1$. For every $\beta _1, \beta _2, \gamma , \delta \in \mathbb {N}_0^n$ with $\beta _1 + \beta _2 = \beta$ and $|\delta | \le N-|\alpha |$ an application of $a \in C^{ \tilde{m} ,s} S^m_{\rho ,0}(\mathbb {R}^n \times \mathbb {R}^n\times \mathbb {R}^n; N)$ provides

$$\begin{aligned}&\bigg |\partial _y^{\gamma } \partial _{\eta }^{\delta } \big (\partial _x^{\beta _1} \partial _{\xi }^{\alpha } \partial _y^{ \beta _2}a \big ) (x', \eta + \xi ', x'+y) \bigg | \\&\quad \le C_{\alpha , \beta , \gamma , \delta } \langle \eta + \xi ' \rangle ^{ m - \rho (| \alpha | + |\delta |)}\le C_{\alpha , \beta , \gamma , \delta } \langle \eta \rangle ^{m} \langle \xi ' \rangle ^{ |m| } \\&\quad \le C_{\alpha , \beta , \gamma , \delta } \langle \eta \rangle ^{m} \langle \xi '- \xi \rangle ^ { |m| } \langle \xi \rangle ^{|m|} \le C_{\alpha , \beta , \gamma , \delta } \langle \eta \rangle ^{m} \langle \xi \rangle ^{ |m| } . \end{aligned}$$

Here $C_{\alpha , \beta , \gamma , \delta }$ is independent of $x', \eta , \xi ', y \in \mathbb {R}^{n}$. This yields the boundedness of

$$\begin{aligned} \bigg \{ \big (\partial _x^{\beta _1} \partial _{\xi }^{\alpha } \partial _y^{\beta _2}a \big ) (x', \eta + \xi ', x'+ y) : x', \xi ' \in \mathbb {R}^{n}\text { with } |x-x'|, |\xi - \xi '| < 1 \bigg \} \end{aligned}$$

in $\mathscr {A}_0^{m, N-|\alpha |}(\mathbb {R}^n_{y} \times \mathbb {R}^n_{\eta })$. Moreover, we obtain for all $y, \eta \in \mathbb {R}^{n}$ and for each $\beta _1, \beta _2, \gamma , \delta \in \mathbb {N}_0^n$ with $\beta _1 + \beta _2 = \beta$ and $|\delta | \le N-|\alpha |$:

$$\begin{aligned} \partial _y^{\gamma } \partial _{\eta }^{\delta } \big (\partial _x^{\beta _1} \partial _{\xi }^{\alpha } \partial _y^{\beta _2}a \big ) (x', \eta + \xi ', x'+y) \xrightarrow [x' \rightarrow x]{\xi ' \rightarrow \xi } \partial _y^{\gamma } \partial _{\eta }^{\delta } \big (\partial _x^{\beta _1} \partial _{\xi }^{\alpha } \partial _y^{\beta _2}a \big ) (x, \eta + \xi , x+y) \end{aligned}$$

due to $a \in C^{ \tilde{m} ,s} S^m_{\rho ,0}(\mathbb {R}^n \times \mathbb {R}^n\times \mathbb {R}^n; N)$. Using Leibniz’s rule and Corollary 2.13 yields

Hence $\partial _x^{\beta } \partial _{\xi }^{\alpha } a_L$ is continuous. $\square$

Now we are in the position to show that $a_L$ is a non-smooth symbol. Unfortunately we loose some smoothness with respect to $\xi$ of the double symbol:

Theorem 4.13

Let $0< s < 1$, $\tilde{m} \in \mathbb {N}_0$ and $m \in \mathbb {R}$. Additionally we choose $N \in \mathbb {N}_0 \cup \{ \infty \}$ with $N > n$. We define $\tilde{N}:= N-(n+1)$. Furthermore, let $\mathscr {B} \subseteq C^{\tilde{m}, s}_*S^{m }_{0,0}(\mathbb {R}^n \times \mathbb {R}^n\times \mathbb {R}^n; N)$ be bounded. If we define for each $a \in \mathscr {B}$ the function $a_L: \mathbb {R}^n \times \mathbb {R}^n\rightarrow \mathbb {C}$ as in Lemma 4.9, we get $a_L \in C^{\tilde{m},s}S^{m}_{0,0}(\mathbb {R}^n \times \mathbb {R}^n; \tilde{N})$ for all $a \in \mathscr {B}$ and the existence of a constant $C_{\beta }$, independent of $a \in \mathscr {B}$, such that

$$\begin{aligned} \bigg \Vert \partial _{\xi }^{\beta } a_L(.,\xi ) \bigg \Vert _{ C^{\tilde{m}, s} (\mathbb {R}^{n}) } \le C_{\beta } \langle \xi \rangle ^m \quad \text {for all } \xi \in \mathbb {R}^{n}\text { and } \beta \in \mathbb {N}_0^n\text { with } |\beta | \le \tilde{N}. \end{aligned}$$

This implies the boundedness of $\{ a_L: a \in \mathscr {B} \} \subseteq C^{\tilde{m},s} S^{m}_{0,0}(\mathbb {R}^n \times \mathbb {R}^n; \tilde{N})$.

Proof

Due to Lemma 4.12 we have $\partial _x^{\delta } \partial _{\xi }^{\gamma } a_L \in C^{0}(\mathbb {R}^n \times \mathbb {R}^n)$ for every $\gamma , \delta \in \mathbb {N}_0^n$ with $|\gamma | \le \tilde{N}$ and $|\delta | \le \tilde{m}$. Since $a(x, \xi + \eta , x+y)$ is an element of $\mathscr {A}_0^{m^+, N}(\mathbb {R}^{2n}_{(y,y')} \times \mathbb {R}^{2n}_{(\xi ,\eta )})$ and $N-|\alpha | > n$, we derive from Theorem 2.11 for each $\alpha \in \mathbb {N}_0^n$ with $|\alpha | \le \tilde{N}$:

Let $\alpha \in \mathbb {N}_0^n$ with $|\alpha | \le \tilde{N}$. Remark 3.14 and the boundedness of $\mathscr {B}$ implies the boundedness of

$$\begin{aligned} \mathscr {\tilde{B}} := \left\{ \partial _{\xi }^{\alpha } a : a \in \mathscr {B} \right\} \subseteq C^{\tilde{m},s} S^{m}_{0, 0}(\mathbb {R}^n \times \mathbb {R}^n\times \mathbb {R}^{n}, N-|\alpha |). \end{aligned}$$

Hence it remains to show

$$\begin{aligned} \Vert a_L(.,\xi ) \Vert _{C^{\tilde{m},s}(\mathbb {R}^{n})} \le C \langle \xi \rangle ^m \quad \text {for all } \xi \in \mathbb {R}^{n}, a \in \mathscr {\tilde{B}}. \end{aligned}$$

(35)

Inequality (35) implies $\Vert \partial _{\xi }^{\alpha } a_L(.,\xi ) \Vert _{C^{\tilde{m},s}(\mathbb {R}^{n})} \le C_{\alpha } \langle \xi \rangle ^m$ for all $\xi \in \mathbb {R}^{n}$ and $a \in \mathscr {B}$. This yields the boundedness of $\{ a_L : a \in \mathscr {B} \} \subseteq C^{\tilde{m},s} S^m_{0,0}( \mathbb {R}^n \times \mathbb {R}^n\times \mathbb {R}^n; \tilde{N} )$. Now we choose $l \in \mathbb {N}_0$ with $-2l+|m|<-n$ and $l_0:= N-\tilde{N}$. An application of Lemma 4.10 and Theorem 2.10 provides for every $\delta \in \mathbb {N}_0^n$ with $|\delta | \le \tilde{m}$:

(36)

On account Proposition 4.11 and $\left| \partial _y^{\alpha _1} \frac{y_j}{\langle y \rangle } \right| \le 1$ for all $j \in \{ 1, \ldots , n\}$ and $\alpha _1 \in \mathbb {N}_0^n$ we obtain similary to the proof of Remark 2.9 for $\delta \in \mathbb {N}_0^n$ with $|\delta | \le \tilde{m}$:

$$\begin{aligned}&\left| \langle \eta \rangle ^{-2l} \langle D_y \rangle ^{2l} A^{l_0}(D_{\eta }, y) \left\{ \frac{ \partial _{x_1}^{\delta } a(x_1, \xi + \eta , x_1 + y) - \partial _{x_2}^{\delta } a(x_ 2, \xi + \eta , x_2 + y) }{(x_1-x_2)^s} \right\} \right| \\&\qquad \le C \langle y \rangle ^{-l_0} \langle \xi \rangle ^m \langle \eta \rangle ^{-2l+|m|} \end{aligned}$$

for all $x_1, x_2, y, \xi , \eta \in \mathbb {R}^{n}$ with $x_1 \ne x_2$ and all $a \in \mathscr {\tilde{B}}$. Consequently we have for each $\delta \in \mathbb {N}_0^n$ with $|\delta | \le \tilde{m}$:

for all $x_1,x_2, \xi \in \mathbb {R}^{n}$ with $x_1 \ne x_2$ and $a \in \mathscr { \tilde{B}}$. Finally, we only have to use the previous inequality and Lemma 4.9 to get (35). $\square$

We still need to show $a_L(x,D_x)=a(x,D_x,x')$. For this we need:

Proposition 4.14

Let $\tilde{m} \in \mathbb {N}_0$, $m \in \mathbb {R}$, $0<s<1$ and $\chi \in \mathcal {S}(\mathbb {R}^n \times \mathbb {R}^n)$ with $\chi (0,0)=1$. Additionally let $N \in \mathbb {N}_0 \cup \{ \infty \}$ with $N > n$. Moreover, we choose $l,l_0,l_0' \in \mathbb {N}_0$ with

$$\begin{aligned} -2l + m< -n, \quad -2l_0<-n, \; -2l_0' + 2l <-n. \end{aligned}$$

Assuming $0< \varepsilon ' < 1$, $a \in C^{ \tilde{m} ,s} S^m_{0,0}(\mathbb {R}^n \times \mathbb {R}^n\times \mathbb {R}^n; N)$ and $u \in \mathcal {S}(\mathbb {R}^n)$ we define for every $0< \varepsilon < 1$ the functions $a_0, \hat{a}, a_{\varepsilon }, \tilde{a}_{0}: \mathbb {R}^{5n} \rightarrow \mathbb {C}$ by

$$\begin{aligned} a_0 (x,x', x'', \xi , \xi ')&:= \chi (\varepsilon ' x'', \varepsilon ' \xi ') a(x,\xi , x') u(x'')\\ \hat{a}(x,x', x'', \xi , \xi ')&:= \langle x'-x'' \rangle ^{-2l_0} \langle D_{\xi '} \rangle ^{2l_0} \left[ \langle \xi ' \rangle ^{-2l_0'} \langle D_{x''} \rangle ^{2l_0'} a_0 (x,x', x'', \xi , \xi ') \right] , \\ a_{\varepsilon }(x,x', x'', \xi , \xi ')&:= \chi (\varepsilon x', \varepsilon \xi ) \hat{a}(x,x', x'', \xi , \xi '), \\ \tilde{a}_0 (x,x', x'', \xi , \xi ')&:= \langle - \xi + \xi ' \rangle ^{-2l} \langle D_ {x'} \rangle ^{2l} \hat{a} (x,x', x'', \xi , \xi '). \end{aligned}$$

for all $x,x', x'', \xi , \xi ' \in \mathbb {R}^{n}$. Then

Proof

First of all we define for each $0< \varepsilon < 1$ the function $\tilde{a}_{\varepsilon }: \mathbb {R}^{5n} \rightarrow \mathbb {C}$ by

$$\begin{aligned} \tilde{a}_{\varepsilon } (x,x', x'', \xi , \xi ')&:= \langle - \xi + \xi ' \rangle ^{-2l} \langle D_{x'} \rangle ^{2l} a_{\varepsilon }(x,x', x'', \xi , \xi ') \end{aligned}$$

for all $x,x', x'', \xi , \xi ' \in \mathbb {R}^{n}$. By means of the Leibniz rule, $a \in C^{ \tilde{m} ,s} S^m_{0,0}(\mathbb {R}^n \times \mathbb {R}^n\times \mathbb {R}^n; N)$, $u \in \mathcal {S}(\mathbb {R}^n)$ and $\chi \in \mathcal {S}(\mathbb {R}^n \times \mathbb {R}^n)$ one can show for a fixed $x \in \mathbb {R}^{n}$ and for arbitrary $M, M_1, M _2 \in \mathbb {N}_0$ with $-2l_0 - M_1 < -n$ and $m-M_2 < -n$:

$$\begin{aligned} \left| a_{\varepsilon } (x,x', x'', \xi , \xi ') \right|&\le C |\chi (\varepsilon x', \varepsilon \xi )| \langle x'-x'' \rangle ^{-2l_0} \langle \xi ' \rangle ^{-2l_0'} \langle \xi \rangle ^{m}\langle x'' \rangle ^{-M} \nonumber \\&\le C_{\varepsilon } \langle x' \rangle ^{-2l_0-M_1} \langle \xi \rangle ^{m-M_2} \langle x'' \rangle ^{2l_0-M} \langle \xi ' \rangle ^{-2l_0'}, \end{aligned}$$

(37)

$$\begin{aligned} \left| \tilde{a}_{\varepsilon } (x,x', x'', \xi , \xi ') \right|&\le C \langle -\xi + \xi ' \rangle ^{-2l} \langle x'-x'' \rangle ^{-2l_0} \langle \xi ' \rangle ^{-2l_0'} \langle \xi \rangle ^{m} \langle x'' \rangle ^{ -M} \nonumber \\&\le C \langle x' \rangle ^{-2l_0} \langle x'' \rangle ^{2l_0-M} \langle \xi ' \rangle ^{-2l_0' + 2l} \langle \xi \rangle ^{-2l + m}, \end{aligned}$$

(38)

$$\begin{aligned} \left| \tilde{a}_{0} (x,x', x'', \xi , \xi ') \right|&\le C \langle -\xi + \xi ' \rangle ^{-2l} \langle x'-x'' \rangle ^{-2l_0} \langle \xi ' \rangle ^{-2l_0'} \langle \xi \rangle ^{m} \langle x'' \rangle ^{- M} \nonumber \\&\le C \langle x' \rangle ^{-2l_0} \langle x'' \rangle ^{2l_0-M} \langle \xi ' \rangle ^{-2l_0' + 2l} \langle \xi \rangle ^{-2l + m}, \end{aligned}$$

(39)

where C is independent of $x,x', x'', \xi , \xi ' \in \mathbb {R}^{n}$ and of $0< \varepsilon < 1$. Now we choose $M \in \mathbb {N}$ with $2l_0-M <-n$. Then we have $a_{\varepsilon } (x,x', x'', \xi , \xi ') \in L^1(\mathbb {R}^n_{x''} \times \mathbb {R}^n_{\xi '} \times \mathbb {R}^n_{ x'} \times \mathbb {R}^n_{\xi })$ and $\tilde{a}_{\varepsilon } (x,x', x'', \xi , \xi ') \in L^1(\mathbb {R}^n_{x' } \times \mathbb {R}^n_{\xi })$ for every fixed $x, x'', \xi ' \in \mathbb {R}^{n}$ and $0< \varepsilon < 1$. Hence we are able to use Fubini’s theorem first and integrate by parts with respect to $x'$ and $\xi$ afterwards and get

(40)

Because of (38) we have for every $x \in \mathbb {R}^{n}$

$$\begin{aligned}&|e^{-ix'' \cdot \xi '-ix'\cdot \xi + ix'\cdot \xi ' + ix\cdot \xi } \tilde{a}_{\varepsilon } (x,x', x'', \xi , \xi ')| \nonumber \\&\qquad \le C_x \langle x' \rangle ^{-2l_0} \langle x'' \rangle ^{2l_ 0-M} \langle \xi ' \rangle ^{-2l_0' + 2l} \langle \xi \rangle ^{-2l + m} \in L^1\big (\mathbb {R}^n_{x'} \times \mathbb {R}^n_{\xi } \times \mathbb {R}^n_{x''} \times \mathbb {R}^n_{\xi '} \big ), \end{aligned}$$

(41)

where $C_x$ is independent of $x', x'', \xi , \xi ' \in \mathbb {R}^{n}$ and of $0<\varepsilon < 1$. Making use of Fubini’s theorem in (40) provides:

(42)

It remains to calculate the limit $\varepsilon \rightarrow 0$. Due to (39 ) we know that the function $e^{-ix'' \cdot \xi '} e^{-ix'\cdot \xi + ix'\cdot \xi ' + ix\cdot \xi } \tilde{a}_0 (x,x', x'', \xi , \xi ')$ is an element of $L^1(\mathbb {R}^n_{x'} \times \mathbb {R}^n_{\xi } \times \mathbb {R}^n_{x''} \times \mathbb {R}^n_{\xi '} )$. Using the definition of $\langle D_{x'} \rangle ^{2l}$ and the Leibniz rule, one easily obtains by the pointwise convergence of $\chi _{\varepsilon }$ to 1 and the pointwise convergence of every derivative of $\chi _{\varepsilon }$ to 0, see e.g. [11, Lemma 6.3]:

$$\begin{aligned}&e^{-ix'' \cdot \xi '} e^{-ix'\cdot \xi + ix'\cdot \xi ' + ix\cdot \xi } \tilde{a}_{\varepsilon } (x,x', x'', \xi , \xi ') \\&\quad \rightarrow e^{-ix'' \cdot \xi '} e^{-ix'\cdot \xi + ix'\cdot \xi ' + ix\cdot \xi } \tilde{a}_0 (x,x', x'', \xi , \xi ') \end{aligned}$$

for all $x,x', x'', \xi , \xi ' \in \mathbb {R}^{n}$ if $\varepsilon \rightarrow 0$. We conclude the claim by applying Lebesgue’s theorem to (42) which is possible due to the previous convergence and (41). $\square$

Combining the previous results we obtain:

Theorem 4.15

Let $N \in \mathbb {N}_0 \cup \{ \infty \}$ with $N > n$. We define $\tilde{N}:= N-(n+1)$. Assuming an $a \in C^{ \tilde{m} ,s} S^m_{0,0}(\mathbb {R}^n \times \mathbb {R}^n\times \mathbb {R}^n; N)$ with $\tilde{m} \in \mathbb {N}_0$, $m \in \mathbb {R}$ and $0<s<1$, we define $a_L:\mathbb {R}^n \times \mathbb {R}^n\rightarrow \mathbb {C}$ by

for all $x, \xi \in \mathbb {R}^{n}$. Then we have for every $u \in \mathcal {S}(\mathbb {R}^n)$

$$\begin{aligned} a(x,D_x, x') u = a_L(x,D_x) u. \end{aligned}$$

Proof

We already know that $a_L \in C^{\tilde{m}, s} S^m_{0,0} (\mathbb {R}^n \times \mathbb {R}^n; \tilde{N})$ due to Theorem 4.13. Now we choose $u \in \mathcal {S}(\mathbb {R}^n)$ and $l,l_0,l_0' \in \mathbb {N}_0^n$ with the property

$$\begin{aligned} -2l+ m< -n, \quad&\quad -2l_0<-n, \quad&\quad&\quad -2l_0' + 2l <-n. \end{aligned}$$

(43)

On account of $a_L(x,\xi ')u(x'') \in \mathscr {A}^{m, \tilde{N}}_{-k}(\mathbb {R}^n_{x''} \times \mathbb {R}^n_{\xi '})$ for all $k \in \mathbb {N}_0$, Theorem 2.10 yields the existence of $a_L(x,D_x)u$. Because of $a(x, \eta + \xi ', x+y) \in \mathscr {A}^{m, N}_{0}(\mathbb {R}^n_{y} \times \mathbb {R}^n_{\eta })$ for every fixed $x, \xi ' \in \mathbb {R}^{n}$ we can apply Theorem 2.14 and get

(44)

where $\chi \in \mathcal {S}(\mathbb {R}^n \times \mathbb {R}^n)$ with $\chi (0,0)=1$. Integration by parts yields for arbitrary $0< \varepsilon < 1$ and $k, k' \in \mathbb {N}_0$ with $-N \le -k<-n$ and $-2k'+m < -n$ on account of Remark 2.8:

(45)

where

$$\begin{aligned}&b_{\varepsilon }(x,\xi ,x', \xi ',x'') \\&\quad := A^k(D_ {\xi }, x'-x) \left[ \langle \xi - \xi ' \rangle ^{-2k'} \langle D_{x'} \rangle ^{2k '} \chi (\varepsilon x', \varepsilon \xi ) a(x,\xi , x') u(x'') \right] \end{aligned}$$

for all $x, \xi , x', \xi ', x'' \in \mathbb {R}^{n}$ and each $0 \le \varepsilon < 1$. We choose $M_1, M_2 \in \mathbb {N}$ with $-M_2 < -2n$ and $-M_1 + M_2 < -n$. Using Leibniz’s rule and Petree’s inequality we obtain for arbitrary but fixed $x, \xi ', x'' \in \mathbb {R}^{n}$ if we use Remark 2.9:

$$\begin{aligned} |b_{\varepsilon }(x,\xi ,x', \xi ',x'')|&\le C \langle x'-x \rangle ^{-k} \langle \xi - \xi ' \rangle ^{-2k'} \langle \xi \rangle ^m \langle x'' \rangle ^{-M_1 } \nonumber \\&\le C_x \langle x' \rangle ^{-k} \langle \xi ' \rangle ^{2k'} \langle \xi \rangle ^{m-2k'} \langle x'' \rangle ^{-M_1} \in L^1 \big (\mathbb {R}^n_{x'} \times \mathbb {R}^n_{\xi }\big ), \end{aligned}$$

(46)

where $C_x$ is independent of $0< \varepsilon < 1$ and $x', \xi , \xi ', x'' \in \mathbb {R}^{n}$. Since $b_{\varepsilon }(x,\xi ,x', \xi ',x'')$ converges to $b_{0}(x,\xi ,x', \xi ',x '')$ as $\varepsilon \rightarrow 0$ and (46) holds we are able to apply Lebesgue’s theorem to (45) and get

for every $x, x'', \xi ' \in \mathbb {R}^{n}$ and $0< \varepsilon ' < 1$. Additionally using (45), (46) and $\chi \in \mathcal {S}(\mathbb {R}^n \times \mathbb {R}^n)$ yields for fixed but arbitrary $0< \varepsilon ' <1$:

Applying Lebesgue’s theorem we obtain because of (44):

(47)

Now we define

$$\begin{aligned} \tilde{a}_{\varepsilon '}(x,x', x'', \xi , \xi ')&:= \chi (\varepsilon ' x'', \varepsilon ' \xi ') a (x,\xi , x') u(x'') , \\ a_{\varepsilon '}(x,x', x'', \xi , \xi ')&:= \langle x'-x'' \rangle ^{-2l_0} \langle D_{\xi '} \rangle ^{2l_0} \left[ \langle \xi ' \rangle ^{-2l_0'} \langle D_{x''} \rangle ^{2l_0'} \tilde{a}_{\varepsilon '}(x,x ', x'', \xi , \xi ') \right] , \\ \hat{a}_{\varepsilon '}(x,x', x'', \xi , \xi ')&:= \langle -\xi + \xi ' \rangle ^{-2l} \langle D_ {x'} \rangle ^{2l} a_{\varepsilon '}(x,x', x'', \xi , \xi ') \end{aligned}$$

for all $x,x', x'', \xi , \xi ' \in \mathbb {R}^{n}$ and $0< \varepsilon < 1$. Integrating by parts in (47) provides:

(48)

due to Proposition 4.14. We define

$$\begin{aligned} \hat{a}(x,x', x'', \xi , \xi ')&:= \langle -\xi + \xi ' \rangle ^{-2l} \langle D_{x'} \rangle ^{2l} a_0(x,x', x'', \xi , \xi '), \\ a_0 (x,x', x'', \xi , \xi ')&:= \langle x'-x'' \rangle ^{-2l_0} \langle D_{\xi '} \rangle ^{2 l_0} \left[ \langle \xi ' \rangle ^{-2l_0'} \langle D_{x''} \rangle ^{2l_0'} a(x,\xi , x') u(x'') \right] \end{aligned}$$

for all $x,x', x'', \xi , \xi ' \in \mathbb {R}^{n}$. Then

$$\begin{aligned} \hat{a}_{\varepsilon '}(x,x', x'', \xi , \xi ') \xrightarrow []{ \varepsilon ' \rightarrow 0 } \hat{a}(x,x', x'', \xi , \xi ') \end{aligned}$$

for all $x,x', x'', \xi , \xi ' \in \mathbb {R}^{n}$. Similarly to (46) we get due to Leibniz’s rule and Petree’s inequality:

$$\begin{aligned} |\hat{a}_{\varepsilon '}(x,x', x'', \xi , \xi ')|&\le C \langle x' \rangle ^{ - 2l_0} \langle x'' \rangle ^{2l_0-M} \langle \xi \rangle ^{m-2l} \langle \xi ' \rangle ^{2l- 2l_0'} \\&\quad \in L^1(\mathbb {R}^{n}_{x'}\times \mathbb {R}^{n}_{\xi } \times \mathbb {R}^{n}_{x''} \times \mathbb {R}^{n}_{\xi ' }). \end{aligned}$$

Here the constant C is independent of $0< \varepsilon ' <1$ and $x, x', x'', \xi , \xi ' \in \mathbb {R}^{n}$. An application of Lebesgue’s theorem to (48) provides:

Hence we get the claim by using Lemma 4.4. $\square$

4.3 Properties of the Operator $T_{\varepsilon }$

Besides the results of Subsection 4.1 and Subsection 4.2 we need an approximation $( T_{\varepsilon } )_{\varepsilon \in (0,1]}$ for a given operator $T \in \mathcal {A}^{0, M}_{0,0} (\tilde{m},q)$ such that

• $T_{\varepsilon }: \mathcal {S'}(\mathbb {R}^n)\rightarrow \mathcal {S}(\mathbb {R}^n)$ is continuous,

• The iterated commutators of $T_{\varepsilon }$ are uniformly bounded with respect to $\varepsilon$ as maps from $L^q(\mathbb {R}^{n})$ to $L^q(\mathbb {R}^{n})$,

• $T_{\varepsilon }$ converges pointwise to T if $\varepsilon \rightarrow 0$.

Throughout this subsection we assume: Let $1< q < \infty$ and $M \in \mathbb {N}_ 0 \cup \{ \infty \}$ be arbitrary. Moreover, let $T \in \mathcal {A}^{0, M }_{0,0}(\tilde{m},q)$ with $\tilde{m} \in \mathbb {N}_0$ and $\varphi \in C^{\infty }_c(\mathbb {R}^n)$ with $\varphi (x)=1$ for all $|x|\le \frac{1}{2}$ and $\varphi (x)=0$ for all $|x|\ge 1$. Then we define for $0< \varepsilon \le 1$

$$\begin{aligned} P_{\varepsilon } := \tilde{p}_{\varepsilon }(x,D_x) \quad \text { and } \; Q_{\varepsilon } : = q_{\varepsilon }(x,D_x), \end{aligned}$$

where the symbols $\tilde{p}_{\varepsilon }$ and $q_{\varepsilon }$ are defined as $\tilde{p}_{\varepsilon }(x,\xi ):= \varphi (\varepsilon x)$ and $q_{\varepsilon }(x,\xi ):=\varphi (\varepsilon \xi )$. Then $\{\tilde{p}_ {\varepsilon }| 0 < \varepsilon \le 1 \}$ and $\{q_{\varepsilon }|0 < \varepsilon \le 1 \}$ are bounded subsets of $S^0_{1, 0}(\mathbb {R}^n \times \mathbb {R}^n)$. Note that for $u \in \mathcal {S}(\mathbb {R}^n)$ we have $P_{\varepsilon } u= \tilde{p}_{\varepsilon } u$. Additionally the continuity of multiplication operators with $C^{\infty }_c$-functions imply the continuity of $P_{\varepsilon } : C^{\infty }(\mathbb {R}^{n}) \rightarrow C^{\infty }_c(\mathbb {R}^{n})$. Moreover, we define

$$\begin{aligned} T_{\varepsilon } := P_{\varepsilon } Q_{\varepsilon } T P_{\varepsilon } Q_{\varepsilon }. \end{aligned}$$

Since $P_{\varepsilon }Q_{\varepsilon }$ is continuous as a map from $\mathcal {S}(\mathbb {R}^n)'$ to $\mathcal {S}(\mathbb {R}^n)$, $T_{\varepsilon }$ is continuous as a map from $\mathcal {S}(\mathbb {R}^n)'$ to $\mathcal {S}(\mathbb {R}^n)$, too. For the following we need:

Lemma 4.16

For all $u \in L^q(\mathbb {R}^{n})$ we have the following convergence:

$$\begin{aligned} L^q- \lim _{\varepsilon \rightarrow 0} T_{\varepsilon }u = Tu. \end{aligned}$$

Proof

With the theorem of Banach-Steinhaus at hand, we easily can show

$$\begin{aligned} Q_{\varepsilon }u \xrightarrow []{\varepsilon \rightarrow 0} u \quad \text {and} \; P _{\varepsilon }u \xrightarrow []{\varepsilon \rightarrow 0} u \; \text {in } L^q(\mathbb {R}^{n}). \end{aligned}$$

(49)

For more details, see [17, Proof of Lemma 5.27]. By means of (49) and Theorem 3.6 we get for all $u \in L^q(\mathbb {R}^{n})$:

$$\begin{aligned} \Vert P_{\varepsilon } Q_{\varepsilon }u - u \Vert _{L^q(\mathbb {R}^{n})} \le C \Vert Q_{\varepsilon }u - u \Vert _{L^q(\mathbb {R}^{n})} + \Vert P_{\varepsilon }u - u \Vert _{L^q(\mathbb {R}^{n})} \xrightarrow []{\varepsilon \rightarrow 0} 0. \end{aligned}$$

An application of Theorem 3.6 gives us for all $u \in L^q(\mathbb {R}^{n})$:

$$\begin{aligned} \Vert T_{\varepsilon } u - T u \Vert _{L^q(\mathbb {R}^{n})}&\le \Vert P_{\varepsilon } Q_{\varepsilon } T P_{\varepsilon } Q_{\varepsilon } u - P_{\varepsilon } Q_{\varepsilon } T u \Vert _{L^q( \mathbb {R}^{n})} + \Vert P_{\varepsilon } Q_{\varepsilon } T u - T u \Vert _{L^q(\mathbb {R}^{n})} \\&\le C \Vert P_{\varepsilon } Q_{\varepsilon } u - u \Vert _{L^q(\mathbb {R}^{n})} + \Vert P_{\varepsilon } Q_{\varepsilon } T u - T u \Vert _{L^q(\mathbb {R}^{n})} \xrightarrow []{\varepsilon \rightarrow 0} 0. \end{aligned}$$

$\square$

Lemma 4.17

Let $\alpha , \beta \in \mathbb {N}_0^n$ with $|\beta | \le \tilde{m}$ and $|\alpha | \le M$. Then

$$\begin{aligned} \Vert {{\mathrm{ad}}}(-ix)^{\alpha } {{\mathrm{ad}}}(D_x)^{\beta } T_{\varepsilon } \Vert _{ \mathscr {L}(L^q(\mathbb {R}^{n})) } \le C_{\alpha , \beta } \quad \text {for all } 0 < \varepsilon \le 1. \end{aligned}$$

Proof

Let $\alpha , \beta \in \mathbb {N}_0^n$ with $|\beta | \le \tilde{m}$ and $|\alpha | \le M$. We define

$$\begin{aligned} R^{\beta _1, \beta _2, \beta _3}_{\alpha _1, \alpha _2, \alpha _3}:= \left[ {{\mathrm{ad}}}(D_{x})^{\beta _1} P_{\varepsilon } \right] \left[ {{\mathrm{ad}}}(-ix)^{\alpha _1} Q_{\varepsilon } \right] T^{\alpha _2,\beta _2} \left[ {{\mathrm{ad}}}(D_{x})^{\beta _3} P_{\varepsilon } \right] \left[ {{\mathrm{ad}}}(-ix)^{\alpha _3} Q_{\varepsilon } \right] , \end{aligned}$$

where $T^{\alpha _2,\beta _2}:= {{\mathrm{ad}}}(-ix)^{\alpha _2}{{\mathrm{ad}}}(D_{x})^{\beta _2} T$. Then we obtain for all $u \in \mathcal {S}(\mathbb {R}^n)$

$$\begin{aligned} {{\mathrm{ad}}}(-ix)^{\alpha } {{\mathrm{ad}}}(D_x)^{\beta } T_{\varepsilon } u = \sum _{ \begin{array}{c} \alpha _1 + \alpha _2 + \alpha _3 = \alpha \\ \beta _1 + \beta _2 + \beta _3 = \beta \end{array} } C_{\alpha _1, \alpha _2, \beta _1, \beta _2} R^{\beta _1, \beta _2, \beta _3}_{\alpha _1, \alpha _2, \alpha _3} u. \end{aligned}$$

Due to Remark 2.5 we get ${{\mathrm{ad}}}(D_{x})^{ \gamma } P_{\varepsilon } \in OPS^0_{1,0}$ and ${{\mathrm{ad}}}(-ix)^{\delta } Q_{\varepsilon } \in OPS ^{-|\delta |}_{1,0} \subseteq OPS^0_{1,0}$ for each $\gamma , \delta \in \mathbb {N}_0^n$. On account of Theorem 3.6, the boundedness of $\{\tilde{p}_{\varepsilon }| 0 < \varepsilon \le 1 \}$ and $\{q _{\varepsilon }|0 < \varepsilon \le 1 \}$ in $S^0_{1, 0}(\mathbb {R}^n \times \mathbb {R}^n)$ and of $T \in \mathcal {A}^{0,M}_{0,0}(\tilde{m},q)$ we obtain

$$\begin{aligned} \big \Vert {{\mathrm{ad}}}(-ix)^{\alpha } {{\mathrm{ad}}}(D_x)^{\beta } T_{\varepsilon } u \big \Vert _{L^q} \le C_{\alpha , \beta , q} \Vert u \Vert _{L^q} \quad \text {for all } u \in \mathcal {S}(\mathbb {R}^n). \end{aligned}$$

$\square$

Proposition 4.18

Let $g \in \mathcal {S}(\mathbb {R}^n)$ and $0 < \varepsilon \le 1$. For each $y \in \mathbb {R}^{n}$ we define $g_y := \tau _y(g)$. Moreover, we define

$$\begin{aligned} p_{\varepsilon ,0} (x,\xi ,y):= e^{-ix \cdot \xi } T_{\varepsilon } ( e_{\xi } g_y)(x) \quad \text {for all } (x,\xi ,y) \in \mathbb {R}^n \times \mathbb {R}^n\times \mathbb {R}^n. \end{aligned}$$

Then $p_{\varepsilon ,0} \in C^{\infty }(\mathbb {R}^n \times \mathbb {R}^n\times \mathbb {R}^n)$.

In order to prove the previous proposition, we need:

Definition 4.19

For $k \in \mathbb {N}_0$ we define the normed space $L^q_k(\mathbb {R}^{n})$ as

$$\begin{aligned} L^q_k(\mathbb {R}^{n}):= \left\{ f\in L^q(\mathbb {R}^{n}) : \Vert f \Vert _{L^q_k} := \Vert \langle x \rangle ^ {k+1} f(x) \Vert _{L^q(\mathbb {R}^{n}_x)} < \infty \right\} . \end{aligned}$$

Sketch of the proof of Proposition 4.18. Let $k \in \mathbb {N}_0$ be arbitrary but fixed. For every $x,\xi \in \mathbb {R}^{n}$, $f \in C^{k+1}_b(\mathbb {R}^{n})$ and each $h \in L^q_k(\mathbb {R}^{n})$ we define $\delta _x(f):= f(x)$ and $M_{\xi }(h):= e_{\xi }h$. We define $\tilde{\delta }: \mathbb {R}^n \times \mathbb {R}^n\times \mathbb {R}^n\rightarrow \mathscr {L} (C^{k+1}_b(\mathbb {R}^{n}), \mathbb {C})$, $\tilde{G}: \mathbb {R}^n \times \mathbb {R}^n\times \mathbb {R}^n\rightarrow L^q_k(\mathbb {R}^{n})$ and $\tilde{M}: \mathbb {R}^n \times \mathbb {R}^n\times \mathbb {R}^n\rightarrow \mathscr {L} ( L^q_k(\mathbb {R}^{n}), L^q(\mathbb {R}^{n}) )$ by

$$\begin{aligned} \tilde{\delta }(x,y,\xi ):= \delta _{x}, \tilde{G} (x,y,\xi ):= g_ y, \tilde{M}(x,y,\xi ):= M_{\xi } \quad \text {for all } x,y,\xi \in \mathbb {R}^{n}. \end{aligned}$$

One can show that $\tilde{G}$ is a smooth function and that $\tilde{\delta }, \tilde{M}$ are k-times continuous differentiable, cf. [17, Proposition 5.33 and Proposition 5.34]. On account of the product rule we get

$$\begin{aligned} \tilde{M}(x,y,\xi ) \circ \tilde{G}(x,y,\xi ) \in C^k \big (\mathbb {R}^{n}_x \times \mathbb {R}^{n}_y \times \mathbb {R}^{n}_{\xi }, L^q(\mathbb {R}^{n}) \big ). \end{aligned}$$

(50)

Since $T_{\varepsilon }$ is continuous as map from $\mathcal {S'}(\mathbb {R}^n)$ to $\mathcal {S}(\mathbb {R}^n)$, we have $T_{\varepsilon } \in \mathscr {L} (L^q(\mathbb {R}^{n}), C_b^{k+1}(\mathbb {R}^{n}))$ and hence we obtain

$$\begin{aligned} T_{\varepsilon } (\tilde{M}(x,y,\xi ) \circ \tilde{G}(x,y,\xi )) \in C^k \big ( \mathbb {R}^{n}_x \times \mathbb {R}^{n}_y \times \mathbb {R}^{n}_{\xi }, C_b^{k+1}(\mathbb {R}^{n}) \big ) \end{aligned}$$

due to (50). Applying the product rule again yields

$$\begin{aligned} p_{\varepsilon ,0} (x,y,\xi )&=e^{-ix \cdot \xi } \tilde{\delta }(x,y,\xi ) \circ T_{\varepsilon } (\tilde{M}(x,y,\xi ) \circ \tilde{G}(x,y,\xi )) \\&\quad \in C^k \big ( \mathbb {R}^{n}_x \times \mathbb {R}^{n}_y \times \mathbb {R}^{n}_{\xi } \big ). \end{aligned}$$

$\square$

For more details, see [17, Proposition 5.31].

4.4 Characterization of Pseudodifferential Operators with Symbols in $C^s S^m_{0,0}$

In this subsection we will first prove the characterization of pseudodifferential operators with symbols of the symbol-class $C^s S^{0}_{0,0}(\mathbb {R}^{n}\times \mathbb {R}^{n}; M)$. Then the result is extended to non-smooth pseudodifferential operators of the class $C^s S^{m}_{0,0}(\mathbb {R}^{n}\times \mathbb {R}^{n}; M)$ of the order m. In the non-smooth case, one is confronted with the following problem: In general we do not have the continuity of non-smooth pseudodifferential operators with coefficients in a Hölder space as a map from $H^m_q(\mathbb {R}^{n})$ to $L^q(\mathbb {R}^{n})$. But every element of the set $\mathcal {A}^ {m,M}_{0,0}(\tilde{m},q)$ is a linear and bounded map from $H^m_q(\mathbb {R}^{n})$ t o $L^q(\mathbb {R}^{n})$. Hence this ansatz just provides a characterization of those non-smooth pseudodifferential operators which are linear and bounded as maps from $H^m_q(\mathbb {R}^{n})$ to $L^q(\mathbb {R}^{n})$. As already mentioned, the proof relies on the main idea of the proof in the smooth case by Ueberberg [22].

Theorem 4.20

Let $1<q<\infty$ and $m \in \mathbb {N}_0$ with $m>n/q$. Additionally let $M \in \mathbb {N}\cup \{ \infty \}$ with $M >n$. We define $\tilde{M}:= M-(n+1)$. Considering $T \in \mathcal {A}^{0,M}_{0,0}(m,q)$ and $\tilde{M}\ge 1$, we get for all $0< \tau \le m- n/q$ with $\tau \notin \mathbb {N}_0$

$$\begin{aligned} T\in OPC^{\tau }S^0_{0,0}(\mathbb {R}^n \times \mathbb {R}^n; \tilde{M}-1) \cap \mathscr {L}(L^q(\mathbb {R}^{n})). \end{aligned}$$

Proof

Let $\tau \in (0,m-n/q]$ with $\tau \notin \mathbb {N}$ be arbitrary but fixed. Let $T_{\varepsilon }$, $\varepsilon \in (0,1]$, be as in Sect. 4.3. Then $T_{\varepsilon } : \mathcal {S'}(\mathbb {R}^n)\rightarrow \mathcal {S}(\mathbb {R}^n)$ is continuous. The proof of this theorem is divided into three different parts. First we write $T_{\varepsilon }$ as a pseudodifferential operator with a double symbol. In step two we reduce the double symbol to an ordinary symbol $p_{\varepsilon }$ of $T_{\varepsilon }$. Finally, we conclude the proof in part three. Here we use the pointwise convergence of a subsequence of $( p_{\varepsilon } )_{\varepsilon > 0}$ to get a symbol p with the property $p(x,D_x)u = Tu$ for all $u \in \mathcal {S}(\mathbb {R}^n)$.

We begin with step one: Since $T_{\varepsilon } : \mathcal {S'}(\mathbb {R}^n)\rightarrow \mathcal {S}(\mathbb {R}^n)$ is continuous, Theorem 2.2 gives us the existence of a Schwartz-kernel $t_{\varepsilon } \in \mathcal {S}(\mathbb {R}^n \times \mathbb {R}^n)$ of $T_{\varepsilon }$. Thus

$$\begin{aligned} T_{\varepsilon } u(x) = \int t_{\varepsilon }(x,y) u(y) dy \quad \text {for all } u \in \mathcal {S}(\mathbb {R}^n)\text { and all } x\in \mathbb {R}^{n}. \end{aligned}$$

(51)

Now we choose $u,g \in \mathcal {S}(\mathbb {R}^n)$ with $g(0)=1$ and $g(-x) = g(x)$ for all $x\in \mathbb {R}$. We define $g_y: \mathbb {R}^{n}\rightarrow \mathbb {C}$ for $y \in \mathbb {R}^{n}$ by $g_y:= \tau _y(g)$. Next let $x \in \mathbb {R}^{n}$ be arbitrary, but fixed. Then we define

$$\begin{aligned} h(z) := u(z) g_z(x) \quad \text {for all } z \in \mathbb {R}^{n}. \end{aligned}$$

Using the inversion formula, cf. e.g. [2, Example 3.11], we obtain

If we first insert the previous equality in (51) and use the definition of the oscillatory integrals, integration by parts with respect to y and Lebesgues theorem afterwards, we get

where $\chi \in \mathcal {S}(\mathbb {R}^n \times \mathbb {R}^n)$ with $\chi (0,0)=1$. Defining $p_{\varepsilon ,0}: \mathbb {R}^n \times \mathbb {R}^n\times \mathbb {R}^n\rightarrow \mathbb {C}$ by

$$\begin{aligned} p_{\varepsilon ,0}(x,\xi ,y) := e^{-ix\cdot \xi } T_{\varepsilon }( e_{\xi } g_y)(x) \quad \text { for all } x, \xi , y \in \mathbb {R}^{n}, \end{aligned}$$

we conclude

Here $p_{\varepsilon ,0}$ is the double symbol of $T_{\varepsilon }$, cf. Lemma 3.13, as we will see in step two.

Secondly we want to construct for all $0< \varepsilon \le 1$ symbols $p_{\varepsilon }\ \in C^{\tau } S^{0}_{0,0}(\mathbb {R}^{n}\times \mathbb {R}^{n}; \tilde{M})$, with

• (i) $T_{\varepsilon } u = p_{\varepsilon }(x,D_x) u$ for all $u \in \mathcal {S}(\mathbb {R}^n)$,

• (ii) $( p_{\varepsilon })_{ 0< \varepsilon \le 1 }$ is a bounded sequence of $C^ {\tau } S^{0}_{0,0}(\mathbb {R}^{n}\times \mathbb {R}^{n}; \tilde{M})$.

Since $T_{\varepsilon }: \mathcal {S'}(\mathbb {R}^n)\rightarrow \mathcal {S}(\mathbb {R}^n)$ is linear and continuous and because of Proposition 4.18, we can apply Lemmas 2.6 and 4.17 and get for $\alpha , \gamma \in \mathbb {N}_0^n$ with $|\alpha | \le M$:

$$\begin{aligned} \left\| \partial _{\xi }^{\alpha } D_y^{\gamma } p_{\varepsilon ,0}(.,\xi ,y) \right\| ^q_{ C^{ \tau } }&\le \left\| \partial _{\xi }^{\alpha } D_y^{\gamma } p_{\varepsilon ,0}(.,\xi ,y) \right\| ^ q_{ H^m_q } \\&\le \sum _{ |\beta | \le m } \left\| \partial _{\xi }^{\alpha } D_x^{\beta } D_y^{\gamma } p_{\varepsilon ,0}(x,\xi ,y) \right\| ^q_{ L^q(\mathbb {R}^{n}_x) } \\&\le \sum _{ |\beta | \le m } \sum _{ \beta _1 + \beta _2 =\beta } \left\| C_{\beta _1,\beta _2} \left[ {{\mathrm{ad}}}(-ix)^{\alpha } {{\mathrm{ad}}}(D_x)^{\beta _ 1} T_{\varepsilon } \right] \right. \\&\quad \left. \left( e^{ix \cdot \xi } D_x^{\beta _2 + \gamma } g_y \right) (x) \right\| ^q_{ L^q(\mathbb {R}^{n}_x) }\\&\le C_{\alpha ,m,\gamma } \end{aligned}$$

for all $\xi , y \in \mathbb {R}^{n}$ and $0< \varepsilon \le 1$. Hence $\{ p_{\varepsilon ,0} : 0< \varepsilon \le 1\}$ is a bounded subset of $C^{\tau } S^{0}_{0,0}(\mathbb {R}^n \times \mathbb {R}^n\times \mathbb {R}^{n};M)$. Now we define

An application of Theorems 4.15 and 4.13 yields the properties i) and ii). So we can turn to step three now.

On account of ii) it is possible to apply Lemma 4.3 which yields the existence of a subsequence $( p_{\varepsilon _k} )_{k \in \mathbb {N}}$ of $( p_{\varepsilon })_{0 < \varepsilon \le 1}$ with $\varepsilon _k \rightarrow 0$ if $k \rightarrow \infty$ such that

$$\begin{aligned} p_{\varepsilon _k} \xrightarrow []{k \rightarrow \infty } p \quad \text {pointwise}, \end{aligned}$$

(52)

where $p \in C^{\tau } S^{0}_{0,0}(\mathbb {R}^{n}\times \mathbb {R}^{n}; \tilde{M}-1 )$. Let $u \in \mathcal {S}(\mathbb {R}^n)$ be arbitrary. Because of (52) and the boundedness of $( p_{\varepsilon _k} )_{k \in \mathbb {N}} \subseteq C^{\tau } S^{0}_{0,0}(\mathbb {R}^{n}\times \mathbb {R}^{n}; \tilde{M})$, we get

$$\begin{aligned} p_{\varepsilon _k}(x,D_x)&u \xrightarrow []{k \rightarrow \infty } p(x,D_x) u \end{aligned}$$

(53)

pointwise due to Lebesgue’s theorem. Choosing $N \in \mathbb {N}$ with $n< 2N \le M$ we get by Lemma 4.5:

$$\begin{aligned}&|p_{\varepsilon _k}(x,D_x)u(x) - p(x,D_x)u(x)|^q \\&\quad \le \left( |p_{\varepsilon _k}(x,D_x)u (x)| + \lim _{k \rightarrow \infty }|p_{\varepsilon _k}(x,D_x)u(x)| \right) ^q \\&\quad \le C_{N,n} \langle x \rangle ^{-2Nq} \in L^1(\mathbb {R}^{n}_x) \end{aligned}$$

for all $k \in \mathbb {N}$. Together with (53) we can apply Lebesgue’s theorem and obtain

$$\begin{aligned}&\Vert p_{\varepsilon _k}(x,D_x)u - p(x,D_x)u \Vert ^q_{L^q(\mathbb {R}^{n})}\\&\quad = \int \limits _{\mathbb {R}^{n}}| p_{\varepsilon _k}(x,D_x)u(x) - p(x,D_x)u(x) |^q dx \xrightarrow [] {k \rightarrow \infty } 0 \end{aligned}$$

Together with i) and Lemma 4.16 we conclude

$$\begin{aligned} p(x,D_x)u = L^q-\lim _{k \rightarrow \infty } p_{\varepsilon _k}(x,D_x)u = L^ q-\lim _{k \rightarrow \infty } T_{\varepsilon _k} u = Tu. \end{aligned}$$

$\square$

By means of order reducing operators we can extend the previous characterization to the class $C^s S^m_{0,0}$ for general m:

Lemma 4.21

Let $m\in \mathbb {R}$, $1<q<\infty$, $\tilde{m} \in \mathbb {N}_0$ with $\tilde{m}>n/q$. Additionally let $M \in \mathbb {N}_0 \cup \{ \infty \}$ with $M >n$. We define $\tilde{M}:=M-(n+1)$. Considering $T \in \mathcal {A}^{m,M}_{0,0}(\tilde{m},q)$ and $\tilde{M} \ge 1$ we have for $s \in (0, \tilde{m}-n/q]$ with $s\notin \mathbb {N}_0$:

$$\begin{aligned} T \in OPC^s S^m_{0,0}(\mathbb {R}^n \times \mathbb {R}^n; \tilde{M}-1) \cap \mathscr {L} \big (H^m_q(\mathbb {R}^{n}),L^q(\mathbb {R}^{n})\big ). \end{aligned}$$

Proof

Let $s \in ( 0, \tilde{m}-n/q ]$ with $s\notin \mathbb {N}_0$ and $\delta \in \mathbb {N}_0^n$. Moreover we define for every $\alpha \in \mathbb {R}$ the order reducing pseudodifferential operator $\Lambda ^{\alpha }:=\lambda ^{\alpha }(D_x)$, where $\lambda ^{\alpha }(\xi ):=\langle \xi \rangle ^{\alpha } \in S^{\alpha }_{1,0}(\mathbb {R}^n \times \mathbb {R}^n)$. Due to Remark 2.5 and Theorem 3.6 we get that

$$\begin{aligned} {{\mathrm{ad}}}(-ix)^{\delta }\Lambda ^{-m}: L^q(\mathbb {R}^{n}) \rightarrow H_q^{m+|\delta |}( \mathbb {R}^{n}) \subseteq H_q^{m}(\mathbb {R}^{n}) \text { is continuous}. \end{aligned}$$

(54)

Now let $l\in \mathbb {N}_0$, $\alpha _1,\ldots , \alpha _l \in \mathbb {N}_0^n$ and $\beta _ 1,\ldots , \beta _l \in \mathbb {N}_0^n$ such that $|\beta |\le \tilde{m}$ and $|\alpha | \le M$, where $\beta := \beta _1 + \cdots + \beta _l$ and $\alpha := \alpha _1 + \cdots + \alpha _l$. Since ${{\mathrm{ad}}}(-ix)^{\tau _2}{{\mathrm{ad}}}(D_x)^{\delta } \Lambda ^{-m} \equiv 0$ for every $\tau _2, \delta \in \mathbb {N}_0^n$ with $|\delta | \ne 0$ due to Remark 2.5, we can iteratively show

$$\begin{aligned}&{{\mathrm{ad}}}(-ix)^{\alpha _1}{{\mathrm{ad}}}(D_x)^{\beta _1}\ldots {{\mathrm{ad}}}(-ix)^{\alpha _l}{{\mathrm{ad}}}(D_x)^{\beta _l} (T\Lambda ^{-m}) \\&\quad = \sum _{ \begin{array}{c} \gamma _1 + \delta _1 = \alpha _1 \\ \vdots \\ \gamma _l + \delta _l = \alpha _l \end{array} } C_{\gamma _1,\ldots ,\gamma _l} [{{\mathrm{ad}}}(-ix)^{\gamma _1}{{\mathrm{ad}}}(D_x)^{\beta _1}\ldots {{\mathrm{ad}}}(-ix)^{\gamma _l}{{\mathrm{ad}}}(D_x)^{\beta _l} T][{{\mathrm{ad}}}(-ix)^{\delta } \Lambda ^{-m}], \end{aligned}$$

where $\delta$ is defined by $\delta :=\delta _1+ \cdots + \delta _l$. Combining (54) and $T \in \mathcal {A}^{m,M}_{0,0}(\tilde{m},q)$ we obtain the continuity of

$$\begin{aligned} {{\mathrm{ad}}}(-ix)^{\alpha _1}{{\mathrm{ad}}}(D_x)^{\beta _1}\ldots {{\mathrm{ad}}}(-ix)^{\alpha _l}{{\mathrm{ad}}}(D_x)^{\beta _l} (T\Lambda ^{-m}) : L^q(\mathbb {R}^{n}) \rightarrow L^q(\mathbb {R}^{n}). \end{aligned}$$

Therefore $T\Lambda ^{-m} \in \mathcal {A}^{0,M}_{0,0}(\tilde{m},q)$. If we use Theorem 4.20, we get

$$\begin{aligned} T\Lambda ^{-m} \in OPC^{s} S^0_{0,0}(\mathbb {R}^n \times \mathbb {R}^n; \tilde{M}-1) \cap \mathscr {L}(L^q(\mathbb {R}^{n})). \end{aligned}$$

On account of $\Lambda ^{m} \in OPS^m_{1,0}(\mathbb {R}^n \times \mathbb {R}^n)$ and Theorem 3.6 we have

$$\begin{aligned} T \in OPC^{s} S^m_{0,0}(\mathbb {R}^n \times \mathbb {R}^n; \tilde{M}-1) \cap \mathscr {L} \big (H^m_q (\mathbb {R}^{n}),L^q(\mathbb {R}^{n}) \big ). \end{aligned}$$

$\square$

4.5 Characterization of Pseudodifferential Operators with Symbols in $C^s S^m_{1,0}$

In applications to nonlinear partial differential equations the pseudodifferential operators are often in the class $C^s S^m_{1,0}(\mathbb {R}^n \times \mathbb {R}^n)$. As we have seen in Example 1.2, these operators are elements of the set $\mathcal {A}^{m}_{1,0}(\lfloor s \rfloor ,q)$ with $1< q < \infty$. In the present subsection we show that operators belonging to the set $\mathcal {A}^{m, M}_{1,0}(\tilde{m},q)$ for sufficiently large $\tilde{m}$ are non-smooth pseudodifferential operators of the order m whose coefficients are in a Hölder space. As an ingredient we use $\mathcal {A}^{m, M}_{1,0}(\tilde{m},q) \subseteq \mathcal { A}^{m, M}_{0,0}(\tilde{m},q)$. Consequently we may apply the characterization of the pseudodifferential operators of the class $C^s S^m_{0,0}(\mathbb {R}^n \times \mathbb {R}^n, M)$ in order to obtain the following main result of this section:

Theorem 4.22

Let $m\in \mathbb {R}$, $1<q<\infty$ and $\tilde{m} \in \mathbb {N}_0$ with $\tilde{m}>n/q$. Additionally let $M \in \mathbb {N}_0$ with $M>n$. We define $\tilde{M}:=M- (n+1)$. Assuming $P \in \mathcal {A}^{m, M}_{1,0}(\tilde{m},q)$ and $\tilde{M} \ge 1$, we obtain for all $\tau \in \left( 0,\tilde{m}-n/q \right]$ with $\tau \notin \mathbb {N}_0$:

$$\begin{aligned} P \in OPC^{\tau } S^m_{1,0}(\mathbb {R}^n \times \mathbb {R}^n; \tilde{M}-1) \cap \mathscr {L}\big (H^m _q(\mathbb {R}^{n}),L^q(\mathbb {R}^{n})\big ). \end{aligned}$$

Proof

Let $\tilde{m}- n/q \ge \tau > 0$ with $\tau \notin \mathbb {N}_0$ and $P \in \mathcal {A}^{m, M}_{1,0}(\tilde{m},q)$. Due to Lemma 1.3 we have $P \in \mathcal {A}^{m, M}_{1,0}(\tilde{m},q) \subseteq \mathcal {A}^{m , M}_{0,0}(\tilde{m},q).$ Hence we get by means of Lemma 4.21:

$$\begin{aligned} P \in OPC^{\tau } S^m_{0,0}(\mathbb {R}^n \times \mathbb {R}^n; \tilde{M}-1) \cap \mathscr {L}\big (H^m _q(\mathbb {R}^{n}),L^q(\mathbb {R}^{n})\big ). \end{aligned}$$

Let $\alpha \in \mathbb {N}_0^n$ with $|\alpha | \le \tilde{M}-1$. Then ${{\mathrm{ad}}}(-ix)^{\alpha } P \in \mathcal {A}^{m-|\alpha |, M-|\alpha |}_{1,0 }(\tilde{m},q)$. Because of Lemmas 1.3 and 4.21, we obtain

$$\begin{aligned} {{\mathrm{ad}}}(-ix)^{\alpha } P \in OPC^{\tau } S^{m-|\alpha |}_{0,0}(\mathbb {R}^n \times \mathbb {R}^n; \tilde{M}-|\alpha |-1). \end{aligned}$$

Due to Remark 2.5 the symbol of ${{\mathrm{ad}}}(-ix)^{\alpha } P$ is $\partial _{\xi }^{\alpha } p(x,\xi )$ if p is the symbol of P. Hence

$$\begin{aligned} \big \Vert \partial _{\xi }^{\alpha } p(.,\xi ) \big \Vert _{ C^{\tau } (\mathbb {R}^{n}) } \le C_{\alpha } \langle \xi \rangle ^{m-|\alpha |} \quad \text {for all } \xi \in \mathbb {R}^n. \end{aligned}$$

Consequently p is an element of $C^{\tau } S^{m}_{1,0}(\mathbb {R}^n \times \mathbb {R}^n; \tilde{M }-1)$. $\square$

In the case $\tilde{M}-1>\max \{n/2, n/q\}$, $1<q< \infty$, every pseudodifferential operator whose symbol is in the class $C^{\tau } S^m_{1,0}(\mathbb {R}^n \times \mathbb {R}^n; \tilde{M}-1)$, where $\tau >0$ and $m \in \mathbb {R}$, is an element of $\mathscr {L}(H^m_q(\mathbb {R}^{n}),L^q(\mathbb {R}^{n}))$ due to Theorem 3.7. Therefore we have in this case

$$\begin{aligned}&OPC^{\tau } S^m_{1,0}(\mathbb {R}^n \times \mathbb {R}^n; \tilde{M}-1) \cap \mathscr {L} \big (H^m_q(\mathbb {R}^{n}), L^q(\mathbb {R}^{n})\big ) \\&\quad = OPC^{\tau } S^m_{1,0}(\mathbb {R}^n \times \mathbb {R}^n; \tilde{M}-1). \end{aligned}$$

5 Composition of Pseudodifferential Operators

A calculus for non-smooth pseudodifferential operators in the non-smooth symbol-class $C^{\tau } S^{m}_{1,\delta }(\mathbb {R}^{n}\times \mathbb {R}^{n})$ was first developed by Kumano-Go and Nagase in [12]. Meyer and Marschall improved this calculus in [16] and [14, Chapter  6]. Later Marschall adapted the arguments given there to obtain a calculus for the general case $C^{\tau }S^{m}_{\rho ,\delta }(\mathbb {R}^{n}\times \mathbb {R}^{n}; N)$ in [15]. Moreover a calculus for operator-valued pseudodifferential operators with non-smooth symbols of class $C^{\tau }S^{m}_{1,0}(\mathbb {R}^{n}\times \mathbb {R}^{n}; \mathscr {L}(X_1,X_2))$ was treated by A. in [1].

We recall that the composition of two smooth pseudodifferential operators is also a smooth pseudodifferential operator, cf. e. g. [2, Theorem 3.16]. But in contrast to the smooth case, the composition of two non-smooth pseudodifferential operators is in general not a pseudodifferential operator with the same regularity with respect to its coefficient, cf.[1, p.1465]. To illustrate this, let $p\in C^{\tau } S^{m}_{1,0}(\mathbb {R}^{n}_x \times \mathbb {R}^{n} _{\xi })$ with $\tau \in (0,1)$, $m \in \mathbb {R}$ and $p(x,\xi ) \notin C^1(\mathbb {R}^{n}_x )$ for all $\xi \in \mathbb {R}^{n}$. Moreover let $j \in \{ 1, \ldots , n\}$. Then $p (x,D_x)D_{x_j} =OP(p(x,\xi )\xi _j) \in OPC^{\tau } S^{m+1}_{1,0}(\mathbb {R}^ {n}\times \mathbb {R}^{n})$ is a pseudodifferential operator. But $OP(\xi _j)p(x,D _x)$ is no pseudodifferential operator, otherwise the iterated commutator

$$\begin{aligned} {{\mathrm{ad}}}(D_{x_j})p(x,D_x) = OP(\xi _j)p(x,D_x) - p(x,D_x) OP(\xi _j)= ( \partial _{x_j} p)(x,D_x) \end{aligned}$$

(55)

would be a pseudodifferential operator, too. Hence $\partial _{x_j} p \in C^0(\mathbb {R}^{n})$ for all $j \in \{ 1, \ldots , n\}$ and $p(.,\xi ) \in C^1(\mathbb {R}^{n})$ for all $\xi \in \mathbb {R}^{n}$, which contradicts the assumptions.

With the characterization of non-smooth pseudodifferential operators at hand, we can prove a result for the composition of two non-smooth pseudodifferential operators:

Theorem 5.1

Let $m_i \in \mathbb {R}$, $M_i \in \mathbb {N}\cup \{ \infty \}$ and $\rho _i \in \{0,1\}$ for $i \in \{ 1,2 \}$. Additionally let $0< \tau _i < 1$ and $\tilde{m} _i \in \mathbb {N}_0$ be such that $\tau _i + \tilde{m}_i > (1-\rho _i)n/2$ for $i \in \{ 1,2 \}$. We define $k_{i}:=(1-\rho _i)n/2$ for $i \in \{ 1,2\}$, $\rho := \min \{\rho _1; \rho _2 \}$ and $m:= m_1+ m_2 + k_{1} + k_{2}$. Moreover, let $\tilde{m}, M \in \mathbb {N}$ and $1< q < \infty$ be such that

• (i) $M \le \min \left\{ M_i- \max \{ n/q; n/2 \}: i \in \{1,2\} \right\}$,

• (ii) $n/q < \tilde{m} \le \min \{ \tilde{m}_1 ; \tilde{m}_2\}$,

• (iii) $\tilde{m} < \tilde{m}_2 + \tau _2 - m_1 - k_{1}$,

• (iv) $\rho M + \tilde{m} < \tilde{m}_2 + \tau _2 + m_1 + k_{1}$ ,

• (v) $\tilde{M} \ge 1$, where $\tilde{M}:= M-(n+1)$,

• (vi) $q=2$ in the case $(\rho _1, \rho _2) \ne (1,1)$.

Considering two symbols $p_i \in C^{\tilde{m}_i, \tau _i} S^{m_i}_{\rho _i,0}(\mathbb {R}^{n}\times \mathbb {R}^{n}; M_i)$, $i \in \{1,2\}$, we obtain

$$\begin{aligned} p_1(x,D_x) p_2(x,D_x) \in OPC^{\tilde{m}-n/q} S^{m}_{\rho ,0}(\mathbb {R}^{n}\times \mathbb {R}^{n}; \tilde{M}-1). \end{aligned}$$

Proof

Let $l \in \mathbb {N}$, $\tilde{\alpha }_1, \ldots , \tilde{\alpha }_{l} \in \mathbb {N}_0^n$ and $\tilde{\beta }_1, \ldots , \tilde{\beta }_{l} \in \mathbb {N}_0^n$ with $|\tilde{\alpha }| \le M$, $|\tilde{\beta }| \le \tilde{m}$ and $|\tilde{\alpha }_1| + |\tilde{\beta }_1| = \cdots = |\tilde{\alpha }_l| + |\tilde{\beta }_l| = 1$ be arbitrary. Here $\tilde{\alpha }:= \tilde{\alpha }_1 + \cdots + \tilde{\alpha }_{l}$ and $\tilde{\beta }:= \tilde{\beta }_1 + \cdots + \tilde{\beta }_{l}$. Due to Remark 2.5, i) and ii) we know that

$$\begin{aligned} {{\mathrm{ad}}}(-ix)^{\tilde{\alpha }_l} {{\mathrm{ad}}}(D_x)^{\tilde{\beta }_l} \ldots {{\mathrm{ad}}}(-ix )^{\tilde{\alpha }_1} {{\mathrm{ad}}}(D_x)^{\tilde{\beta }_1} p_i(x,D_x) \end{aligned}$$

is a pseudodifferential operator with symbol

$$\begin{aligned} \partial _{\xi }^{\tilde{\alpha }} D_x^{\tilde{\beta }} p_i \in C^{\tilde{m}_i-|\tilde{\beta }|, \tau _i} S^{m_i-\rho _i |\tilde{\alpha }|}_{\rho _i,0}(\mathbb {R}^{n}\times \mathbb {R}^{n}; M_i-|\tilde{\alpha }|) \end{aligned}$$

for $i \in \{1,2\}$. Since $0 < \tilde{m}_1 - |\tilde{\beta }| + \tau _1$ because of ii), an application of Theorem 3.7 if $\rho _1=1$ and Theorem 3.9 else provides for all elements u of $H^{m_1-\rho |\tilde{\alpha }|+ k_{1}}_q(\mathbb {R}^{n})$:

$$\begin{aligned} \Vert {{\mathrm{ad}}}(-ix)^{\tilde{\alpha }_l} {{\mathrm{ad}}}(D_x)^{\tilde{\beta }_l} \ldots&{{\mathrm{ad}}}(-ix)^{\tilde{\alpha }_1} {{\mathrm{ad}}}(D_x)^{\tilde{\beta }_1} p_1(x,D_x) u\Vert _{L^q} \le C \Vert u\Vert _{H^{m_1-\rho _1|\tilde{\alpha }|+ k_{1}}_q} \nonumber \\&\qquad \qquad \le C \Vert u\Vert _{H^{m_1-\rho |\tilde{\alpha }|+ k_{1}}_q}. \end{aligned}$$

(56)

Now let $k \in \mathbb {N}_0$ with $k \le M$ be arbitrary. Making use of estimate iii) and iv) we are able to verify that the assumptions of Theorem 3.7 hold. An application of Theorem 3.7 in the case $\rho _1=1$ and Theorem 3.9 else yields for all $u \in H^{m-\rho (k+|\tilde{ \alpha }|)}_q(\mathbb {R}^{n})$:

$$\begin{aligned} \Vert {{\mathrm{ad}}}(-ix)^{\tilde{\alpha }_l} {{\mathrm{ad}}}(D_x)^{\tilde{\beta }_l} \ldots&{{\mathrm{ad}}}(-ix)^{\tilde{\alpha }_1} {{\mathrm{ad}}}(D_x)^{\tilde{\beta }_1} p_2(x,D_x) u\Vert _{H^{m_ 1-\rho k+ k_{1}}_q} \nonumber \\&\le C \Vert u\Vert _{H^{m-\rho k - \rho _2 |\tilde{\alpha }|}_q} \le C \Vert u\Vert _{H^{m-\rho (|\tilde{\alpha }| + k)}_q}. \end{aligned}$$

(57)

We assume arbitrary $l \in \mathbb {N}$, $\alpha _1, \ldots , \alpha _{l} \in \mathbb {N}_0^n$ and $\beta _1, \ldots , \beta _{l} \in \mathbb {N}_0^n$ with $|\alpha | \le M$, $|\beta | \le \tilde{m}$ and $|\alpha _1| + |\beta _1| = \ldots = |\alpha _l| + |\beta _l| = 1$. Here $\alpha := \alpha _1 + \cdots + \alpha _{l}$ and $\beta := \beta _1 + \cdots + \beta _{l}$. Using (56) and (57) we obtain iteratively

$$\begin{aligned}&\left\| {{\mathrm{ad}}}(-ix)^{\alpha _l} {{\mathrm{ad}}}(D_x)^{\beta _l} \ldots {{\mathrm{ad}}}(-ix)^{\alpha _1} {{\mathrm{ad}}}(D_x)^{\beta _1} [ p_1(x,D_x) p_2(x,D_x) ] u \right\| _{L^q} \\&\qquad \le C \sum _{ \begin{array}{c} \tilde{\alpha }_j + \gamma _j = \alpha _j \\ \tilde{\beta }_j + \delta _j = \beta _j \end{array} } \Vert [{{\mathrm{ad}}}(-ix)^{\tilde{\alpha }_l} {{\mathrm{ad}}}(D_x)^{\tilde{\beta }_l} \ldots \ ad(-ix)^{\tilde{\alpha }_1} {{\mathrm{ad}}}(D_x)^{\tilde{\beta }_1} p_1(x,D_x)] \\&\qquad \qquad \qquad \quad \qquad \left. [{{\mathrm{ad}}}(-ix)^{\gamma _l} {{\mathrm{ad}}}( D_x)^{\delta _l} \ldots {{\mathrm{ad}}}(-ix)^{\gamma _1} {{\mathrm{ad}}}(D_x)^{\delta _1} p_2(x,D_x) ]u \right\| _{L^q} \\&\qquad \le C \Vert u\Vert _{H^{m-\rho |\alpha |}_q} \; \text {for all } u \in H^{m-\rho |\alpha |}_q(\mathbb {R}^{n}). \end{aligned}$$

Consequently $p_1(x,D_x) p_2(x,D_x) \in \mathcal {A}^{m,M}_{\rho ,0}(\tilde{m},q)$. Due to ii) and v) we can apply Theorem 4.22 in the case $\rho =1$ and Lemma 4.21 if $\rho = 0$ and we get the claim. $\square$