1 Introduction

From 1950s which the modern techniques of analysis began to develop, Fourier analysis became one of the main branches. This area, particularly relates to the study of Fourier integral operators, pseudo-differential operators, generalized functions, and the class of symbols. In the microlocal analysis, usually symbols lie in a certain class of infinitely differentiable functions [5, 6]. But in the physics observations we often deal with functions which vary with respect to time and thus make nets of functions. Here, we deal with a class of symbols that vary in respect of time and are integrable with respect to a given measure introduced in [2]. Using this class of symbols we obtain generalized notion of supersymbols and supersingular pseudo-differential operators. In the sequel, suppose that \(N\) is a fixed natural number.

Definition 1.1

Let \( \psi (x,\theta )\in C^{\infty }({\mathbb {R}}^N\times {\mathbb {R}}^N \)). By \(\psi \) lies in \( S^m_{\rho ,\delta } \) we mean that for all \((\alpha ,\beta ,n)\in {\mathbb {Z}}^N_+\times {\mathbb {Z}}^N_+\times {\mathbb {Z}}_+\),

$$\begin{aligned} \Vert \psi :\alpha ,\beta ,n:S^m_{\rho ,\delta }\Vert :=\sup _{|x|\le n}\sup _{\theta \in {\mathbb {R}}^N} |\partial ^{\alpha }_{\theta } \partial ^{\beta }_{x}\psi (x,\theta )|\langle \theta \rangle ^ {-m+\rho |\alpha |-\delta |\beta |}<\infty , \end{aligned}$$

where \(\langle \theta \rangle =(1+|\theta |^2)^{\frac{1}{2}}\). When \(\rho =1 \) and \(\delta =0 \), we simply write \( S^m \) instead of \( S^m_{1,0} \).

Pseudo-differential operators (\(\psi \) dos) are generated from \(S^m_{\rho ,\delta }\) symbols by the formula

$$\begin{aligned} \begin{array}{c} \displaystyle \text {OP}\psi :\mathcal E'({\mathbb {R}}^N) \longrightarrow \mathcal D'({\mathbb {R}}^N),\\ \displaystyle f\longmapsto \int e^{ix\theta }\psi (x,\theta )\hat{f}(\theta )d\theta . \end{array} \end{aligned}$$

(Recall that \(f(\theta )=\int e^{-iy\theta }f(y)dy \) is the Fourier transform of \(f\)). A \(\psi \) do is continuous on Sobolev and Lebesgue spaces:

$$\begin{aligned} \begin{array}{c} \displaystyle \text {OP}:S^m\rightarrow {\mathcal {L}}(H^s_{comp}({\mathbb {R}}^N),H^{s-m}_{loc}({\mathbb {R}}^N)),\quad s\in {\mathbb {R}},\\ \displaystyle \text {OP}:S^0\rightarrow {\mathcal {L}}(L^p_{comp}({\mathbb {R}}^N),H^p_{loc}({\mathbb {R}}^N)),\quad 1/p\in (0,1). \end{array} \end{aligned}$$

Moreover, both of OP above are continuous [4], the former on all Freshet spaces \(S^m\), \(m\in {\mathbb {R}}\), with seminorms \(\Vert \psi :\alpha ,\beta ,n:S^m_{\rho ,\delta }\Vert \).

Definition 1.2

Let \(\sigma \) be a complex (possibly infinite) measure on \({\mathbb {R}}^N\). A map \(\psi {:}\,{\mathbb {R}}^N\rightarrow S^m_{\rho , \delta }\) for \(m\in {\mathbb {R}}\), is called a supersymbol if for all \((\alpha ,\beta ,n)\in {\mathbb {Z}}^N_+\times {\mathbb {Z}}^N_+\times {\mathbb {Z}}_+ \),

$$\begin{aligned} \int \Vert \psi (t):\alpha ,\beta ,n;S^m_{\rho ,\delta }\Vert d\sigma (t)<\infty . \end{aligned}$$

Then we write \(\psi \in SS^m_{\rho ,\delta }(\sigma )\), or \((\psi ,\sigma )\in SS^m_{\rho ,\delta }\). Each supersymbol together with the measure generates a supersingular pseudo-differential operator (abbr. s \(\psi \) do),

$$\begin{aligned} T(\psi ,\sigma ):=\int \text {OP}\psi (t)\circ Sh_td\sigma (t). \end{aligned}$$

(Here, \(Sh_t:f(x)\mapsto f(x-t)\) is the translation or shift operator).

As usual, notations \(\text {OP}SS^m_{\rho ,\delta }(\sigma )\) and \(\text {OP}SS^{-\infty }\) etc. stand for the spaces of operators generated by the corresponding spaces of supersymbols, i.e., \(SS^m_{\rho ,\delta }(\sigma )\) and \(SS^{-\infty }\equiv \cap _mSS^m_{\rho ,\delta }.\)

As a trivial case, when \(\sigma \) is the unit measure \(\delta (t)\) supported at the origin, \(T(\psi ,\sigma )\) is the pseudo-differential operator \(\text {OP}\psi (0)\).

Theorem 1.3

Each s \(\psi \) do \(T(\psi ,\sigma )\) is continuous on Sobolev and Lebesgue spaces;

$$\begin{aligned}&T(\psi ,\sigma ):H^s_{comp}({\mathbb {R}}^N)\rightarrow H^{s-m}_{loc}({\mathbb {R}}^N),\qquad \psi \in SS^m(\sigma ),\quad \in {\mathbb {R}},\\&T(\psi ,\sigma ):L^p_{comp}({\mathbb {R}}^N)\rightarrow L^p_{loc}({\mathbb {R}}^N),\qquad \psi \in SS^0(\sigma ),\quad 1/p\in (0,1). \end{aligned}$$

Applying the s \(\psi \) do \(T(\psi ,\sigma )\) on a smooth function \(f\),

$$\begin{aligned} T(\psi ,\sigma )(f)&=\int e^{ix\theta }\psi (t)(x,\theta )(\widehat{Sh_tf})(\theta )d\theta d\sigma (t)\\&=\int e^{ix\theta }\psi (t)(x,\theta ) e^{-iy\theta }f(y-t)dy d\theta d\sigma (t) \end{aligned}$$

which with a change of variable, it is equal to

$$\begin{aligned} \int e^{i(x-y-t)\theta }\psi (t)(x,\theta )f(y) dy d\theta d\sigma (t), \end{aligned}$$

which is another form of s \(\psi \) do.

2 Main results

Recall that the Fourier integral operators are of the form

$$\begin{aligned} {\textsc {f}}(f)(x)=\int e^{i\phi (x,y,\theta )}\psi (x,\theta )f(y)dy d\theta , \end{aligned}$$

where \(\phi (x,y,\theta )\) is a phase function and \(f(y)\) belongs to the Schwartz space \({\mathcal {S}}({\mathbb {R}}^N)\) [1].

In a special case, it is well known that Fourier integral operator can be a pseudo-differential operator under some conditions. Our aim is to do for a Fourier integral operator but with supersymbol.

Definition 2.1

Let \(\psi \) be a supersymbol and \(\phi \) a phase function such that \(Sh_t\circ \phi (x,y,\theta )=\phi (x-t,y,\theta )\). Then we define a Fourier integral operator with supersymbole \(\psi \) and shifted phase function \(\phi \) (denoted by sfio) as follow

$$\begin{aligned} {\textsc {sf}}(f)(x):=\int e^{i.Sh_t\circ \phi (x,y,\theta )}\psi (t)(x,\theta )dy\, d\theta \, d\sigma (t). \end{aligned}$$

Applying Lemma 4.1 of [3] in which \(\Phi \) is replaced by the phase function \(Sh_t\circ \phi \), the required result will be achieved. Here, we establishe that change of variable is well-defined.

Note that, it makes sense to say that \(\psi (t)(x,\theta )\in S^m_{\rho ,\delta }(U)\), where \(U\) is an arbitrary region in \({\mathbb {R}}^N\times {\mathbb {R}}^N\), which is conic with respect to \(\theta \). Indeed, \(\psi (t)(x,\theta )\in S^m_{\rho ,\delta } (U)\), if for any compact set \(K\subset ({\mathbb {R}}^N\times S^{N-1})\cap U\) (\(S^{N-1}\) is the unit sphere in \({\mathbb {R}}^N\)) and for arbitrary multi-indices \(\alpha \) and \(\beta \) there is a constant \(C_{\alpha , \beta , K}>0\) such that

$$\begin{aligned} |\partial _{\theta }^{\alpha }\partial _x^{\beta }\psi (t)(x,\theta )|\le C_{\alpha , \beta ,K}\langle \theta \rangle ^{m-\rho |\alpha |+\delta |\beta |} \end{aligned}$$

for \((x,\theta /|\theta |)\in K\) and \(|\theta |\ge 1\). For a diffeomorphism from a conical region \(V\subset {\mathbb {R}}^N\times {\mathbb {R}}^N\) onto the conical region \(U\subset {\mathbb {R}}^N\times {\mathbb {R}}^N\), commutating with the natural action of the multiplication group \({\mathbb {R}}_+\) of positive numbers, i.e., the diffeomorphism maps a point \((y,\eta )\in V\) to another point

$$\begin{aligned} (x(y,\eta ),\theta (y,\eta ))\in U, \end{aligned}$$
(2.1)

where \(x(y,\eta )\) and \(\theta (y,\eta )\) are positively homogeneous in \(\eta \) of degree \(0\) and \(1\), respectively. Then changing the variables in \(\psi (t)(x,\theta )\):

$$\begin{aligned} \zeta (t)(y,\eta )=\psi (t)(x(y,\eta ),\theta (y,\eta )), \end{aligned}$$
(2.2)

yields that \( \zeta (t)(y,\eta )\) belongs to the class of \(S^m_{\rho ,\delta }(V)\). A similar argument could be given in the the case of \(SS^m_{\rho ,\delta }\).

Here is the main theorem in which we want to reduce sfio to s \(\psi \) do.

Theorem 2.2

Let \(\phi _t\) be a phase function in \(X\times X\times {\mathbb {R}}^n\), such that

1.:

\(\phi _t(x,y,\theta )\) is linear in \(\theta \),

2.:

\((\phi _t)'_{\theta }(x,y,\theta )=0\Leftrightarrow x-t=y\).

Let \(F\) be a sfio with phase function \(\phi _t(x,y,\theta )\) and \(\psi (t)(x,\theta )\in SS^m_{\rho ,\delta }(X\times {\mathbb {R}}^N)\) where

$$\begin{aligned} 1-\rho \le \delta <\rho . \end{aligned}$$
(2.3)

Then \(F\) is also a s \(\psi \) do with symbol in \(SS^m_{\rho , \delta }\).

To prove this theorem first we need the following proposition which makes it easy to do. At the last section a review of the proof of the theorem could be found.

Proposition 2.3

Let \(\psi (t)(x,\theta )\in SS^m_{\rho ,\delta }(U)\) and assume that one of the following three conditions holds:

1.:

\(\rho +\delta \le 1\),

2.:

\(\rho +\delta \ge 1,\) and \(x=x(y)\) does not depend on \(\eta \), where \(\eta \) is mentioned in (2.1),

3.:

\(x=x(y), \theta =\theta (\eta ).\)

Then \(\zeta (t)(y,\eta )\in SS^m_{\rho ,\delta }(V)\), where \(\zeta \) is mentioned in (2.2).

Proof

First, suppose that \( |\alpha +\beta |\le 1 \). Differentiating \(\zeta (t)(y,\eta )\), we get

$$\begin{aligned} \frac{\partial \zeta (t)}{\partial \eta _r}= & {} \sum _i \psi ^{(i)} \frac{\partial x_i}{\partial \eta _r}+\sum _j \psi _{(j)} \frac{\partial \theta _j}{\partial \eta _r},\\ \frac{\partial \zeta (t)}{\partial y_s}= & {} \sum _i\psi ^{(i)} \frac{\partial x_i}{\partial y_s}+\sum _j \psi _{(j)} \frac{\partial \theta _j}{\partial y_s}, \end{aligned}$$

where \(\psi ^{(i)}=\frac{\partial \psi (t)}{\partial x_i} (x(y,\eta ),\theta (y,\eta ))\) and \(\psi _{(j)}=\frac{\partial \psi (t)}{\partial \theta _j} (x(y,\eta ),\theta (y,\eta ))\). The functions \(\frac{\partial x_i}{\partial \eta _r}\), \(\frac{\partial \theta _j}{\partial \eta _r}\), \(\frac{\partial x_i}{\partial y_s} \), and \(\frac{\partial \theta _j}{\partial y_s}\) are positively homogeneous in \(\theta \) of degree \(-1\), \(0\), \(0\), and, \(1\), respectively, so they belong to the classes \(S^{-1}\), \(S^0\), \(S^0\), and, \(S^1\), respectively in \(V\) (since \(x(y,\eta )\) and \(\theta (y,\eta )\) are homogeneous in \(\eta \) of degree \(0\) and \(1\), respectively, so they belong to the classes \(S^0\) and \(S^1\), respectively). We need to estimate the integral

$$\begin{aligned} \int \sup _{|y|<n}\sup _{\eta \in {\mathbb {R}}^N}|\partial ^\alpha _\eta \partial ^\beta _y\zeta (t)(y,\eta )|\langle \eta \rangle ^{-m+\rho |\alpha | -\delta |\beta |}d\sigma (t). \end{aligned}$$

Using the definition of \(S^m\) and estimating the derivatives of \(\psi \), for \(|\eta |\ge 1\) we easily obtain

$$\begin{aligned} \left| \frac{\partial \zeta (t)}{\partial \eta _r}\right|\le & {} C_K\left( |\eta |^{m+\delta -1}+|\eta |^{m-\rho }\right) ,\qquad \left( y,\frac{\eta }{|\eta |}\right) \in K, \\ \left| \frac{\partial \zeta (t)}{\partial y_s}\right|\le & {} C_K\left( |\eta |^{m+\delta }+|\eta |^{m-\rho +1}\right) ,\qquad \left( y,\frac{\eta }{|\eta |}\right) \in K, \end{aligned}$$

where \(K\) is a compact set in \(V\). If \(m+\delta -1\le m+\rho \), i.e., \(\rho +\delta \le 1\), then \(\left| \frac{\partial \zeta (t)}{\partial \eta _r}\right| \le 2C_K\langle \eta \rangle ^{m-\rho } \). If \( x=x(y) \) then \(\frac{\partial x_k}{\partial \eta _r}=0 \) and we do not need the condition \(\rho +\delta \le 1 \).

Similarly, if \( m-\rho +1\le m+\delta \), i.e., \(\rho +\delta \ge 1 \), it follows that \( \left| \frac{\partial \zeta (t)}{\partial y_s}\right| \le 2C_K\langle \eta \rangle ^{m+\delta }\) and if \( \theta =\theta (\eta )\), the estimate is obtained without another assumption. Thus \(\zeta (t)(y,\eta )\) would lie in \( S^m_{\rho ,\delta }\) and it only remains to confirm the condition of being supersymbol. Since \( \psi (t)(x,\theta )\in S^m \), we get \(|\partial _{\theta }^{\alpha } \partial _x^{\beta }\psi (t)(x,\theta )| \le C(t)\langle \theta \rangle ^{m-\rho |\alpha |+\delta |\beta |} \). But \( \psi (t)(x,\theta )\in SS^m \), so \(\int \Vert \psi (t):\alpha ,\beta ,n;S^m_{\rho ,\delta }\Vert d\sigma (t)<\infty \), in which by considering \( C(t) \) as infimum of the upper bound of the norm,

$$\begin{aligned}&\int \sup _{|x|<n}\sup _{\theta \in {\mathbb {R}}^N}|\partial ^\alpha _\theta \partial ^\beta _x\xi (t)(x,\theta )|\langle \theta \rangle ^{-m+\rho |\alpha | -\delta | \beta |}d\sigma (t)\\&\quad \le \int C(t) \langle \theta \rangle ^{m-\rho |\alpha |+\delta |\beta |} \langle \theta \rangle ^{-m+\rho |\alpha | -\delta |\beta |}d\sigma (t)\\&\quad =\int C(t) d\sigma (t)<\infty . \end{aligned}$$

There is some constant \(A\) such that

$$\begin{aligned} \left| \frac{\partial \zeta (t)}{\partial \eta _r}\right| \le AC(t)\langle \eta \rangle ^{m-\rho }, \end{aligned}$$

and also some constant \(B\) such that

$$\begin{aligned} \left| \frac{\partial \zeta (t)}{\partial y_s}\right| \le BC(t)\langle \eta \rangle ^{m+\delta }. \end{aligned}$$

Thus \(\zeta (t) \) belongs to class of s \(\psi \) dos, which proves the statement in the case of \(|\alpha +\beta |\le 1\).

Now suppose that the estimates hold when \(|\alpha +\beta |\le k\) for arbitrary \(\psi (t)\in SS^m_{\rho ,\delta }(U)\). In particular, for the derivatives of order \({\le } k\) of \( \psi ^{(i)}\) and \( \psi _{(i)}\) the estimates of the classes \(S^{m-\rho }_{\rho , \delta }(V)\) and \(S^{m+\delta }_{\rho , \delta }(V)\) hold, respectively. By a similar argument, these estimates hold when derivatives of order \({\le } (k+1)\) and for arbitrary \(\psi (t)\in SS^m_{\rho ,\delta }(U)\). \(\square \)

Now it is easy to prove the main theorem by using the above statement.

3 Proof of Theorem 2.2

Proof

In view of Proposition 2.1 and Exercise 2.4 of [3], we may assume that \(\psi (t)(x,\theta )=0\) and also \((x,y)\notin \Omega '\) where \(\Omega '\) is any neighbourhood of the diagonal \(\Delta \subset X\times X\). Making the change of variable \(\theta =\rho (x,y)(\xi )\) in the Fourier integral operator, and replacing \(x\) by \(x-t\), we have

$$\begin{aligned} F(f)(x)=\int e^{i(x-y-t)\xi }\psi (t)(x,\rho (x,y)\xi )|\det \rho (x,y)| f(y)dy d\xi d\sigma (t). \end{aligned}$$

Finally, we have to show \(\psi _1(t)(x,\xi )=\psi (t)(x,\rho (x,y)\xi )\) belongs to the class of supersymbols. Now by using Proposition 2.3 it follows from the condition (2.3) that \( \psi _1(t)(x,\xi )\in SS^m_{\rho , \delta }\) and this proves the theorem. \(\square \)