On the boundedness of periodic pseudo-differential operators

Abstract

In this paper we investigate the $L^p$-boundedness of certain classes of periodic pseudo-differential operators. The operators considered arise from the study of symbols on ${\mathbb {T}}^n\times {\mathbb {Z}}^n$ with limited regularity.

1 Introduction

In this work we obtain $L^p$-boundedness theorems for pseudo-differential operators with symbols defined on ${\mathbb {T}}^n\times {\mathbb {Z}}^n$. For some recent work on boundedness results of periodic pseudo-differential operators in $L^p$-spaces we refer the reader to [5, 8, 20, 23, 24]. Pseudo-differential operators on ${\mathbb {R}}^n$ are generalizations of differential operators and singular integrals. They are formally defined by

$$\begin{aligned} T_{\sigma }u(x)=\int e^{i2\pi \langle x,\eta \rangle }\sigma (x,\eta )\widehat{u}(\eta )d\eta . \end{aligned}$$

The function $\sigma$ is usually called the symbol of the corresponding operator $T_{\sigma }.$ Symbols are classified according to their behavior and the behavior of their derivatives. For $m\in \mathbb {R}$ and $0\le \rho ,\delta \le 1,$ the $(\rho ,\delta )$-Hörmander class $S^{m}_{\rho ,\delta }(\mathbb {R}^n\times \mathbb {R}^n)$ consists of those functions which are smooth in $(x,\eta )$ and which satisfy symbols inequalities

$$\begin{aligned} \left| \partial ^{\beta }_{x}\partial _{\eta }^{\alpha }\sigma (x,\eta )\right| \le C_{\alpha ,\beta }\langle \eta \rangle ^{m-\rho |\alpha |+\delta |\beta |}. \end{aligned}$$

The corresponding set of operators with symbols in $(\rho ,\delta )$-classes will be denoted by $\Psi ^{m}_{\rho ,\delta }(\mathbb {R}^n\times \mathbb {R}^n).$ A remarkable result due to A. P. Calderón and R. Vaillancourt, gives us that a pseudo-differential operator with symbol in $S^{0}_{\rho ,\rho }(\mathbb {R}^n\times \mathbb {R}^n)$ for some $0\le \rho <1$ is bounded on $L^2(\mathbb {R}^n),$ (see [3, 4]). Their result is false when $\rho =1:$ there exists symbols in $S^{0}_{1,1}(\mathbb {R}^n\times \mathbb {R}^n)$ whose associated pseudo-differential operators are not bounded on $L^2(\mathbb {R}^n),$ (see [10]). For general $1<p<\infty$, we have the following theorem (see Fefferman [12]): if $m\le -m_{p}= -n(1-\rho )\left| \frac{1}{p}-\frac{1}{2}\right|$, $m_{p}<n(1-\rho )/2,$ then $T_{\sigma }\in \Psi ^{m}_{\rho ,\delta }(\mathbb {R}^n\times \mathbb {R}^n)$ is a $L^p$-bounded operator. It is well known that for every $m>m_p$ there exists $T_{\sigma }\in \Psi ^{m}_{\rho ,\delta }(\mathbb {R}^n\times \mathbb {R}^n)$ which is not bounded on $L^{p}(\mathbb {R}^n).$ The historical development of the problem about the $L^p$-boundedness of pseudo-differential operators on $\mathbb {R}^n$ can be found in [22, 28].

Pseudo-differential operators with symbols in Hörmander classes can be defined on $C^{\infty }$-manifolds by using local charts. In 1979, Agranovich (see [1]) gives a global definition of pseudo-differential operators on the circle $\mathbb {S}^1,$ (instead of the local formulation on the circle as a manifold). By using the Fourier transform Agranovich’s definition was readily generalizable to the n-dimensional torus $\mathbb {T}^n.$ It is a non-trivial result, that the definition of pseudo-differential operators with symbols on $(\rho ,\delta )$-classes by Agranovich and Hörmander are equivalent. This fact is known as the equivalence theorem of McLean (see [19]). Important consequences of this equivalence are the following periodic versions of the Calderón–Vaillancourt theorem and the Fefferman theorem:

Theorem 1.1

Let $0\le \delta <\rho \le 1$ and let $a{:}\mathbb {T}^n\times \mathbb {Z}^n\rightarrow \mathbb {C}$ be a symbol with corresponding periodic pseudo-differential operator defined on $C^{\infty }(\mathbb {T}^n)$ by

$$\begin{aligned} a(x,D)f(x)=\sum _{\xi \in \mathbb {Z}^n } e^{ i2\pi \langle x,\xi \rangle }a(x,\xi )\widehat{f}(\xi ). \end{aligned}$$

(1.1)

If $\alpha , \beta \in \mathbb {N}^n,$ under one of the following two conditions,

• 1. (Calderon–Vaillancourt condition, periodic version).

  $$\begin{aligned} \left| \partial _{x}^{\beta }\Delta _{\xi }^{\alpha }a(x,\xi )\right| \le C_{\alpha ,\beta } \langle \xi \rangle ^{\rho (|\beta |-|\alpha |)},\quad (x,\xi )\in \mathbb {T}^n\times \mathbb {Z}^n, \end{aligned}$$

• 2. (Fefferman condition, periodic version).

  $$\begin{aligned} \left| \partial _{x}^{\beta }\Delta _{\xi }^{\alpha }a(x,\xi )\right| \le C_{\alpha ,\beta } \langle \xi \rangle ^{-m_{p}-\rho |\alpha |+\delta |\beta | },\quad (x,\xi )\in \mathbb {T}^n\times \mathbb {Z}^n,\quad \delta <1, \end{aligned}$$

  the operator a(x, D) is bounded on $L^p(\mathbb {T}^n)$, for all $1<p<\infty .$

Theorem 1.1 is a consequence of the McLean equivalence theorem, the Calderón–Vaillancourt theorem and the Fefferman theorem. Moreover, from the proofs of these results it is easy to see that symbols with limited regularity satisfying conditions in Theorem 1.1 give rise to $L^p$-bounded pseudo-differential operators (under the condition $0\le \delta <\rho \le 1$) (see [2]). With this in mind, in this paper we provide some new results about the $L^p$ boundedness of periodic pseudo-differential operators associated to toroidal symbols with limited regularity. Now, we first recall some recent $L^p$-theorems of periodic operators and later we present our results.

Theorem 1.2

[20, Wong–Molahajloo] Let a(x, D) be a periodic pseudo-differential operator on $\mathbb {T}^1\equiv \mathbb {S}^1$. If $\sigma :\mathbb {T}\times \mathbb {Z}\rightarrow \mathbb {C}$ satisfies

$$\begin{aligned} |\partial _{x}^{\beta }\Delta _{\xi }^{\alpha }a(x,\xi ) |\le C_{\alpha ,\beta }\langle \xi \rangle ^{-|\alpha |},\,\,\,(x,\xi )\in \mathbb {T}^n\times \mathbb {Z}^n, \end{aligned}$$

(1.2)

with $|\alpha |,|\beta |\le 1,$ then, $a(x,D):L^p(\mathbb {T}^1)\rightarrow L^p(\mathbb {T}^1)$ is a bounded operator for all $1<p<\infty$.

Theorem 1.3

[23, Ruzhansky–Turunen] Let $n\in \mathbb {N},$ $k:=[\frac{n}{2}]+1$ and $a:\mathbb {T}^n\times \mathbb {Z}^n\rightarrow \mathbb {C}$ such that $\left| \partial ^{\beta }_{x}a(x,\xi )\right| \le C_{\beta },\,\,\, |\beta |\le k.$ Then, $a(x,D){:}L^2(\mathbb {T}^n)\rightarrow L^2(\mathbb {T}^n)$ is a bounded operator.

We note that compared with several well-known theorems on $L^2$-boundedness of pseudo-differential operators (Calderon–Vaillancourt theorem, for example), the Theorem 1.3 does not require any regularity with respect to the $\xi$-variable. As a consequence of real interpolation and the $L^2$-estimate by Ruzhansky and Turunen, Delgado [8] established the following sharp $L^p$-theorem.

Theorem 1.4

[8, Delgado] Let $0<\varepsilon <1$ and $k:=\left[ \frac{n}{2}\right] +1,$ let $a:\mathbb {T}^n\times \mathbb {Z}^n\rightarrow \mathbb {C}$ be a symbol such that $\left| \Delta _{\xi }^{\alpha }a(x,\xi )\right| \le C_{\alpha }\langle \xi \rangle ^{-\frac{n}{2}\varepsilon -(1-\varepsilon )|\alpha |},$ $\left| \partial _{x}^{\beta }a(x,\xi )\right| \le C_{\beta }\langle \xi \rangle ^{-\frac{n}{2}\varepsilon },$ for $|\alpha |,|\beta |\le k.$ Then $a(x,D):L^p(\mathbb {T}^n)\rightarrow L^p(\mathbb {T}^n)$ is a bounded linear operator, for $2\le p<\infty .$

In the recent paper [9] the authors have considered the $L^{p}$-boundedness of pseudo-differential operators on Compact Lie groups. Although, the original theorem is valid for general compact Lie groups, we present the following periodic version.

Theorem 1.5

[9, Delgado–Ruzhansky] Let $0\le \delta ,\rho \le 1$ and $1<p<\infty .$ Denote by $\kappa$ the smallest even integer larger than $\frac{n}{2}.$ Let $\sigma (x,D)$ be a pseudo-differential operator with symbol $\sigma$ satisfying

$$\begin{aligned} \left| \partial _{x}^{\beta }\Delta _{\xi }^{\alpha }\sigma (x,\xi )\right| \le C\langle \xi \rangle ^{-m_0-\rho |\alpha |+\delta |\beta |}, \end{aligned}$$

(1.3)

for all $|\alpha |\le \kappa$ and $|\beta |\le \left[ \frac{n}{p}\right] +1$, where $m_{0}:=\kappa (1-\rho ) |1/p-1/2|+\delta \left( \left[ \frac{n}{p}\right] +1\right)$. Then $\sigma (x,D)$ extends to a bounded operator on $L^p(\mathbb {T}^n)$ for all $1<p<\infty .$

Now, we present the main results of this paper. Our starting point is the following result which consider the problem about the $L^{p}(\mathbb {T}^n)$-boundedness for $1\le p<\infty$, in contrast with the previous results where the case $p=1$ can be not considered because, as it is well known the boundedness of pseudo-differential operators with symbols in $(\rho ,\delta )$-classes fails on $L^1$. We denote $m_{p}:=n(1-\rho )\left| \frac{1}{p}-\frac{1}{2}\right|$, for all $1\le p<\infty$, $n\in \mathbb {N}$ and $0\le \rho \le 1$.

Theorem IA.  Let $0<\varepsilon \le 1$, $0\le \rho \le 1$ and let $\omega :[0,\infty )\rightarrow [0,\infty )$  be a non-decreasing and bounded function such that

$$\begin{aligned} \int _{0}^1\omega (t)t^{-1}dt<\infty . \end{aligned}$$

(1.4)

Let $\sigma (x,\xi ):\mathbb {T}^n\times \mathbb {Z}^n\rightarrow \mathbb {C}$ be a symbol satisfying

$$\begin{aligned} \left| \partial _{x}^{\beta }\Delta ^{\alpha }_{\xi }\sigma (x,\xi )\right| \le C\omega (\langle \xi \rangle ^{-\varepsilon })\langle \xi \rangle ^{-\frac{n}{2}(1-\rho )-\rho |\alpha |} \end{aligned}$$

(1.5)

for $|\alpha |\le \left[ \frac{n}{2}\right] +1$, $|\beta |\le \left[ \frac{n}{p}\right] +1,$ then the corresponding periodic operator $\sigma (x,D)$ extends to a bounded operator on $L^p(\mathbb {T}^n)$ for all $1\le p<\infty$.

The following theorem is related with the result proved by Delgado and Ruzhansky mentioned above. We discuss such relation in Remark 3.5.

Theorem IB.  Let $0< \rho \le 1,$ and $1< p<\infty .$ Let $\sigma :\mathbb {T}^n\times \mathbb {Z}^n\rightarrow \mathbb {C}$ be a symbol satisfying

$$\begin{aligned} \left| \partial _{x}^\beta \Delta ^{\alpha }_{\xi }\sigma (x,\xi )\right| \le C\langle \xi \rangle ^{-m_p-\rho |\alpha |}, \end{aligned}$$

(1.6)

for all $\alpha$ and $\beta$ in $\mathbb {N}^n$ such that $|\alpha |\le \left[ \frac{n}{2}\right] +1$, $|\beta |\le \left[ \frac{n}{p}\right] +1$, then the corresponding periodic operator $\sigma (x,D)$ extends to a bounded operator on $L^p(\mathbb {T}^n)$.

Remark 1.6

We observe that from classical results (e.g., Lemma 2.2 of [22]) together with suitable versions of the McLean equivalence theorem (see Proposition 2.4 below) the $L^p$-boundedness of a periodic operator satisfying (1.6) for $|\alpha |\le \left[ \frac{n}{2}\right] +1$, $|\beta |\le \left[ \frac{n}{p}\right] +1$, $2\le p<\infty$ can be proved. For the case of $\mathbb {R}^n$ it is well known that symbols with $\left[ \frac{n}{2}\right] +1$-derivatives in $\xi$ satisfying $(\rho ,\delta )$-estimates does not always imply $L^p$-boundedness, $1<p\le 2$. However, in order to assure boundedness it is suffice to consider for this case $(n+1)$-derivatives in $\xi$. However, with our approach, we only need $k:=\left[ \frac{n}{2}\right] +1$-derivatives in $\xi$ for a periodic symbol improving the immediate result that we could have if we apply suitable versions of the McLean equivalence theorem (see Proposition 2.4) (together with Lemma 2.2 of [22]). It is important to mention that the previous result can be not deduced from Theorem 1.4. In the following result we consider the boundedness of multipliers on $L^{\nu }$, $0<\nu \le 1$.

Theorem IC. Let $k:=\left[ \frac{n}{2}\right] +1$. Let $\sigma (\xi )$ be a periodic symbol on $\mathbb {Z}^n$ satisfying

$$\begin{aligned} \left| \Delta ^{\alpha }_{\xi }\sigma (\xi )\right| \le C_{\alpha }\langle \xi \rangle ^{-|\alpha |},\quad |\alpha |\le k, \end{aligned}$$

(1.7)

then, the corresponding periodic operator $\sigma (D)$ is bounded from $L^1(\mathbb {T}^n)$ into $L^p(\mathbb {T}^n)$ for all $0<p<1.$

Interpolation via Riesz–Thorin Theorem allow us to obtain the next $(L^p,L^r)$-theorem:

Theorem II. Let $2\le p<\infty ,$ $0<\varepsilon <1$ and $k:=\left[ \frac{n}{2}\right] +1$. Let $a:\mathbb {T}^n\times \mathbb {Z}^n\rightarrow \mathbb {C}$ be a symbol such that $\left| \Delta _{\xi }^{\alpha }a(x,\xi )\right| \le C_{\alpha }\langle \xi \rangle ^{-\frac{n}{2}\varepsilon -(1-\varepsilon )|\alpha |}$, $\left| \partial _{x}^{\beta }a(x,\xi )\right| \le C_{\beta }\langle \xi \rangle ^{-\frac{n}{2}\varepsilon },$ for $|\alpha |,|\beta |\le k$, then $\sigma (x,D):L^p(\mathbb {T}^n)\rightarrow L^r(\mathbb {T}^n)$ is a bounded linear operator for all $1<q\le r\le p<\infty$, where $\frac{1}{p}+\frac{1}{q}=1$.

An operator T is positive if $f\ge 0$ implies that, the function Tf is non-negative. In the following theorem we study positive and periodic amplitude operators (see Eq. (2.9) for the definition of amplitude operator):

Theorem III. Let $0<\varepsilon ,\delta <1.$ If a(x, y, D) is a positive amplitude operator with symbol satisfying the following inequalities

$$\begin{aligned} \left| \partial _{x}^{\beta }\partial _{y}^{\alpha }a(x,y,\xi )\right| \le C_{\alpha ,\beta }\langle \xi \rangle ^{\delta |\beta |-\varepsilon |\alpha |},\quad |\alpha |,|\beta | \le \mu :=\left[ \frac{n}{2}(1-\delta )^{-1}\right] +1, \end{aligned}$$

(1.8)

then a(x, y, D) is bounded on $L^1(\mathbb {T}^n).$

Same as in Theorem 1.3, symbols considered in Theorem III does not require any regularity condition on the Fourier variable. It is important to mention that there exists a connection between the $L^p$ boundedness of periodic operators and its continuity on Besov spaces. This relation has been studied by the author on general compact Lie groups in [6, Section 3]. Although some results in this paper consider the $L^p$-boundedness of pseudo-differential operators on the torus (for $1\le p<\infty$), this problem has been addressed on general compact Lie groups in the references [9, 13] for all $1<p<\infty .$ Finally, we refer the reader to the references [14,15,16, 21, 30] for other properties of periodic operators on $L^p$-spaces.

2 Preliminaries

We use the standard notation of pseudo-differential operators (see [11, 17, 18, 23, 29]). The Schwartz space $\mathcal {S}(\mathbb {Z}^n)$ denote the space of functions $\phi :\mathbb {Z}^n\rightarrow \mathbb {C}$ such that

$$\begin{aligned} \forall M\in \mathbb {R}, \exists C_{M}>0,\quad |\phi (\xi )|\le C_{M}\langle \xi \rangle ^M, \end{aligned}$$

(2.1)

where $\langle \xi \rangle =(1+|\xi |^2)^{\frac{1}{2}}.$ The toroidal Fourier transform is defined for any $f\in C^{\infty }(\mathbb {T}^n)$ by $\widehat{f}(\xi )=\int _{}e^{-i2\pi \langle x,\xi \rangle }f(x)dx,\,\,\xi \in \mathbb {Z}^n.$ The inversion formula is given by $f(x)=\sum _{}e^{i2\pi \langle x,\xi \rangle }\widehat{u}(\xi ),\,\,x\in \mathbb {T}^n.$ The periodic Hörmander class $S^m_{\rho ,\delta }(\mathbb {T}^n\times \mathbb {R}^n), \,\, 0\le \rho ,\delta \le 1,$ consists of those functions $a(x,\xi )$ which are smooth in $(x,\xi )\in \mathbb {T}^n\times \mathbb {R}^n$ and which satisfy toroidal symbols inequalities

$$\begin{aligned} \left| \partial ^{\beta }_{x}\partial ^{\alpha }_{\xi }a(x,\xi )\right| \le C_{\alpha ,\beta }\langle \xi \rangle ^{m-\rho |\alpha |+\delta |\beta |}. \end{aligned}$$

(2.2)

Symbols in $S^m_{\rho ,\delta }(\mathbb {T}^n\times \mathbb {R}^n)$ are symbols in $S^m_{\rho ,\delta }(\mathbb {R}^n\times \mathbb {R}^n)$ (see [17, 23]) of order m which are 1-periodic in x. If $a(x,\xi )\in S^{m}_{\rho ,\delta }(\mathbb {T}^n\times \mathbb {R}^n),$ the corresponding pseudo-differential operator is defined by

$$\begin{aligned} a(x,D)u(x)=\int _{\mathbb {T}^n}\int _{\mathbb {R}^n}e^{i2\pi \langle x-y,\xi \rangle }a(x,\xi )u(y)d\xi dy,\quad u\in C^{\infty }(\mathbb {T}^n). \end{aligned}$$

(2.3)

The set $S^m_{\rho ,\delta }(\mathbb {T}^n\times \mathbb {Z}^n),\, 0\le \rho ,\delta \le 1,$ consists of those functions $a(x, \xi )$ which are smooth in x for all $\xi \in \mathbb {Z}^n$ and which satisfy

$$\begin{aligned} \forall \alpha ,\beta \in \mathbb {N}^n,\exists C_{\alpha ,\beta }>0,\quad \left| \Delta ^{\alpha }_{\xi }\partial ^{\beta }_{x}a(x,\xi )\right| \le C_{\alpha ,\beta }\langle \xi \rangle ^{m-\rho |\alpha |+\delta |\beta |}. \end{aligned}$$

(2.4)

The operator $\Delta _\xi ^\alpha$ in (2.4) is the difference operator which is defined as follows. First, if $f:\mathbb {Z}^n\rightarrow \mathbb {C}$ is a discrete function and $(e_j)_{1\le j\le n}$ is the canonical basis of $\mathbb {R}^n,$

$$\begin{aligned} (\Delta _{\xi _{j}} f)(\xi )=f(\xi +e_{j})-f(\xi ). \end{aligned}$$

(2.5)

If $k\in \mathbb {N},$ denote by $\Delta ^k_{\xi _{j}}$ the composition of $\Delta _{\xi _{j}}$ with itself k-times. Finally, if $\alpha \in \mathbb {N}^n,$ $\Delta ^{\alpha }_{\xi }= \Delta ^{\alpha _1}_{\xi _{1}}\cdots \Delta ^{\alpha _n}_{\xi _{n}}.$ The toroidal operator (or periodic operator) with symbol $a(x,\xi )$ is defined as

$$\begin{aligned} a(x,D)u(x)=\sum _{\xi \in \mathbb {Z}^n}e^{i 2\pi \langle x,\xi \rangle }a(x,\xi )\widehat{u}(\xi ),\quad u\in C^{\infty }(\mathbb {T}^n). \end{aligned}$$

(2.6)

There exists a process to interpolate the second argument of symbols on $\mathbb {T}^n\times \mathbb {Z}^n$ in a smooth way to get a symbol defined on $\mathbb {T}^n\times \mathbb {R}^n.$

Proposition 2.1

Let $0\le \delta \le 1,$ $0< \rho \le 1.$ The symbol $a\in S^m_{\rho ,\delta }(\mathbb {T}^n\times \mathbb {Z}^n)$ if only if there exists a Euclidean symbol $a'\in S^m_{\rho ,\delta }(\mathbb {T}^n\times \mathbb {R}^n)$ such that $a=a'|_{\mathbb {T}^n\times \mathbb {Z}^n}.$

Proof

The proof can be found in [19, 23]. $\square$

It is a non trivial fact, however, that the definition of pseudo-differential operator on a torus given by Agranovich (Eq. 2.6) and Hörmander (Eq. 2.3) are equivalent. McLean (see [19]) prove this for all the Hörmander classes $S^m_{\rho ,\delta }(\mathbb {T}^n\times \mathbb {Z}^n).$ A different proof to this fact can be found in [23], Corollary 4.6.13.

Proposition 2.2

(Equality of Operators Classes). For $0\le \delta \le 1,$ $0<\rho \le 1$ we have $\Psi ^{m}_{\rho ,\delta }(\mathbb {T}^n\times \mathbb {Z}^n)=\Psi ^{m}_{\rho ,\delta }(\mathbb {T}^n\times \mathbb {R}^n).$

A look at the proof (based in Theorem 4.5.3 of [23]) of the Proposition 2.2 shows us that a more general version is still valid for symbols with limited regularity as follows (see Corollary 4.5.7 of [23]):

Corollary 2.3

Let $0\le \delta \le 1,$ $0\le \rho <1.$ Let $a:\mathbb {T}^n\times \mathbb {R}^n\rightarrow \mathbb {C}$ satisfying $(\rho ,\delta )$-inequalities for $|\alpha |\le N_{1}$ and $|\beta |\le N_{2}$. Then the restriction $\tilde{a}=a|_{\mathbb {T}^n\times \mathbb {Z}^n}$ satisfies $(\rho ,\delta )$-estimates for $|\alpha |\le N_{1}$ and $|\beta |\le N_{2}.$ The converse holds true, i.e., if every symbol on $\mathbb {T}^n\times \mathbb {Z}^n$ satisfying $(\rho ,\delta )$-inequalities (as in (2.4)) is the restriction of a symbol on $\mathbb {T}^n\times \mathbb {R}^n$ satisfying $(\rho ,\delta )$ inequalities as in (2.2).

Let us denote $\Psi ^{m}_{\rho ,\delta ,N_1,N_2}(\mathbb {T}^n\times \mathbb {Z}^n)$ to the set of operators associated to symbols satisfying (2.4) for all $|\alpha |\le N_{1}$ and $|\beta |\le N_2,$ and $\Psi ^{m}_{\rho ,\delta ,N_1,N_2}(\mathbb {T}^n\times \mathbb {R}^n)$ defined similarly. Then we have (see Theorem 2.14 of [8]):

Proposition 2.4

(Equality of Operators Classes). For $0\le \delta \le 1,$ $0<\rho \le 1$ we have $\Psi ^{m}_{\rho ,\delta ,N_1,N_2}(\mathbb {T}^n\times \mathbb {Z}^n)=\Psi ^{m}_{\rho ,\delta ,N_1,N_2}(\mathbb {T}^n\times \mathbb {R}^n).$

The toroidal calculus is closed under adjoint operators. The corresponding announcement is the following.

Proposition 2.5

Let $0\le \delta <\rho \le 1.$ Let a(x, D) be a operator with symbol $a(X,\xi )\in S^m_{\rho ,\delta }(\mathbb {T}^n\times \mathbb {Z}^n).$ Then, the adjoint $a^{*}(x,D)$ of a(x, D),  has symbol $a^{*}(x,\xi )\in S^m_{\rho ,\delta }(\mathbb {T}^n\times \mathbb {Z}^n).$ The symbol $a^{*}(x,\xi )$ has the following asymptotic expansion:

$$\begin{aligned} \sigma ^{*}(x,\xi )\approx \sum _{\alpha \ge 0}\frac{1}{\alpha !}\Delta ^{\alpha }_{\xi }D^{(\alpha )}_{x}\overline{\sigma (x,\xi )}. \end{aligned}$$

(2.7)

Moreover, if $n_{0}\in \mathbb {N},$ then

$$\begin{aligned} \sigma ^{*}(x,\xi )- \sum _{|\alpha |<n_{0} }\frac{1}{\alpha !}\Delta ^{\alpha }_{\xi }D^{(\alpha )}_{x}\overline{\sigma (x,\xi )}\,\,\, \in S^{m-n_{0}(\rho -\delta )}_{\rho ,\delta }(\mathbb {T}^n\times \mathbb {Z}^n). \end{aligned}$$

(2.8)

In order to establish our result on positive operators, we introduce amplitude operators. The periodic amplitudes are functions $a(x,y,\xi )$ defined on $\mathbb {T}^n\times \mathbb {T}^n\times \mathbb {Z}^n.$ The corresponding amplitude operators are defined as

$$\begin{aligned} a(x,y,D)u(x)=\int \limits _{\mathbb {T}^n\times \mathbb {R}^n} e^{i2\pi \langle x-y,\xi \rangle }a(x,y,\xi )u(y)dy\,d\xi . \end{aligned}$$

(2.9)

If the symbol depends only on $(x,\xi )$-variables then $p(x,y,D)=p(x,D).$ Moreover, if a symbol $\sigma (x,\xi )=\sigma (\xi )$ depends only on the Fourier variable $\xi ,$ the corresponding pseudo-differential operator $\sigma (x,D)=\sigma (D)$ is called a Fourier multiplier. An instrumental result on Fourier multipliers in the proof of our main results is the following: (see, Theorem 3.8 of Stein [27]).

Proposition 2.6

Suppose $1\le p\le \infty$ and $T_\sigma$ be a Fourier multiplier on $\mathbb {R}^n$ with symbol $\sigma (\xi )$. If $\sigma (\xi )$ is continuous at each point of $\mathbb {Z}^n$ then the periodic operator defined by

$$\begin{aligned} \sigma (D)f(x)=\sum _{\xi \in \mathbb {Z}^n}e^{i2\pi \langle x,\xi \rangle }\sigma (\xi )\widehat{u}(\xi ), \end{aligned}$$

(2.10)

is a bounded operator from $L^p(\mathbb {T}^n)$ into $L^p(\mathbb {T}^n).$

The following results will help clarify the nature of the conditions that will be imposed on periodic symbols in order to obtain $L^p$ theorems for periodic operators with symbols of limited regularity.

Proposition 2.7

Let $0\le \rho \le 1$ and $0<\varepsilon \le 1,$ and suppose that the symbol $\sigma (x,\xi )$ on $\mathbb {R}^n\times \mathbb {R}^n$ satisfies

$$\begin{aligned} \left| \partial _{\xi }^{\alpha }\sigma (x,\xi )\right| \le C\omega (\langle \xi \rangle ^{-\varepsilon } )\langle \xi \rangle ^{-\frac{n}{2}(1-\rho )-\rho |\alpha |},\quad |\alpha |\le \left[ \frac{n}{2}\right] +1, \end{aligned}$$

(2.11)

where $\omega$ is a non-decreasing, bounded and non-negative function on $[0,\infty )$ satisfying

$$\begin{aligned} \int _{0}^{1}\omega (t)t^{-1}dt <\infty . \end{aligned}$$

Then $T_{\sigma }$ is a bounded operator on $L^p(\mathbb {R}^n)$ for all $2\le p\le \infty .$

Proof

See Theorem 4.4 and Corollary 4.2 of [2]. $\square$

The following is a version of the Fefferman theorem but symbols are considered with limited smoothness.

Proposition 2.8

Let $2\le p<\infty$ and $0\le \delta \le \rho \le 1,$ $\delta <1.$ Let $\sigma (x,\xi )$ be a symbol satisfying

$$\begin{aligned} \left| \partial _{x}^{\beta }\partial _{\xi }^{\alpha }\sigma (x,\xi )\right| \le C\langle \xi \rangle ^{-m_{p}-\rho |\alpha |+\delta |\beta |},\quad |\alpha |,|\beta |\le \left[ \frac{n}{2}\right] +1, \end{aligned}$$

(2.12)

where $m_{p}=n(1-\rho )\left| \frac{1}{p}-\frac{1}{2}\right|$. Then $T_\sigma :L^p(\mathbb {R}^n)\rightarrow L^p(\mathbb {R}^n)$ is a bounded operator.

Proof

See Theorem 5.1 and Corollary 5.1 of [2]. $\square$

The following theorem is the particular case of one proved in [25] for compact Lie groups.

Theorem 2.9

Let $k>\frac{n}{2}$ be an even integer. If $\sigma (\xi )$ is a periodic symbol on $\mathbb {Z}^n$ satisfying

$$\begin{aligned} \left| \Delta ^{\alpha }_{\xi }\sigma (\xi )\right| \le C_{\alpha }\langle \xi \rangle ^{-|\alpha |},\quad |\alpha |\le k, \end{aligned}$$

(2.13)

then, the corresponding periodic operator $\sigma (D)$ is of weak type (1,1) and $L^p$-bounded for all $1<p<\infty .$

Also, weak(1,1) boundedness of periodic operators has been considered by the author in [5]. We end this section with the following result proved in [9]. Although, the original theorem is valid for general compact Lie groups, we present the periodic version for simplicity.

Theorem 2.10

Let $0\le \delta ,\rho \le 1$ and $1<p<\infty .$ Denote by $\kappa$ the smallest even integer larger than $\frac{n}{2}.$ Let $\sigma (x,D)$ be a pseudo-differential operator with symbol $\sigma$ satisfying

$$\begin{aligned} \left| \partial _{x}^{\beta }\Delta _{\xi }^{\alpha }\sigma (x,\xi )\right| \le C\langle \xi \rangle ^{-m_0-\rho |\alpha |+\delta |\beta |}, \end{aligned}$$

(2.14)

for all $|\alpha |\le \kappa$ and $|\beta |\le \left[ \frac{n}{p}\right] +1,$ where $m_{0}:=\kappa (1-\rho ) |1/p-1/2|+\delta \left( \left[ \frac{n}{p}\right] +1\right)$. Then $\sigma (x,D)$ extends to a bounded operator on $L^p(\mathbb {T}^n)$ for all $1<p<\infty .$

3 Main results-proofs

In this section we prove our main results. we discuss that conditions on the periodic symbol $\sigma (x,\xi )$ guarantee the $L^p$-boundedness of the corresponding pseudo-differential operator.

Lemma 3.1

Let $0<\varepsilon \le 1,$ $0\le \rho \le 1,$ $1\le p\le \infty$ and $\omega :[0,\infty )\rightarrow [0,\infty )$ be a non-decreasing function such that

$$\begin{aligned} \int _{0}^1\omega (t)t^{-1}dt<\infty . \end{aligned}$$

(3.1)

If $\sigma _{1}: \mathbb {R}^n\rightarrow \mathbb {C}$ is a symbol satisfying

$$\begin{aligned} \left| \partial ^{\alpha }_{\xi }\sigma _{1}(\xi )\right| \le C\omega (\langle \xi \rangle ^{-\varepsilon })\langle \xi \rangle ^{-\frac{n}{2}(1-\rho )-\rho |\alpha |} \end{aligned}$$

(3.2)

for all $\alpha$ with $|\alpha |\le \left[ \frac{n}{2}\right] +1,$ then the corresponding periodic operator $\sigma (D)$ with symbol $\sigma (\xi )=\sigma _{1}(\xi )|_{ \mathbb {Z}^n}$ is a bounded operator on $L^p(\mathbb {T}^n).$

Proof

Proposition 2.7 provides the $L^p(\mathbb {R}^n)$-boundedness of $T_{\sigma _1}$. Now, by Proposition 2.6, the pseudo-differential operator with symbol $\sigma (\xi )$ is $L^{p}(\mathbb {T}^n)$-bounded. $\square$

Theorem 3.2

Let $0<\varepsilon \le 1,$ $0\le \rho \le 1,$ $1\le p<\infty$ and $\omega :[0,\infty )\rightarrow [0,\infty )$ be a bounded and non-decreasing function such that

$$\begin{aligned} \int _{0}^1\omega (t)t^{-1}dt<\infty . \end{aligned}$$

(3.3)

If $\sigma :\mathbb {T}^n\times \mathbb {R}^n\rightarrow \mathbb {C}$ is a symbol satisfying

$$\begin{aligned} \left| \partial ^{\alpha }_{\xi } \partial _{x}^\beta \sigma (x,\xi )\right| \le C\omega (\langle \xi \rangle ^{-\varepsilon })\langle \xi \rangle ^{-\frac{n}{2}(1-\rho )-\rho |\alpha |} \end{aligned}$$

(3.4)

for $|\alpha |\le \left[ \frac{n}{2}\right] +1,$ $|\beta |\le \left[ \frac{n}{p}\right] +1$, then the corresponding periodic operator $\sigma (x,D)$ is a bounded linear operator on $L^p(\mathbb {T}^n)$.

Proof

Let us consider the Schwartz kernel K(x, z) of $\sigma (x,D)$ which is given by $K(x,y)=r(x-y,x)$ where $r(x,z)=\int _{\mathbb {R}^n}e^{i2\pi \langle x,\xi \rangle }\sigma (z,\xi ),$ is understood in the distributional sense. For every $z\in \mathbb {T}^n$ fixed, $r(z)(\cdot )=r(\cdot ,z)$ is a distribution and the map $f\mapsto f*r(\cdot ,z)=f*r(z)(\cdot )$ is a pseudo-differential operator with symbol $\sigma _{z}:\xi \mapsto \sigma (z,\xi ).$ If $x\in \mathbb {T}^n$ and $f\in C^{\infty }(\mathbb {T}^n),$ $\sigma (x,D)f(x)=(f*r(x))(x).$ Moreover, for any $\beta \in \mathbb {N}^n$ with $|\beta |\le \left[ \frac{n}{p}\right] +1,$ the pseudo-differential operator $f*\left[ \partial ^{\beta }_{x}r(\cdot ,x)\right] |_{x=z}:=f*\left[ \partial ^{\beta }_{z}r(\cdot ,z)\right]$ is a pseudo-differential operator with symbol $\left[ \partial ^{\beta }_{x}\sigma (\cdot ,x)\right] |_{x=z}:=\partial ^{\beta }_{z}\sigma (z,\cdot )$. By Lemma 3.1, every pseudo-differential operator $\sigma _{z,\beta }(D)$ with symbol $\sigma _{z,\beta }(\xi )=\partial ^{\beta }_{z}\sigma (z,\xi )$ is $L^p(\mathbb {T}^n)$-bounded for $1\le p< \infty$. Now, by the Sobolev embedding theorem, for $1\le p<\infty ,$

$$\begin{aligned} |\sigma _(x,D)f(x)|^p \le \sup _{z\in \mathbb {T}^n} |(f*r(z))(x)|^p \le \sum _{|\beta |\le \left[ \frac{n}{p}\right] +1}\int _{\mathbb {T}^n}\left| \left( f*\partial ^{\beta }_{z}r(z)\right) (x)\right| ^pdz \end{aligned}$$

Hence, by application of the Fubini theorem we get

$$\begin{aligned} \Vert \sigma (x,D)f \Vert ^{p}_{L^p(\mathbb {T}^n)}&\le C^p \sum _{|\beta |\le \left[ \frac{n}{p}\right] +1}\int _{\mathbb {T}^n}\int _{\mathbb {T}^n}\left| \left( f*\partial ^{\beta }_{z}r(z)\right) (x)\right| ^pdzdx\\&\le C^p\sum _{|\beta |\le \left[ \frac{n}{p}\right] +1}\int _{\mathbb {T}^n}\int _{\mathbb {T}^n}\left| \left( f*\partial ^{\beta }_{z}r(z)\right) (x)\right| ^pdxdz\\&\le C^p \sum _{|\beta |\le \left[ \frac{n}{p}\right] +1 }\sup _{z\in \mathbb {T}^n} \int _{\mathbb {T}^n}\left| \left( f*\partial ^{\beta }_{z}r(z)\right) (x)\right| ^pdx\\&=C^p\sum _{|\beta |\le \left[ \frac{n}{p}\right] +1 }\sup _{z\in \mathbb {T}^n}\Vert \sigma _{z,\beta }(D)f \Vert ^p_{L^p(\mathbb {T}^n)} \\&\le C^p\left( \sum _{|\beta |\le \left[ \frac{n}{p}\right] +1 }\sup _{z\in \mathbb {T}^n}\Vert \sigma _{z,\beta }(D) \Vert ^p_{B(L^p,L^p)} \right) \Vert f\Vert ^p_{L^p(\mathbb {T}^n)}. \end{aligned}$$

Thus,

$$\begin{aligned} \Vert \sigma (x,D)f \Vert _{L^p(\mathbb {T}^n)}\le C \left( \sum _{|\beta |\le \left[ \frac{n}{p}\right] +1 }\sup _{z\in \mathbb {T}^n}\Vert \sigma _{z,\beta }(D) \Vert ^p_{B(L^p,L^p)} \right) ^{\frac{1}{p}}\Vert f \Vert _{L^p(\mathbb {T}^n)}. \end{aligned}$$

$\square$

Lemma 3.3

Let $k\in \mathbb {R}$, $\varepsilon >0$, $\omega$ be a function as in Proposition 2.7 and $\sigma :\mathbb {T}^n\times \mathbb {R}^n\rightarrow \mathbb {C}$ be a symbol satisfying

$$\begin{aligned} \left| \partial ^{\alpha }_{\xi } \partial _{x}^\beta \sigma (x,\xi )\right| \le C\omega (\langle \xi \rangle ^{-\varepsilon })\langle \xi \rangle ^{k}, \end{aligned}$$

(3.5)

for all $|\alpha |\le N_1$ and $|\beta |\le N_2.$ Let $\tilde{a}(x,\xi ):=\sigma (x,\xi )|_{\mathbb {T}^{n}\times \mathbb {Z}^n}.$ Then

$$\begin{aligned} \left| \Delta ^{\alpha }_{\xi } \partial _{x}^\beta \tilde{\sigma }(x,\xi )\right| \le C'\omega (\langle \xi \rangle ^{-\varepsilon })\langle \xi \rangle ^{k}, \end{aligned}$$

(3.6)

for all $|\alpha |\le N_1$ and $|\beta |\le N_2.$ Moreover, every symbol satisfying (3.6) is the restriction of a symbol on $\mathbb {T}^n\times \mathbb {R}^n$ satisfying (3.5).

Proof

Let us consider $\sigma$ as in (3.5). By the mean value theorem, if $|\alpha |=1$ we have

$$\begin{aligned} \Delta _{\xi }^{\alpha }\partial ^{\beta }_{x}\tilde{\sigma }(x,\xi ) =\Delta _{\xi }^{\alpha }\partial ^{\beta }_{x}{\sigma }(x,\xi ) =\partial _{\xi }^{\alpha }\partial ^{\beta }_{x}{\sigma }(x,\xi )|_{\xi =\eta } \end{aligned}$$

where $\eta$ is on the line $[\xi ,\xi +\alpha ].$ For a general multi-index $\alpha \in \mathbb {N}^n,$ it can proved by induction that

$$\begin{aligned} \Delta _{\xi }^{\alpha }\partial ^{\beta }_{x}\tilde{\sigma }(x,\xi ) =\partial _{\xi }^{\alpha }\partial ^{\beta }_{x}{\sigma }(x,\xi )|_{\xi =\eta } \end{aligned}$$

for some $\eta \in Q:=[\xi _{1}\times \xi _{1}+\alpha _1]\times \cdots [\xi _{n}\times \xi _{n}+\alpha _n].$ Hence, we have

$$\begin{aligned} \left| \Delta _{\xi }^{\alpha }\partial _{x}^{\beta }\tilde{\sigma }(x,\xi )\right|&=\left| \partial _{\xi }^{\alpha }\partial ^{\beta }_{x}{\sigma }(x,\xi )|_{\xi =\eta \in Q}\right| \\&\le C\omega (\langle \eta \rangle ^{-\varepsilon })\langle \eta \rangle ^{k}\le C'\omega (\langle \xi \rangle ^{-\varepsilon })\langle \xi \rangle ^{k}. \end{aligned}$$

So, we have proved the first part of the theorem. Now, let us consider a symbol $\tilde{\sigma }$ on $\mathbb {T}^n\times \mathbb {Z}^n$ satisfying (3.6). Let us consider $\theta$ as in Lemma 4.5.1 of [23]. Define the symbol $\sigma$ on $\mathbb {T}^n\times \mathbb {R}^n$ by

$$\begin{aligned} \sigma (x,\xi )=\sum _{\eta \in \mathbb {Z}^n}(\mathcal {F}_{\mathbb {R}^n}\theta ) (\xi -\eta )\tilde{\sigma }(x,\eta ). \end{aligned}$$

(3.7)

Same as in the proof of Theorem 4.5.3 of [23], pag. 359, we have

$$\begin{aligned} \left| \partial _{x}^{\beta }\partial _{\xi }^{\alpha }\sigma (x,\xi )\right| =\left| \sum _{\eta \in \mathbb {Z}^n}\phi _{\alpha }(\xi -\eta )\partial _{x}^{\beta } \Delta _{\xi }^{\alpha }\tilde{\sigma }(x,\eta ) \right| \end{aligned}$$

where every $\phi _{\alpha }\in \mathcal {S}(\mathbb {R}^n)$ is a function as in Lemma 4.5.1 of [23]. So, we obtain

$$\begin{aligned} \left| \partial _{x}^{\beta }\partial _{\xi }^{\alpha }\sigma (x,\xi )\right| \le \sum _{\eta \in \mathbb {Z}^n}\left| \phi _{\alpha }(\eta )\omega (\langle \xi -\eta \rangle ^{-\varepsilon })\langle \xi -\eta \rangle ^{k}\right| \end{aligned}$$

(3.8)

Since $\omega$ is bounded, for some $M>0,$ and all $\alpha \ge 1$ we have

$$\begin{aligned} |\omega (\alpha t)|\le \alpha \omega (t),\quad t>M. \end{aligned}$$

(3.9)

By (3.9), the Peetre inequality and using that $\omega$ is increasing we have

$$\begin{aligned} \left| \partial _{x}^{\beta }\partial _{\xi }^{\alpha }\sigma (x,\xi )\right|&\le \sum _{\eta \in \mathbb {Z}^n}\left| \phi _{\alpha }(\eta ) \omega (\langle \xi -\eta \rangle ^{-\varepsilon })\langle \xi -\eta \rangle ^{k}\right| \\&\le \sum _{\eta \in \mathbb {Z}^n}\left| \phi _{\alpha }(\eta ) \omega (\langle \xi \rangle ^{-\varepsilon }\langle \eta \rangle ^{\varepsilon }) \langle \xi \rangle ^{k}\langle \eta \rangle ^{k}\right| \\&\le C \omega (\langle \xi \rangle ^{-\varepsilon })\langle \xi \rangle ^{k}\sum _{\eta \in \mathbb {Z}^n}\left| \phi _{\alpha }(\eta )\langle \eta \rangle ^{\varepsilon +k}\right| . \end{aligned}$$

Since every $\phi _{\alpha }$ is a function in the Schwartz class we obtain

$$\begin{aligned} \left| \partial _{x}^{\beta }\partial _{\xi }^{\alpha }\sigma (x,\xi )\right| \le C'\omega (\langle \xi \rangle ^{-\varepsilon })\langle \xi \rangle ^{k}. \end{aligned}$$

(3.10)

So, we end the proof. $\square$

As a consequence of the results above we obtain the following result.

Theorem IA

Let $0<\varepsilon \le 1,$ $0\le \rho \le 1$ and let $\omega :[0,\infty )\rightarrow [0,\infty )$ be a non-decreasing and bounded function such that

$$\begin{aligned} \int _{0}^1\omega (t)t^{-1}dt<\infty . \end{aligned}$$

(3.11)

Let $\sigma (x,\xi ):\mathbb {T}^n\times \mathbb {Z}^n\rightarrow \mathbb {C}$ be a symbol satisfying

$$\begin{aligned} \left| \partial _{x}^{\beta }\Delta ^{\alpha }_{\xi }\sigma (x,\xi )\right| \le C\omega (\langle \xi \rangle ^{-\varepsilon })\langle \xi \rangle ^{-\frac{n}{2}(1-\rho )-\rho |\alpha |} \end{aligned}$$

(3.12)

for $\alpha ,\beta \in \mathbb {N}^n$ such that $|\alpha |\le \left[ \frac{n}{2}\right] +1$, $|\beta |\le [\frac{n}{p}]+1$, then the corresponding periodic operator $\sigma (x,D)$ extends to a bounded operator on $L^p(\mathbb {T}^n)$ for all $1\le p<\infty .$

Proof

First, let us denote by $\Psi ^{m,\omega }_{\rho ,\delta , N_1,N_2}(\mathbb {T}^n\times \mathbb {R}^n)$ and $\Psi ^{m,\omega }_{\rho ,\delta , N_1,N_2}(\mathbb {T}^n\times \mathbb {Z}^n)$ to the set of operators with symbols satisfying (3.5) and (3.6) respectively. If we combine the Theorem 4.6.12 of [23] and Lemma 3.3 we obtain the equality of classes

$$\begin{aligned} \Psi ^{m,\omega }_{\rho ,\delta , N_1,N_2}(\mathbb {T}^n\times \mathbb {R}^n)=\Psi ^{m,\omega }_{\rho ,\delta , N_1,N_2}(\mathbb {T}^n\times \mathbb {Z}^n). \end{aligned}$$

(3.13)

Hence, in order to proof the boundedness of $\sigma (x,D)$ we only need to proof that

$$\begin{aligned} \Psi ^{-\frac{n}{2}(1-\rho ),\,\omega }_{\rho ,0, \left[ \frac{n}{2}\right] +1,\left[ \frac{n}{p}\right] +1}(\mathbb {T}^n\times \mathbb {R}^n)\subset \mathcal {L}(L^{p}(\mathbb {T}^n)), \end{aligned}$$

(3.14)

but, this fact has been proved in Theorem 3.2.

Lemma 3.4

Let $0\le \rho \le 1,$ and $1< p<\infty .$ Let $\sigma _{1}: \mathbb {R}^n\rightarrow \mathbb {C}$ be a symbol satisfying

$$\begin{aligned} \left| \partial ^{\alpha }_{\xi }\sigma _{1}(\xi )\right| \le C\langle \xi \rangle ^{-m_p-\rho |\alpha |} \end{aligned}$$

(3.15)

for all $\alpha$ and $\beta$ with $|\alpha |\le [\frac{n}{2}]+1,$ then the corresponding periodic operator $\sigma (D)$ with symbol $\sigma (\xi )=\sigma _{1}(\xi )|_{ \mathbb {Z}^n}$ is a bounded operator on $L^p(\mathbb {T}^n).$

Proof

From Proposition 2.8, we deduce the $L^p(\mathbb {R}^n)$-boundedness of $T_{\sigma _1}.$ Finally, by Proposition 2.6, $\sigma (D)$ is a bounded operator on $L^{p}(\mathbb {T}^n).$

Theorem IB

Let $0< \rho \le 1,$ and $1< p<\infty .$ Let $\sigma :\mathbb {T}^n\times \mathbb {Z}^n\rightarrow \mathbb {C}$ be a symbol satisfying

$$\begin{aligned} \left| \partial _{x}^{\beta }\Delta _{\xi }^{\alpha }\sigma (x,\xi )\right| \le C\langle \xi \rangle ^{-m_p-\rho |\alpha |}, \end{aligned}$$

(3.16)

for all $\alpha$ and $\beta$ with $|\alpha |\le \left[ \frac{n}{2}\right] +1,$ $|\beta |\le \left[ \frac{n}{p}\right] +1,$ then $\sigma (x,D)$ extends to a bounded operator on $L^p(\mathbb {T}^n).$

Proof

By Proposition 2.4 we only need to proof that

$$\begin{aligned} \Psi ^{-m_p}_{\rho ,0,\left[ \frac{n}{2}\right] +1,\left[ \frac{n}{p}\right] +1}(\mathbb {T}^n\times \mathbb {R}^n)\subset \mathcal {L}(L^{p}(\mathbb {T}^n)),\quad 1<p<\infty . \end{aligned}$$

(3.17)

Hence, let us consider a symbol $\sigma _{1}$ on $\mathbb {T}^n\times \mathbb {R}^n$ satisfying

$$\begin{aligned} \left| \partial _{x}^\beta \partial ^{\alpha }_{\xi }\sigma _{1}(x,\xi )\right| \le C\langle \xi \rangle ^{-m_p-\rho |\alpha |}, \end{aligned}$$

(3.18)

for all $|\alpha |\le \left[ \frac{n}{2}\right] +1$, $|\beta |\le \left[ \frac{n}{p}\right] +1$. From Lemma 3.4, we obtain the $L^{p}$-boundedness of every operator ${\sigma _1}_{z,\beta }(D)$ with symbol ${\sigma _1}_{z,\beta }(\xi )=\left( \partial _{x}^{\beta }\sigma _{1}(x,\xi )\right) |_{x=z},$ $z\in \mathbb {T}^n.$ As in the proof of Theorem IA, an application of the Sobolev embedding theorem gives

$$\begin{aligned} \Vert \sigma _{1}(x,D)f \Vert _{L^p(\mathbb {T}^n)}\le C \left( \sum _{|\beta |\le \left[ \frac{n}{p}\right] +1 }\sup _{z\in \mathbb {T}^n}\Vert {\sigma _1}_{z,\beta }(D) \Vert ^p_{B(L^p,L^p)} \right) ^{\frac{1}{p}}\Vert f \Vert _{L^p(\mathbb {T}^n)}. \end{aligned}$$

Hence $\sigma _1(x,D)$ is $L^p$-bounded. So we end the proof. $\square$

Remark 3.5

We observe that conditions on the number of derivatives in Theorems 2.10 and IB agree, but the behavior of the symbol derivatives in every theorem is different. In fact, (3.16) can be written as

$$\begin{aligned} \left| \partial _{x}^{\beta }\Delta _{\xi }^{\alpha }\sigma (x,\xi )\right| \le C\langle \xi \rangle ^{-n(1-\rho )|1/p-1/2|-\rho |\alpha |}, \end{aligned}$$

(3.19)

in contrast with (2.14):

$$\begin{aligned} \left| \partial _{x}^{\beta }\Delta _{\xi }^{\alpha }\sigma (x,\xi )\right| \le C\langle \xi \rangle ^{-\kappa (1-\rho )|1/p-1/2|-\rho |\alpha |+\delta \left( |\beta |-\left( \left[ \frac{n}{p}\right] +1\right) \right) }, \end{aligned}$$

(3.20)

where $|\beta |-\left( \left[ \frac{n}{p}\right] +1\right) \le 0$.

Lemma 3.6

(Kolmogorov’s lemma) Given an operator S which is weak(1,1), $0<v<1,$ and a set E of finite measure, there exists a $C>0$ such that

$$\begin{aligned} \int _{E}|Sf(x)|^v\,dx\le C\mu (E)^{1-v}\Vert f\Vert ^v_{L^1(\mathbb {R}^n)}. \end{aligned}$$

(3.21)

Proof

For the proof of this result, see Lemma 5.16 of [10]. $\square$

Theorem IC

Let $k:=\left[ \frac{n}{2}\right] +1$. If $\sigma (\xi )$ is a periodic symbol on $\mathbb {Z}^n$ satisfying

$$\begin{aligned} \left| \Delta ^{\alpha }_{\xi }\sigma (\xi )\right| \le C_{\alpha }\langle \xi \rangle ^{-|\alpha |},\quad |\alpha |\le k, \end{aligned}$$

(3.22)

then, the corresponding periodic operator $\sigma (D)$ is bounded from $L^1(\mathbb {T}^n)$ into $L^p(\mathbb {T}^n)$ for all $0<p<1.$

Proof

By Theorem 2.4 we only need to prove that an operator $\sigma _{1}(D)$ with symbol $\sigma _{1}$ on $\mathbb {T}^n\times \mathbb {R}^n$ satisfying

$$\begin{aligned} \left| \partial ^{\alpha }_{\xi }\sigma _1(\xi )\right| \le C_{\alpha }\langle \xi \rangle ^{-|\alpha |},\quad |\alpha |\le k, \end{aligned}$$

(3.23)

is a bounded operator from $L^1(\mathbb {T}^n)$ into $L^p(\mathbb {T}^n)$ for all $0<p<1.$ The multiplier $\sigma (D)$ is of weak type (1,1) on $\mathbb {R}^n$ (see [26, 27]). Now, by Kolmogorov’s lemma with $E=\mathbb {T}^n,$ for $f\in L^{p}(\mathbb {T}^n)$ (which can be considered as a function on $\mathbb {R}^n$ equal to zero in $\mathbb {R}^n-\mathbb {T}^n$) we have

$$\begin{aligned} \int _{\mathbb {T}^n}|\sigma _{1}(x,D)f(x)|^p\,dx\le C\Vert f\Vert ^p_{L^1(\mathbb {R}^n)}=C\Vert f\Vert ^p_{L^1(\mathbb {T}^n)} \end{aligned}$$

which proves the boundedness of $\sigma _{1}(x,D).$ So, we end the proof. $\square$

Lemma 3.7

Let $2\le p<\infty ,$ and $k=\left[ \frac{n}{2}\right] +1,$ let $p(x,\xi )$ be a symbol such that

$$\begin{aligned} \left| \partial _{x}^{\beta }p(x,\xi )\right| \le C_{k},\quad |\beta |\le k. \end{aligned}$$

(3.24)

Then $p(x,D):L^p(\mathbb {T}^n)\rightarrow L^q(\mathbb {T}^n)$ is a bounded operator, where $\frac{1}{p}+\frac{1}{q}=1.$

Proof

By the definition of periodic pseudo-differential operator, integration by parts and inversion formula, we have

$$\begin{aligned} \langle p(x,D_{x})u, g \rangle _{L^2(\mathbb {T}^n)}&=\int _{T^n}\sum _{\xi \in \mathbb {Z}^n}e^{i2\pi x\xi }p(x,\xi )\widehat{u}(\xi )g(x)dx\\&=\int _{T^n}\sum _{\xi \in \mathbb {Z}^n}\sum _{\eta \in \mathbb {Z}^n}e^{i2\pi x(\xi -\eta )}p(x,\xi )\widehat{u}(\xi )\overline{\widehat{g}(\eta )}dx\\&=\sum _{\xi \in \mathbb {Z}^n}\sum _{\eta \in \mathbb {Z}^n}\int _{T^n}e^{i2\pi x(\xi -\eta )}p(x,\xi )\widehat{u}(\xi )\overline{\widehat{g}(\eta )}dx\\&=\sum _{\xi \in \mathbb {Z}^n}\sum _{\eta \in \mathbb {Z}^n}\int _{T^n}\langle \xi -\eta \rangle ^{-2k}e^{i2\pi x(\xi -\eta )}L^{k}_{x}p(x,\xi )\widehat{u}(\xi )\overline{\widehat{g}(\eta )}dx, \end{aligned}$$

where $L^q_{x}=(I-\frac{1}{4\pi ^2}\mathcal {L}_{x})^q$ and $\mathcal {L}_{x}$ is the Laplacian in x-variables. Using the Young Inequality, we get

$$\begin{aligned} |\langle p(x,D_{x})u, g \rangle _{L^2(\mathbb {T}^n)}|&\le \sum _{\xi \in \mathbb {Z}^n}\sum _{\eta \in \mathbb {Z}^n}\langle \xi -\eta \rangle ^{-2k}\left| L^{k}_{x}p(x,\xi )||\widehat{u}(\xi )||\widehat{g}(\eta )\right| \\&\le \sum _{\xi \in \mathbb {Z}^n}\sum _{\eta \in \mathbb {Z}^n}\langle \xi -\eta \rangle ^{-2k}C_{k}\widehat{u}(\xi )||\widehat{g}(\eta )|\\&= \sum _{\xi \in \mathbb {Z}^n}|\widehat{u}(\xi )|\sum _{\eta \in \mathbb {Z}^n}\langle \xi -\eta \rangle ^{-2k}|\widehat{g}(\eta )|C_{k}\\&= \sum _{\xi \in \mathbb {Z}^n}C_{k}|\widehat{u}(\xi )|\langle \cdot \rangle ^{-2q}*|\widehat{g}(\cdot )|(\xi )C_{k}\\&\le C_{k} \Vert \widehat{u}\Vert _{L^2(\mathbb {Z}^n)} \Vert \langle \cdot \rangle ^{-2k}|\widehat{g}| \Vert _{L^2(\mathbb {Z}^n)}\\&\le C_{k}\Vert u\Vert _{L^2(\mathbb {T}^n)} \Vert \langle \cdot \rangle ^{-2k} \Vert _{L^1(\mathbb {Z})}\Vert {g} \Vert _{L^2(\mathbb {T}^n)} \end{aligned}$$

If $p\ge 2$ then the inclusion map $i:L^p(\mathbb {T}^n)\rightarrow L^2(\mathbb {T}^n)$ is continuous. So, for some $C>0$ we have $\Vert \cdot \Vert _{L^2(\mathbb {T}^n)}\le C\Vert \cdot \Vert _{L^p(\mathbb {T}^n)}.$ So, we get

$$\begin{aligned} |\langle p(x,D_{x})u, g \rangle _{L^2(\mathbb {T}^n)}| \le CC_{k} \Vert \langle \cdot \rangle ^{-2k} \Vert _{L^1(\mathbb {Z})} \Vert u \Vert _{L^p(\mathbb {T}^n)} \Vert g \Vert _{L^p(\mathbb {T}^n)}. \end{aligned}$$

Finally,

$$\begin{aligned} \Vert p(x,D_{x})u \Vert _{L^q(\mathbb {T}^n)}&=\sup \left\{ |\langle p(x,D_{x})u,g \rangle |:{\Vert g \Vert _{L^{p}(\mathbb {T}^n)}}\le 1 \right\} \\&\le CC_{k}\Vert \langle \cdot \rangle ^{-2k} \Vert _{L^1(\mathbb {Z})} \Vert u \Vert _{L^p(\mathbb {T}^n)}. \end{aligned}$$

$\square$

By using the previous lemma we can prove the following theorem on $(L^p,L^r)$-boundedness of periodic operators.

Theorem II

Let $2\le p<\infty .$ If $\sigma (x,\xi )$ satisfies the hypotheses of Theorem 1.4, then $\sigma (x,D):L^p(\mathbb {T}^n)\rightarrow L^r(\mathbb {T}^n)$ is a bounded linear operator for all $1<q\le r\le p<\infty ,$ where $\frac{1}{p}+\frac{1}{q}=1.$

Proof

Theorem 1.4 implies that, $\sigma (x,D):L^p(\mathbb {T}^n)\rightarrow L^p(\mathbb {T}^n)$ is a bounded operator for $1<p<\infty .$ That $\sigma (x,D):L^p(\mathbb {T}^n)\rightarrow L^q(\mathbb {T}^n)$ is a bounded operator is a consequence of Lemma 3.7. Now, by Riesz–Thorin interpolation theorem we deduce that $\sigma (x,D):L^p(\mathbb {T}^n)\rightarrow L^r(\mathbb {T}^n)$ is a bounded operator for all $q\le r\le p<\infty ,$ with $2\le p<\infty .$

In the following theorem we analyze the boundedness of periodic amplitude operators on $L^1.$

Theorem III

Let $0<\varepsilon ,\delta <1.$ If a(x, y, D) is a positive amplitude operator with symbol satisfying the following inequalities

$$\begin{aligned} \left| \partial _{x}^{\beta }\partial _{y}^{\alpha }a(x,y,\xi )\right| \le C_{\alpha ,\beta }\langle \xi \rangle ^{\delta |\beta |-\varepsilon |\alpha |},\quad |\alpha |,\quad |\beta |\le \left[ \frac{n}{2}(1-\delta )^{-1}\right] +1, \end{aligned}$$

(3.25)

then a(x, y, D) is bounded on $L^1(\mathbb {T}^n).$

Proof

If the operator a(x, y, D) is positive and $f\ge 0$ we have that

$$\begin{aligned} a(x,y,D)f\ge 0. \end{aligned}$$

Hence,

$$\begin{aligned} |a (x,y,D)f(x)|&=a(x,y,D)f(x)\\&=\int \limits _{\mathbb {T}^n \times \mathbb {R}^n}e^{i2\pi (x-y)\xi }a(x,y,\xi )f(y)d\xi \,dy\\&=\int \limits _{\mathbb {T}^n\times \mathbb {R}^n\times \mathbb {R}^n}e^{i2\pi (x-y)\xi }e^{i2\pi (\eta )y}a(x,y,\xi )d\xi \,dy\widehat{f}(\eta )d\eta \\&=\int \limits _{\mathbb {T}^n\times \mathbb {R}^n\times \mathbb {R}^n}e^{i2\pi (x)\xi }e^{i2\pi (\eta -\xi )y}a(x,y,\xi )d\xi \,dy\widehat{f}(\eta )d\eta . \end{aligned}$$

Clearly $\langle \xi \rangle ^{-2q}L^q_{x}e^{i2\pi x\xi }=e^{i2\pi x\xi },$ where $L^q_{x}=(I-\frac{1}{4\pi ^2}\mathcal {L}_{x})^q$ and $\mathcal {L}_{x}$ is the Laplacian in x-variables. Using integration by parts successively we obtain,

$$\begin{aligned}&\int \limits _{\mathbb {T}^n}| a(x,y,D)f(x)|dx\\&\quad =\int \limits _{\mathbb {T}^n\times \mathbb {T}^n \times \mathbb {R}^n\times \mathbb {R}^n}e^{i2\pi (x)\xi }e^{i2\pi (\eta -\xi )y}a(x,y,\xi )d\xi \,dy\widehat{f}(\eta )d\eta \,dx\\&\quad =\int \limits _{\mathbb {T}^n\times \mathbb {T}^n \times \mathbb {R}^n\times \mathbb {R}^n}\langle \xi \rangle ^{-2q}\langle \eta -\xi \rangle ^{-2q} e^{i2\pi (x)\xi }e^{i2\pi (\eta -\xi )y} L^q_{x}L^{q}_{y}a(x,y,\xi )d\xi \,dy\widehat{f}(\eta )d\eta \,dx \\&\quad \le \int \limits _{\mathbb {T}^n\times \mathbb {T}^n \times \mathbb {R}^n\times \mathbb {R}^n}\langle \xi \rangle ^{-2q}\langle \eta -\xi \rangle ^{-2q} |L^q_{x}L^{q}_{y}a(x,y,\xi )|d\xi \,dy|\widehat{f}(\eta )|d\eta \,dx\\&\quad \le \int \limits _{\mathbb {T}^n\times \mathbb {T}^n \times \mathbb {R}^n\times \mathbb {R}^n}\langle \xi \rangle ^{-2q}\langle \eta -\xi \rangle ^{-2q} C_{q}\langle \xi \rangle ^{\delta \cdot 2q} d\xi \,dy|\widehat{f}(\eta )|d\eta \,dx\\&\quad \le \int \limits _{\mathbb {T}^n\times \mathbb {T}^n \times \mathbb {R}^n\times \mathbb {R}^n}\langle \xi \rangle ^{2q(\delta -1)}\langle \eta -\xi \rangle ^{-2q} C_{q} d\xi \,dy\Vert f\Vert _{L^1(\mathbb {T}^n)} d\eta \,dx\\&\quad \le C \Vert f\Vert _{L^1(\mathbb {T}^n)}, \end{aligned}$$

where $C=\int \limits _{\mathbb {R}^n} \left( \langle \cdot \rangle ^{2q(\delta -1)}*\langle \cdot \rangle ^{-2q}\right) (\eta ) C_{q} d\eta .$ Clearly C is finite for $\frac{n}{2}(1-\delta )^{-1}<q\le \left[ \frac{n}{2}(1-\delta )^{-1}\right] +1$.

Our proof of the $L^1$-boundedness is for non-negative f,  but this is sufficient since an arbitrary real function can be decomposed into its positive and negative parts, and complex functions into its real and imaginary parts. $\square$

Remark 3.8

Positive operators in the form of Theorem IV arise with Bessel potentials of order m,  $m\in \mathbb {R}.$ We recall that for every $m\in \mathbb {R},$ the Bessel potential of order m,  denoted by $\langle D_{x}\rangle ^m$ is the pseudo-differential operator with symbol $\sigma _{m}(\xi )=\langle \xi \rangle ^m,$ $\xi \in \mathbb {Z}^n.$

Remark 3.9

There exists a connection between the $L^{p}$ boundedness of Fourier multipliers on compact Lie groups and its continuity on Besov spaces $B^r_{p,q}$. This fact was proved by the author in Theorem 1.2 of [7]. In fact, the Lie group structure of the torus $\mathbb {T}^n$ implies that every periodic Fourier multiplier (i.e. a periodic pseudo-differential operator with symbol depending only on the dual variable $\xi \in \mathbb {Z}^n$) bounded from $L^{p_{1}}$ into $L^{p_{2}}$ also is bounded from $B^{r}_{p_1,q}$ into $B^{r}_{p_2,q}$, $r\in \mathbb {R}$ and $0<q\le \infty$. So, restrictions of our results to the case of symbols $\sigma (\xi )$ associated to Fourier multipliers give the boundedness of these operators on every Besov space $B^r_{p,q}(\mathbb {T}^n)$, $1<p<\infty$.