Pseudo-differential operators in the generalized weinstein setting

Abstract

In this paper, we consider the generalized Weinstein operator $\Delta _{W}^{d,\alpha ,n}$. For $n=0,$ we regain the classical Weinstein operator $\Delta _{W}^{\alpha ,d}$. We introduce and study the Sobolev spaces associated with the generalized Weinstein operator and investigate their properties. Next, we introduce a class of symbols and their associated pseudo-differential operators.

1 Introduction

In this paper, we consider the generalized Weinstein operator $\Delta _{W}^{\alpha ,d,n}$ defined on ${\mathbb {R}}_{+}^{d+1}={\mathbb {R}}^{d} \times ] 0,\ +\infty [,$ by:

$$\begin{aligned} \Delta _{W}^{\alpha ,d,n}=\sum _{i=1}^{d+1}\frac{\partial ^{2}}{\partial x_{i} ^{2}}+\frac{2\alpha +1}{x_{d+1}}\frac{\partial }{\partial x_{d+1}} -\frac{4n\left(\alpha +n\right) }{x_{d+1}^{2}}=\Delta _{d}+L_{\alpha ,n} \end{aligned}$$

(1.1)

where $n\in {\mathbb {N}}$, $\alpha >-\frac{1}{2}$, $\Delta _{d}$ is the Laplacian for the d first variables and $L_{\alpha ,n}$ is the second-order singular differential operator on the half line given by:

$$\begin{aligned} L_{\alpha ,n}=\frac{\partial ^{2}}{\partial x_{d+1}^{2}}+\frac{2\alpha +1}{x_{d+1}}\frac{\partial }{\partial x_{d+1}}-\frac{4n\left(\alpha +n\right) }{x_{d+1}^{2}}. \end{aligned}$$

(1.2)

For $n=0,$ we regain the classical Weinstein operator $\Delta _{W}^{\alpha ,d}$ given by:

$$\begin{aligned} \Delta _{W}^{\alpha ,d}=\sum _{i=1}^{d+1}\frac{\partial ^{2}}{\partial x_{i}^{2} }+\frac{2\alpha +1}{x_{d+1}}\frac{\partial }{\partial x_{d+1}}=\Delta _{d}+L_{\alpha } \end{aligned}$$

(1.3)

$L_{\alpha }=L_{\alpha ,0}$ is the Bessel operator. (see [3, 2, 4, 5, 9] and [10]).

The harmonic analysis associated with the generalized Weinstein operator $\Delta _{W}^{\alpha ,d,n}$ is studied by Aboulez et al. (see [1, 6,7,8]).

For all $f\in L^{1}({\mathbb {R}}_{+}^{d+1},d\mu _{\alpha ,d}(x))$, we define the Weinstein transform ${\mathscr {F}}_{W}^{\alpha ,d,n}$ by:

$$\begin{aligned} \forall \uplambda \in {\mathbb {R}}_{+}^{d+1},\,{\mathscr {F}}_{W}^{\alpha ,d,n}\left(f\right) (\uplambda)=\int _{{\mathbb {R}}_{+}^{d+1}}f(x)\Lambda _{\alpha ,d,n}(x,\uplambda)d\mu _{\alpha ,d}(x) \end{aligned}$$

where $\mu _{\alpha ,d}$ is the measure defined on ${\mathbb {R}}_{+}^{d+1}$ by:

$$\begin{aligned} d\mu _{\alpha ,d}(x)=x_{d+1}^{2\alpha +1}dx \end{aligned}$$

(1.4)

and $\Lambda _{\alpha ,d,n}$ is the generalized Weinstein kernel given by:

$$\begin{aligned} \forall x,y\mathbb {\in C}^{d+1},\,\Lambda _{\alpha ,d,n}\left(x,y\right) =x_{d+1}^{2n}e^{-i\left\langle x^{\prime }\text {,}y^{\prime }\right\rangle }j_{\alpha +2n}(x_{d+1}y_{d+1}), \end{aligned}$$

$x=(x^{\prime },x_{d+1}),\,x^{\prime }=\left(x_{1},x_{2},...,x_{d}\right)$ and $j_{\alpha }$ is the normalized Bessel function of index $\alpha \,$defined by:

$$\begin{aligned} \forall \xi \mathbb {\in C},\,j_{\alpha }(\xi)=\Gamma (\alpha +1)\underset{n=0}{\sum ^{\infty }}\frac{(-1)^{n}}{n!\Gamma (n+\alpha +1)}(\frac{\xi }{2})^{2n}. \end{aligned}$$

(1.5)

The generalized Weinstein transform ${\mathscr {F}}_{W}^{\alpha ,d,n}$ can be written in the form

$$\begin{aligned} {\mathscr {F}}_{W}^{\alpha ,d,n}={\mathscr {F}}_{W}^{\alpha +2n,d}\circ {\mathscr {M}}_{n} ^{-1}. \end{aligned}$$

(1.6)

where ${\mathscr {F}}_{W}^{\alpha ,d}={\mathscr {F}}_{W}^{\alpha ,d,0}$ is the classical Weinstein transform and ${\mathscr {M}}_{n}$ is the map defined by:

$$\begin{aligned} \forall x\in {\mathbb {R}}_{+}^{d+1},\ {\mathscr {M}}_{n}\left(f\right) \left(x\right) =x_{d+1}^{2n}f\left(x\right) . \end{aligned}$$

We designe by ${\mathscr {S}}_{*}({\mathbb {R}}^{d+1}),\,$the Schwartz space of rapidly decreasing functions on ${\mathbb {R}}^{d+1}$, even with respect to the last variable and ${\mathscr {S}}_{n,*}({\mathbb {R}}^{d+1})$ the subspace of ${\mathscr {S}}_{*}({\mathbb {R}}^{d+1})$ consisting of functions f such that

$$\begin{aligned} \forall k\in \left\{ 1,...,2n-1\right\} ,\ \frac{\partial ^{k}f}{\partial x_{d+1}^{k}}(x^{\prime },0)=f(x^{\prime },0)=0. \end{aligned}$$

For all $s\in {\mathbb {R}}$, we define the generalized Sobolev-Weinstein space ${\mathscr {H}}^{s,\alpha ,n}({\mathbb {R}}_{+}^{d+1})$ as the set of all $u\in {\mathscr {S}}_{n,*}^{\prime }$ $\left(\text {the strong dual of the space }{\mathscr {S}}_{n,*}({\mathbb {R}}^{d+1})\right)$ such that $\mathscr {F}_{W}^{\alpha ,d,n}(u)$ is a function and

$$\begin{aligned} \int _{{\mathbb {R}}_{+}^{d+1}}(1+\Vert \xi \Vert ^{2})^{s}\left| \mathscr {F}_{W}^{\alpha ,d,n}(u)\left(\xi \right) \right| ^{2}d\mu _{\alpha +2n,d}\left(\xi \right) <\infty . \end{aligned}$$

We investigate the properties of ${\mathscr {H}}^{s,\alpha ,n}({\mathbb {R}}_{+} ^{d+1})$. Moreover, we introduce a class of symbols and their associated pseudo-differential operators.

The contents of the paper is as follows:

In the second section, we recapitulate some results related to the harmonic analysis associated with the generalized Weinstein operator $\Delta _{W}^{\alpha ,d,n}$ given by the relation (1.1).

The section 3 is devoted to define and study the generalized Sobolev-Weinstein space ${\mathscr {H}}^{s,\alpha ,n}({\mathbb {R}}_{+}^{d+1})$.

In the last section, we introduce certain classes of symbols and study their associated pseudo-differential operators.

2 Preliminaires

In this section, we shall collect some results and definitions from the theory of the harmonic analysis associated with the Generalized Weinstein operator $\Delta _{W}^{\alpha ,d,n}$ defined on ${\mathbb {R}}_{+}^{d+1}$ by the relation (1.1).

Notations. In what follows, we need the following notations:

• ${\mathscr {C}}_{*}({\mathbb {R}}^{d+1}),$ the space of continuous functions on ${\mathbb {R}}^{d+1}$, even with respect to the last variable.

• ${\mathscr {E}}_{*}({\mathbb {R}}^{d+1}),$ the space of ${\mathscr {C}}^{\infty }$-functions on ${\mathbb {R}}^{d+1}$, even with respect to the last variable.

• ${\mathscr {S}}_{*}({\mathbb {R}}^{d+1}),$ the Schwartz space of rapidly decreasing functions on ${\mathbb {R}}^{d+1}$, even with respect to the last variable.

• ${\mathscr {D}}_{*}({\mathbb {R}}^{d+1}),$ the space of ${\mathscr {C}}^{\infty }$-functions on ${\mathbb {R}}^{d+1}$ which are of compact support, even with respect to the last variable.

• ${\mathscr {H}}_{*}({\mathbb {C}}^{d+1}\mathbb {)},$ the space of entire functions on ${\mathbb {C}}^{d+1}$, even with respect to the last variable, rapidly decreasing and of exponential type.

• ${\mathscr {M}}_{n}$, the map defined by:

  $$\begin{aligned} \forall x\in {\mathbb {R}}_{+}^{d+1},\ {\mathscr {M}}_{n}\left(f\right) \left(x\right) =x_{d+1}^{2n}f\left(x\right) . \end{aligned}$$

  (2.1)

  where $x=(x^{\prime },x_{d+1})$ and $x^{\prime }=\left(x_{1},x_{2},...,x_{d}\right)$

• $L_{\alpha ,n}^{p}({\mathbb {R}}_{+}^{d+1}),$ $1\le p\le +\infty ,\,$ the space of measurable functions on ${\mathbb {R}} _{+}^{d+1}$ such that

  $$\begin{aligned}{}\begin{array}{lll} \Vert f\Vert _{\alpha ,n,p} &{} = &{} \left[ \int _{{\mathbb {R}}_{+}^{d+1} }|{\mathscr {M}}_{n}^{-1}f(x)|^{p}d\mu _{\alpha +2n,d}(x)\right] ^{\frac{1}{p} }<+\infty ,\text { if }1\le p<+\infty ,\\ \Vert f\Vert _{\alpha ,n,\infty } &{} = &{} \mathrm {ess}\underset{x\in {\mathbb {R}} _{+}^{d+1}}{\sup }\left| {\mathscr {M}}_{n}^{-1}f(x)\right| <+\infty , \end{array} \end{aligned}$$

  where $\mu _{\alpha ,d}$ is the measure given by the relation (1.4).

• $L_{\alpha }^{p}({\mathbb {R}}_{+}^{d+1}):=L_{\alpha ,0} ^{p}({\mathbb {R}}_{+}^{d+1}),$ $1\le p\le +\infty ,\,$and $\Vert f\Vert _{\alpha ,p}:=\Vert f\Vert _{\alpha ,0,p}$.

• ${\mathscr {E}}_{n,*}({\mathbb {R}}^{d+1}),\ {\mathscr {D}}_{n,*}({\mathbb {R}}^{d+1})$ and $\mathscr {S}_{n,*}({\mathbb {R}}^{d+1})$ repespectively stand for the subspace of ${\mathscr {E}}_{*}({\mathbb {R}}^{d+1}),\ {\mathscr {D}}_{*}({\mathbb {R}}^{d+1})$ and ${\mathscr {S}}_{*}({\mathbb {R}}^{d+1})$ consisting of functions f such that

  $$\begin{aligned} \forall k\in \left\{ 1,...,2n-1\right\} ,\ \frac{\partial ^{k}f}{\partial x_{d+1}^{k}}(x^{\prime },0)=f(x^{\prime },0)=0. \end{aligned}$$

  Let us begin by the following result.

Lemma 2.1

(see [1])

i):

The map ${\mathscr {M}}_{n}$ is an isomorphism from ${\mathscr {E}}_{*}({\mathbb {R}}^{d+1}) \left(\text {resp. }{\mathscr {S}}_{*}({\mathbb {R}}^{d+1})\right)$ onto ${\mathscr {E}}_{n,*}({\mathbb {R}}^{d+1})$ $\left(\text {resp. }{\mathscr {S}}_{n,*}({\mathbb {R}} ^{d+1})\right) .$

ii):

For all $f\in {\mathscr {E}}_{*}({\mathbb {R}} ^{d+1})$, we have

$$\begin{aligned} L_{\alpha ,n}\circ {\mathscr {M}}_{n}\left(f\right) ={\mathscr {M}}_{n}\circ L_{\alpha +2n}\left(f\right) . \end{aligned}$$

(2.2)

iii):

For all $f\in {\mathscr {E}}_{*}({\mathbb {R}}^{d+1})$, we have

$$\begin{aligned} \Delta _{W}^{\alpha ,d,n}\circ {\mathscr {M}}_{n}\left(f\right) ={\mathscr {M}}_{n} \circ \Delta _{W}^{\alpha +2n}\left(f\right) . \end{aligned}$$

(2.3)

iv):

For all $f\in {\mathscr {E}}_{*}({\mathbb {R}}^{d+1})$ and $g\in \mathscr {D}_{n,*}({\mathbb {R}}^{d+1})$, we have

$$\begin{aligned} \int _{{\mathbb {R}}_{+}^{d+1}}\Delta _{W}^{\alpha ,d,n}\left(f\right) \left(x\right) g\left(x\right) d\mu _{\alpha ,d}(x)=\int _{{\mathbb {R}}_{+}^{d+1} }f\left(x\right) \Delta _{W}^{\alpha ,d,n}g\left(x\right) d\mu _{\alpha ,d}(x). \end{aligned}$$

(2.4)

Definition 2.1

The generalized Weinstein kernel $\Lambda _{\alpha ,d,n}$ is the function given by:

$$\begin{aligned} \forall x,y\mathbb {\in C}^{d+1},\,\Lambda _{\alpha ,d,n}\left(x,y\right) =x_{d+1}^{2n}e^{-i\left\langle x^{\prime }\text {,}y^{\prime }\right\rangle }j_{\alpha +2n}(x_{d+1}y_{d+1}), \end{aligned}$$

(2.5)

where $x=(x^{\prime },x_{d+1}),\,x^{\prime }=\left(x_{1},x_{2},...,x_{d} \right)$ and $j_{\alpha }$ is the normalized Bessel function of index $\alpha \,$defined by the relation (1.5).

It is easy to see that the generalized Weinstein kernel $\Lambda _{\alpha ,d,n}$ has a unique extention to ${\mathbb {C}}^{d+1}\times {\mathbb {C}}^{d+1}$ and satisifies the following properties.

Proposition 2.1

i):

We have

$$\begin{aligned} \forall x,y\mathbb {\in R}^{d+1},\,\overline{\Lambda _{\alpha ,d,n}\left(x,y\right) }=\Lambda _{\alpha ,d,n}\left(x,-y\right) =\Lambda _{\alpha ,d,n}\left(-x,y\right) \end{aligned}$$

ii):

For all $\beta \in {\mathbb {N}}^{d+1},\,x\in {\mathbb {R}}_{+}^{d+1}$ and $z\in {\mathbb {C}}^{d+1}$, we have

$$\begin{aligned} |D_{z}^{\beta }\Lambda _{\alpha ,d,n}(x,z)|\le x_{d+1}^{2n}\Vert x\Vert ^{\left| \beta \right| }\,\exp (\Vert x\Vert \,\Vert {\text {Im}} z\Vert), \end{aligned}$$

(2.6)

where

$$\begin{aligned} D_{z}^{\beta }=\frac{\partial ^{\beta }}{\partial z_{1}^{\beta _{1}}...\partial z_{d+1}^{\beta _{d+1}}}\text { and }\left| \beta \right| =\beta _{1}+...+\beta _{d+1}. \end{aligned}$$

In particular, we have

$$\begin{aligned} \forall x,y\in {\mathbb {R}}_{+}^{d+1},\,|\Lambda _{\alpha ,d,n}(x,y)|\le x_{d+1}^{2n}. \end{aligned}$$

(2.7)

iii):

The function $x\mapsto \Lambda _{\alpha ,d,n}(x,y)$ satisifies the differential equation

$$\begin{aligned} \triangle _{W}^{\alpha ,d,n}\left(\Lambda _{\alpha ,d,n}(.,y)\right) \left(x\right) =-\left\| y\right\| ^{2}\Lambda _{\alpha ,d,n}(x,y). \end{aligned}$$

(2.8)

iv) For all $x,\,y\in {\mathbb {C}}^{d+1}$, we have

$$\begin{aligned} \Lambda _{\alpha ,d,n}\left(x,y\right) =a_{\alpha +2n}e^{-i\left\langle x^{\prime }\text {,}y^{\prime }\right\rangle }x_{d+1}^{2n}\int _{0}^{1}\left(1-t^{2}\right) ^{\alpha +2n-\frac{1}{2}}\cos (tx_{d+1}y_{d+1})dt \end{aligned}$$

(2.9)

where $a_{\alpha }$ is the constant given by:

$$\begin{aligned} a_{\alpha }=\frac{2\Gamma \left(\alpha +1\right) }{\sqrt{\pi }\Gamma \left(\alpha +\frac{1}{2}\right) }. \end{aligned}$$

(2.10)

Definition 2.2

The generalized Weinstein transform ${\mathscr {F}}_{W}^{\alpha ,d,n}$ is given for $f\in L_{\alpha ,n}^{1}({\mathbb {R}}_{+}^{d+1})$ by:

$$\begin{aligned} \forall \uplambda \in {\mathbb {R}}_{+}^{d+1},\,{\mathscr {F}}_{W}^{\alpha ,d,n} (f)(\uplambda)=\int _{{\mathbb {R}}_{+}^{d+1}}f(x)\Lambda _{\alpha ,d,n} (x,\uplambda)d\mu _{\alpha ,d}(x). \end{aligned}$$

(2.11)

where $\mu _{\alpha ,d}$ is the measure on ${\mathbb {R}}_{+}^{d+1}$ given by the relation (1.4).

Some basic properties of the transform ${\mathscr {F}}_{W}^{\alpha ,d,n}$ are summarized in the following results.

Proposition 2.2

• i) For all $f\in L_{\alpha ,n}^{1}({\mathbb {R}}_{+}^{d+1})$, we have

  $$\begin{aligned} \Vert {\mathscr {F}}_{W}^{\alpha ,d,n}(f)\Vert _{\alpha ,n,\infty }\le \Vert f\Vert _{\alpha ,n,1}. \end{aligned}$$

  (2.12)

• ii) Let $m\in {\mathbb {N}}$ and $f\in {\mathscr {S}}_{n,*}({\mathbb {R}}^{d+1}),\,$we have

  $$\begin{aligned} \forall \uplambda \in {\mathbb {R}}_{+}^{d+1},\,{\mathscr {F}}_{W}^{\alpha ,d,n}\left[ \left(\triangle _{W}^{\alpha ,d,n}\right) ^{m}f\right] (\uplambda)=\left(-1\right) ^{m}\Vert \uplambda \Vert ^{2m}{\mathscr {F}}_{W}^{\alpha ,d,n}(f)(\uplambda). \end{aligned}$$

  (2.13)

• iii) Let $f\in {\mathscr {S}}_{n,*}({\mathbb {R}}^{d+1})$ and $m\in {\mathbb {N}}$. For all $\uplambda \in {\mathbb {R}}_{+}^{d+1}$,we have

  $$\begin{aligned} \left(\triangle _{W}^{\alpha ,d,n}\right) ^{m}\left[ {\mathscr {M}}_{n} {\mathscr {F}}_{W}^{\alpha ,d,n}(f)\right] (\uplambda)={\mathscr {M}}_{n}{\mathscr {F}}_{W} ^{\alpha ,d,n}(P_{m}f)(\uplambda) \end{aligned}$$

  (2.14)

  where $P_{m}(\uplambda)=\left(-1\right) ^{m}\left\| \uplambda \right\| ^{2\,m}.$

Proof

• i) We obtain the result from the relation (2.7).

• ii) Let $f\in {\mathscr {S}}_{n,*}({\mathbb {R}}^{d+1}),$ using the relations (1.6) and (2.3) for all

  $\uplambda \in {\mathbb {R}}_{+}^{d+1},\,$we get

  $$\begin{aligned} {\mathscr {F}}_{W}^{\alpha ,d,n}\left[ \left(\triangle _{W}^{\alpha ,d,n}\right) f\right] (\uplambda)&={\mathscr {F}}_{W}^{\alpha +2n,d}\circ {\mathscr {M}}_{n} ^{-1}\left[ \left(\triangle _{W}^{\alpha ,d,n}\right) f\right] (\uplambda)\\&={\mathscr {F}}_{W}^{\alpha +2n,d}\left[ \Delta _{W}^{\alpha +2n}{\mathscr {M}}_{n} ^{-1}f\right] (\uplambda)\\&=-\left\| \uplambda \right\| ^{2}{\mathscr {F}}_{W}^{\alpha +2n,d}\left[ {\mathscr {M}}_{n}^{-1}f\right] (\uplambda)\\&=-\left\| \uplambda \right\| ^{2}{\mathscr {F}}_{W}^{\alpha ,d,n}\left(f\right) (\uplambda) \end{aligned}$$

  which proves assertion ii).

• iii) The relation (2.8) together with (2.11) give the result. $\square$

Theorem 2.1

• i) Let $f\in L_{\alpha ,n}^{1}({\mathbb {R}}_{+}^{d+1})$. If ${\mathscr {F}}_{W}^{\alpha ,d,n}(f)\in L_{\alpha +2n}^{1}({\mathbb {R}}_{+}^{d+1}),$ then we have

  $$\begin{aligned} f(x)=C_{\alpha +2n,d}^{2}\int _{{\mathbb {R}}_{+}^{d+1}}{\mathscr {F}}_{W}^{\alpha ,d,n}(f)\left(y\right) \Lambda _{\alpha ,d,n}(-x,y)d\mu _{\alpha +2n,d}(y),\,a.e\,x\in {\mathbb {R}}_{+}^{d+1} \end{aligned}$$

  (2.15)

  where $C_{\alpha ,d}$ is the constant given by:

  $$\begin{aligned} C_{\alpha ,d}=\frac{1}{\left(2\pi \right) ^{\frac{d}{2}}2^{\alpha } \Gamma (\alpha +1)}. \end{aligned}$$

  (2.16)

• ii) The Weinstein transform ${\mathscr {F}}_{W}^{\alpha ,d,n}$ is a topological isomorphism from ${\mathscr {S}}_{n,*}({\mathbb {R}}^{d+1})$ onto $\mathscr {S}_{*}({\mathbb {R}}^{d+1})$ and from ${\mathscr {D}}_{n,*}({\mathbb {R}}^{d+1})$ onto ${\mathscr {H}}_{*}({\mathbb {C}}^{d+1}\mathbb {)}$.

Proof

• i) We obtain the result from the relation (1.6) and the fact that

  $$\begin{aligned} \varphi (x)=C_{\alpha ,d}^{2}\int _{{\mathbb {R}}_{+}^{d+1}}{\mathscr {F}}_{W} ^{\alpha ,d}\left(\varphi \right) \left(y\right) \Lambda _{\alpha ,d,0}(-x,y)d\mu _{\alpha ,d}(y),\,a.e\,x\in {\mathbb {R}}_{+}^{d+1} \end{aligned}$$

  where $\varphi ,{\mathscr {F}}_{W}^{\alpha ,d}\left(\varphi \right) \in L_{\alpha }^{1}({\mathbb {R}}_{+}^{d+1})$.

• ii) The transform $\mathscr {F}_{W}^{\alpha ,d}$ is a topological isomorphism from ${\mathscr {S}}_{*}({\mathbb {R}}^{d+1})$ onto itself and from ${\mathscr {D}}_{*}({\mathbb {R}} ^{d+1})$ onto ${\mathscr {H}}_{*}({\mathbb {C}}^{d+1}\mathbb {)}$. Then using the relation (1.6) the assertion ii) is proved. $\square$

The following Theorem is as an immediate consequence of the relation (1.6) and the properties of the transform ${\mathscr {F}}_{W}^{\alpha ,d}$ (see [6,7,8]).

Theorem 2.2

• i) For all $f,g\in {\mathscr {S}}_{n,*}({\mathbb {R}}^{d+1}),$ we have the following Parseval formula

  $$\begin{aligned} \int _{{\mathbb {R}}_{+}^{d+1}}f(x)\overline{g(x)}d\mu _{\alpha ,d}(x)=C_{\alpha +2n,d}^{2}\int _{{\mathbb {R}}_{+}^{d+1}}{\mathscr {F}}_{W}^{\alpha ,d,n} (f)(\uplambda)\overline{{\mathscr {F}}_{W}^{\alpha ,d,n}(g)(\uplambda)}d\mu _{\alpha +2n,d}(\uplambda) \end{aligned}$$

  (2.17)

  where $C_{\alpha ,d}$ is the constant given by the relation (2.16).

• ii) (Plancherel formula).

  For all $f\in \mathscr {S}_{n,*}({\mathbb {R}}^{d+1}),$ we have:

  $$\begin{aligned} \int _{{\mathbb {R}}_{+}^{d+1}}\left| f(x)\right| ^{2}d\mu _{\alpha ,d}(x)=C_{\alpha +2n,d}^{2}\int _{{\mathbb {R}}_{+}^{d+1}}\left| {\mathscr {F}}_{W} ^{\alpha ,d,n}(f)(\uplambda)\right| ^{2}d\mu _{\alpha +2n,d}(\uplambda). \end{aligned}$$

  (2.18)

• iii) (Plancherel Theorem):

  The transform ${\mathscr {F}}_{W} ^{\alpha ,d,n}$ extends uniquely to an isometric isomorphism from $L^{2}({\mathbb {R}}_{+}^{d+1},\ d\mu _{\alpha ,d}(x))$ onto $L^{2}({\mathbb {R}} _{+}^{d+1},\ C_{\alpha +2n,d}^{2}d\mu _{\alpha +2n,d}(x)).$

Definition 2.3

The translation operator $T_{x}^{\alpha ,d,n},\,$ $x\in {\mathbb {R}}_{+}^{d+1}$, associated with the operator $\Delta _{W}^{\alpha ,d,n}$ is defined on ${\mathscr {E}}_{n,*}({\mathbb {R}}_{+}^{d+1})$ by:

$$\begin{aligned} \forall y\in {\mathbb {R}}_{+}^{d+1},\ T_{x}^{\alpha ,d,n}f\left(y\right) =x_{d+1}^{2n}y_{d+1}^{2n}T_{x}^{\alpha +2n,d}{\mathscr {M}}_{n}^{-1}f\left(y\right) \end{aligned}$$

(2.19)

where

$$\begin{aligned} T_{x}^{\alpha ,d}f\left(y\right) =\frac{a_{\alpha }}{2}\int _{0}^{\pi }f\left(x^{\prime }+y^{\prime },\,\sqrt{x_{d+1}^{2}+y_{d+1}^{2}+2x_{d+1}y_{d+1} \cos \theta }\right) \left(\sin \theta \right) ^{2\alpha }d\theta , \end{aligned}$$

(2.20)

$x^{\prime }+y^{\prime }=\left(x_{1}+y_{1},...,x_{d}+y_{d}\right)$ and $a_{\alpha }$ is the constant given by (2.10).

Lemma 2.2

Let $f_{\beta },\,\beta >0,$ be the function defined by:

$$\begin{aligned} \forall \xi \in {\mathbb {R}}_{+}^{d+1},\,f_{\beta }\left(\xi \right) =\left(1+\left\| \xi \right\| ^{2}\right) ^{-\beta }. \end{aligned}$$

Then there exists $k_{\beta }>0$ such that

$$\begin{aligned} \forall x,y\in {\mathbb {R}}_{+}^{d+1},\,T_{x}^{\alpha ,d}\left(f_{\beta }\right) \left(y\right) \le k_{\beta }\left(1+\left\| x\right\| ^{2}\right) ^{-\beta }\left(1+\left\| y\right\| ^{2}\right) ^{-\beta }. \end{aligned}$$

(2.21)

Proof

Using the relation (2.20), for all $x,y\in {\mathbb {R}}_{+}^{d+1},\,$we obtain

$$\begin{aligned} T_{x}^{\alpha ,d}\left(f_{\beta }\right) \left(y\right)&=\frac{a_{\alpha }}{2}\int _{0}^{\pi }\left(1+\left\| x^{\prime }+y^{\prime }\right\| ^{2}+x_{d+1}^{2}+y_{d+1}^{2}+2x_{d+1}y_{d+1}\cos \theta \right) ^{-\beta }\left(\sin \theta \right) ^{2\alpha }d\theta \\&\le k_{\beta }\left(1+\left\| x\right\| ^{2}\right) ^{-\beta }\left(1+\left\| y\right\| ^{2}\right) ^{-\beta }\frac{a_{\alpha }}{2}\int _{0}^{\pi }\left(\sin \theta \right) ^{2\alpha }d\theta \\&\le k_{\beta }\left(1+\left\| x\right\| ^{2}\right) ^{-\beta }\left(1+\left\| y\right\| ^{2}\right) ^{-\beta } \end{aligned}$$

where

$$\begin{aligned} k_{\beta }=\underset{x,y\in {\mathbb {R}}_{+}^{d+1}}{\sup }\left(\frac{1+\left\| x^{\prime }+y^{\prime }\right\| ^{2}+\left(x_{d+1} -y_{d+1}\right) ^{2}}{\left(1+\left\| x\right\| ^{2}\right) \left(1+\left\| y\right\| ^{2}\right) }\right) ^{-\beta }. \end{aligned}$$

$\square$

The following proposition summarizes some properties of the generalized Weinstein translation operator.

Proposition 2.3

• i) For $f\in {\mathscr {E}}_{n,*}({\mathbb {R}}^{d+1})$, we have

  $$\begin{aligned} \forall x,\,y\in {\mathbb {R}}_{+}^{d+1},\,T_{x}^{\alpha ,d,n}f\left(y\right) =T_{y}^{\alpha ,d,n}f\left(x\right) . \end{aligned}$$

• ii) For all $f\in {\mathscr {E}}_{n,*}({\mathbb {R}}^{d+1})$ and $y\in {\mathbb {R}}_{+}^{d+1}$, the function $x\mapsto T_{x}^{\alpha ,d,n}f\left(y\right)$ belongs to ${\mathscr {E}}_{n,*}({\mathbb {R}}^{d+1}).$

• iii) We have

  $$\begin{aligned} \forall x\in {\mathbb {R}}_{+}^{d+1},\,\Delta _{W}^{\alpha ,d,n}\circ T_{x} ^{\alpha ,d,n}=T_{x}^{\alpha ,d,n}\circ \Delta _{W}^{\alpha ,d,n}. \end{aligned}$$

• iv) Let $f\in L_{\alpha ,n}^{p}({\mathbb {R}}_{+}^{d+1}),\,1\le p\le +\infty$ and $x\in {\mathbb {R}}_{+}^{d+1}$. Then $T_{x}^{\alpha ,d,n}f$ belongs to $L_{\alpha ,n}^{p}({\mathbb {R}}_{+}^{d+1})$ and we have

  $$\begin{aligned} \Vert T_{x}^{\alpha ,d,n}f\Vert _{\alpha ,n,p}\le x_{d+1}^{2n}\Vert f\Vert _{\alpha ,n,p}. \end{aligned}$$

  (2.22)

• v) The function $t\mapsto \Lambda _{\alpha ,d,n}\left(t,\uplambda \right) ,$ $\uplambda \in {\mathbb {C}}^{d+1},\,$ satisfies on ${\mathbb {R}}_{+}^{d+1}$ the following product formula:

  $$\begin{aligned} \forall x,y\in {\mathbb {R}}_{+}^{d+1},\,\Lambda _{\alpha ,d,n}\left(x,\uplambda \right) \Lambda _{\alpha ,d,n}\left(y,\uplambda \right) =T_{x} ^{\alpha ,d,n}\left[ \Lambda _{\alpha ,d,n}\left(.,\uplambda \right) \right] \left(y\right) . \end{aligned}$$

  (2.23)

• vi) Let $f\in {\mathscr {S}}_{n,*}({\mathbb {R}}^{d+1})$ and $x\in {\mathbb {R}} _{+}^{d+1}$, we have

  $$\begin{aligned} \forall \uplambda \in {\mathbb {R}}_{+}^{d+1},\,{\mathscr {F}}_{W}^{\alpha ,d,n}\left(T_{x}^{\alpha ,d,n}f\right) \left(\uplambda \right) =\Lambda _{\alpha ,d,n}\left(-x,\uplambda \right) {\mathscr {F}}_{W}^{\alpha ,d,n}\left(f\right) \left(\uplambda \right) . \end{aligned}$$

  (2.24)

Proof

The results can be obtained by a simple calculation by using the relation (2.19). $\square$

Lemma 2.3

Let $f\in {\mathscr {S}}_{n,*}\left({\mathbb {R}}^{d+1}\right)$, for all $x,y\in {\mathbb {R}}_{+}^{d+1},$we hae

$$\begin{aligned} T_{x}^{\alpha ,d,n}\left({\mathscr {M}}_{n}{\mathscr {F}}_{W}^{\alpha ,d,n} {\mathscr {M}}_{n}f\right) (y)=\int _{{\mathbb {R}}_{+}^{d+1}}\Lambda _{\alpha ,d,n}\left(x,\uplambda \right) \Lambda _{\alpha ,d,n}\left(y,\uplambda \right) f\left(\uplambda \right) d\mu _{\alpha +2n,d}(\uplambda). \end{aligned}$$

(2.25)

Proof

Let $f\in {\mathscr {S}}_{n,*}\left({\mathbb {R}}^{d+1}\right)$. Using the the relation (2.15) and (2.24), we obtain

$$\begin{aligned} T_{x}^{\alpha ,d,n}\left({\mathscr {M}}_{n}{\mathscr {F}}_{W}^{\alpha ,d,n} {\mathscr {M}}_{n}f\right) \left(y\right)&=C_{\alpha +2n,d}^{2} \int _{{\mathbb {R}}_{+}^{d+1}}\left\{ \Lambda _{\alpha ,d,n}\left(-y,\uplambda \right) \right. \\&\left. {\mathscr {F}}_{W}^{\alpha ,d,n}\left(T_{x}^{\alpha ,d,n}\left({\mathscr {M}}_{n}{\mathscr {F}}_{W}^{\alpha ,d,n}{\mathscr {M}}_{n}f\right) \right) \left(\uplambda \right) \right\} d\mu _{\alpha +2n,d}(\uplambda)\\&=C_{\alpha +2n,d}^{2}\int _{{\mathbb {R}}_{+}^{d+1}}\Lambda _{\alpha ,d,n}\left(-x,\uplambda \right) \Lambda _{\alpha ,d,n}\left(-y,\uplambda \right) {\mathscr {F}}_{W}^{\alpha ,d,n}\\&\quad \left({\mathscr {M}}_{n}{\mathscr {F}}_{W}^{\alpha ,d,n}{\mathscr {M}}_{n}f\right) \left(\uplambda \right) d\mu _{\alpha +2n,d}(\uplambda)\\&=C_{\alpha +2n,d}^{2}\int _{{\mathbb {R}}_{+}^{d+1}}\Lambda _{\alpha ,d,n}\left(-x,\uplambda \right) \Lambda _{\alpha ,d,n}\left(-y,\uplambda \right) {\mathscr {F}}_{W}^{\alpha +2n,d}\circ {\mathscr {F}}_{W}^{\alpha +2n,d}\left(f\right) \\&\quad \left(\uplambda \right) d\mu _{\alpha +2n,d}(\uplambda)\\&=\int _{{\mathbb {R}}_{+}^{d+1}}\Lambda _{\alpha ,d,n}\left(x,-\uplambda \right) \Lambda _{\alpha ,d,n}\left(y,-\uplambda \right) f\left(-\uplambda \right) d\mu _{\alpha +2n,d}(\uplambda)\\&=\int _{{\mathbb {R}}_{+}^{d+1}}\Lambda _{\alpha ,d,n}\left(x,\uplambda \right) \Lambda _{\alpha ,d,n}\left(y,\uplambda \right) f\left(\uplambda \right) d\mu _{\alpha +2n,d}(\uplambda). \end{aligned}$$

$\square$

Definition 2.4

Let $f,g\in L_{\alpha ,n}^{1}({\mathbb {R}}_{+}^{d+1}).$The generalized Weinstein convolution product of f and g is given by:

$$\begin{aligned} \forall x\in {\mathbb {R}}_{+}^{d+1},\,f*_{\alpha ,n}g\left(x\right) =\int _{{\mathbb {R}}_{+}^{d+1}}T_{x}^{\alpha ,d,n}f\left(-y\right) g\left(y\right) d\mu _{\alpha ,d}(y). \end{aligned}$$

(2.26)

Proposition 2.4

For all $f,g\in L_{\alpha ,n}^{1}({\mathbb {R}}_{+}^{d+1}),\,$we have $f*_{\alpha ,n}g$ $\in L_{\alpha ,n}^{1}({\mathbb {R}}_{+}^{d+1})\,$and

$$\begin{aligned} {\mathscr {F}}_{W}^{\alpha ,d,n}(f*_{\alpha ,n}g)={\mathscr {F}}_{W}^{\alpha ,d,n}(f){\mathscr {F}}_{W}^{\alpha ,d,n}(g). \end{aligned}$$

(2.27)

Proof

Let $f,g\in L_{\alpha ,n}^{1}({\mathbb {R}}_{+}^{d+1})$. We have

$$\begin{aligned} \Vert f*_{\alpha ,n}g\Vert _{\alpha ,n,1}\le \Vert f\Vert _{\alpha ,n,1}\Vert g\Vert _{\alpha ,n,1}. \end{aligned}$$

Now using Fubini’s theorem and the relation (2.24), we obtain

$$\begin{aligned} {\mathscr {F}}_{W}^{\alpha ,d,n}(f*_{\alpha ,n}g)(\uplambda)&=\int _{{\mathbb {R}}_{+}^{d+1}}\left(\int _{{\mathbb {R}}_{+}^{d+1}}T_{x}^{\alpha ,d,n}f(-y)g(y)d\mu _{\alpha ,d}(y)\right) \Lambda _{\alpha ,d,n}(x,\uplambda)d\mu _{\alpha ,d}(x)\\&=\int _{{\mathbb {R}}_{+}^{d+1}}g(y)\left(\int _{{\mathbb {R}}_{+}^{d+1}} T_{-y}^{\alpha ,d,n}f(x)\Lambda _{\alpha ,d,n}(x,\uplambda)d\mu _{\alpha ,d}(x)\right) d\mu _{\alpha ,d}(y)\\&=\int _{{\mathbb {R}}_{+}^{d+1}}g(y){\mathscr {F}}_{W}^{\alpha ,d,n}\left(T_{-y}^{\alpha ,d,n}f\right) \left(\uplambda \right) d\mu _{\alpha ,d}(y)\\&={\mathscr {F}}_{W}^{\alpha ,d}(f)(\uplambda)\int _{{\mathbb {R}}_{+}^{d+1} }g(y)\Lambda _{\alpha ,d,n}(y,\uplambda)d\mu _{\alpha ,d}(y)\\&={\mathscr {F}}_{W}^{\alpha ,d}(f)(\uplambda){\mathscr {F}}_{W}^{\alpha ,d} (g)(\uplambda). \end{aligned}$$

$\square$

Remark 2.1

From the relation (2.27), we deduce that

$$\begin{aligned} f,g\in {\mathscr {S}}_{n,*}({\mathbb {R}}^{d+1})\Rightarrow f*_{\alpha ,n} g\in {\mathscr {S}}_{n,*}({\mathbb {R}}^{d+1}). \end{aligned}$$

Notations. We denoted by:

$\cdot \ {\mathscr {S}}_{*}^{\prime },$ the strong dual of the space ${\mathscr {S}}_{*}({\mathbb {R}}^{d+1}).$

$\cdot \ {\mathscr {S}}_{n,*}^{\prime }$, the strong dual of the space ${\mathscr {S}}_{n,*}({\mathbb {R}}^{d+1}).$

Definition 2.5

The generalized Fourier-Weinstein transform of a distribution $u\in \mathscr {S}_{n,*}^{\prime }$ is defined by:

$$\begin{aligned} \forall \phi \in {\mathscr {S}}_{*}({\mathbb {R}}^{d+1}),\,\langle {\mathscr {F}}_{W} ^{\alpha ,d,n}(u),\,\phi \rangle =\langle u,\,\left({\mathscr {F}}_{W}^{\alpha ,d,n}\right) ^{-1}(\phi)\rangle . \end{aligned}$$

(2.28)

The following proposition is as an immediate consequence of Theorem 2.1.

Proposition 2.5

The transform ${\mathscr {F}}_{W}^{\alpha ,d,n}$ is a topological isomorphism from ${\mathscr {S}}_{n,*}^{\prime }$ onto ${\mathscr {S}}_{*}^{\prime }.$

Lemma 2.4

Let $m\in {\mathbb {N}}$ and $u\in {\mathscr {S}}_{n,*}^{\prime },\,$we have

$$\begin{aligned} \left({\mathscr {F}}_{W}^{\alpha ,d,n}\right) \left[ (\Delta _{W}^{\alpha ,d,n})^{m}u\right] =\left(-1\right) ^{m}\Vert x\Vert ^{2m}\left({\mathscr {F}}_{W}^{\alpha ,d,n}\right) (u) \end{aligned}$$

(2.29)

where

$$\begin{aligned} \forall \phi \in {\mathscr {S}}_{n,*}\left({\mathbb {R}}^{d+1}\right) ,\,\langle \Delta _{W}^{\alpha ,d,n}u,\,\phi \rangle =\langle u,\,\Delta _{W}^{\alpha ,d,n}\phi \rangle . \end{aligned}$$

(2.30)

Proof

Let $m\in {\mathbb {N}}$ and $u\in {\mathscr {S}}_{n,*}^{\prime },\,$by invoking (2.13), (2.28) and (2.30), for all $\phi \in {\mathscr {S}}_{*}\left({\mathbb {R}}^{d+1}\right) ,\,$we can write

$$\begin{aligned} \langle \left({\mathscr {F}}_{W}^{\alpha ,d,n}\right) \left[ (\Delta _{W} ^{\alpha ,d,n})^{m}u\right] ,\ \phi \rangle&=\langle (\Delta _{W}^{\alpha ,d,n})^{m}u,\,\left({\mathscr {F}}_{W}^{\alpha ,d,n}\right) ^{-1}(\phi)\rangle \\&=\langle u,\,(\Delta _{W}^{\alpha ,d,n})^{m}\left({\mathscr {F}}_{W} ^{\alpha ,d,n}\right) ^{-1}(\phi)\rangle \\&=\langle u,\,\left({\mathscr {F}}_{W}^{\alpha ,d,n}\right) ^{-1}(\left(-1\right) ^{m}\Vert x\Vert ^{2m}\phi)\rangle \\&=\langle {\mathscr {F}}_{W}^{\alpha ,d,n}\left(u\right) ,\,\left(-1\right) ^{m}\Vert x\Vert ^{2m}\phi)\rangle \\&=\langle \left(-1\right) ^{m}\Vert x\Vert ^{2m}{\mathscr {F}}_{W}^{\alpha ,d,n}\left(u\right) ,\phi)\rangle . \end{aligned}$$

Which completes the proof. $\square$

3 Sobolev spaces associated with the generalized Weinstein operator

The goal of this section is to introduce and study the Sobolev spaces associated with the generalized Weinstein operator $\Delta _{W}^{\alpha ,d,n}$.

Definition 3.1

For $s\in {\mathbb {R}}$, we define the generalized Sobolev-Weinstein space of order s, that will be denoted $\mathscr {H}^{s,\alpha ,n}({\mathbb {R}}_{+}^{d+1})$, as the set of all $u\in \mathscr {S}_{n,*}^{\prime }$ such that ${\mathscr {F}}_{W}^{\alpha ,d,n}(u)$ is a function and

$$\begin{aligned} \int _{{\mathbb {R}}_{+}^{d+1}}(1+\Vert \uplambda \Vert ^{2})^{s}\left| {\mathscr {F}}_{W}^{\alpha ,d,n}(u)\left(\uplambda \right) \right| ^{2} d\mu _{\alpha +2n,d}\left(\uplambda \right) <+\infty . \end{aligned}$$

(3.1)

We provide ${\mathscr {H}}^{s,\alpha ,n}({\mathbb {R}}_{+}^{d+1})$ with the inner product

$$\begin{aligned} \langle u,v\rangle _{s,\alpha ,n}=C_{\alpha +2n,d}^{2}\int _{{\mathbb {R}}_{+}^{d+1} }(1+\Vert \xi \Vert ^{2})^{s}{\mathscr {F}}_{W}^{\alpha ,d,n}(u)(\xi)\overline{{\mathscr {F}}_{W}^{\alpha ,d,n}(v)(\xi)}d\mu _{\alpha +2n,d}(\xi) \end{aligned}$$

(3.2)

and the norm

$$\begin{aligned} \left\| u\right\| _{{\mathscr {H}}^{s,\alpha ,n}}=\left[ C_{\alpha +2n,d} ^{2}\int _{{\mathbb {R}}_{+}^{d+1}}(1+\Vert \xi \Vert ^{2})^{s}\left| {\mathscr {F}}_{W}^{\alpha ,d,n}(u)\left(\xi \right) \right| ^{2}d\mu _{\alpha +2n,d}\left(\xi \right) \right] ^{\frac{1}{2}}. \end{aligned}$$

(3.3)

The following properties of the spaces ${\mathscr {H}}^{s,\alpha ,n}$ can easily be established.

Proposition 3.1

(i):

For all $s\in {\mathbb {R}},$ we have

$$\begin{aligned} {\mathscr {S}}_{n,*}({\mathbb {R}}^{d+1})\subset {\mathscr {H}}^{s,\alpha ,n} ({\mathbb {R}}_{+}^{d+1}). \end{aligned}$$

(ii):

We have

$$\begin{aligned} {\mathscr {H}}^{0,\alpha ,n}({\mathbb {R}}_{+}^{d+1})=L_{\alpha ,n}^{2}({\mathbb {R}} _{+}^{d+1}). \end{aligned}$$

(iii):

For all s, t $\in {\mathbb {R}},\ t>s$, the space $\mathscr {H}^{t,\alpha ,n}({\mathbb {R}}_{+}^{d+1})$ is continuously contained in $\mathscr {H}^{s,\alpha ,n}({\mathbb {R}}_{+}^{d+1}).$.

Proposition 3.2

The space ${\mathscr {H}}^{s,\alpha ,n}({\mathbb {R}}_{+}^{d+1})$ provided with the norm $\Vert .\Vert _{{\mathscr {H}}^{s,\alpha ,n}}$ is a Banach space.

Proof

Let $\left(f_{m}\right) _{m\in {\mathbb {N}}}$ be a Cauchy sequence of ${\mathscr {H}}^{s,\alpha ,n}({\mathbb {R}}_{+}^{d+1})$. From the definition of the norm $\left\| .\right\| _{{\mathscr {H}}^{s,\alpha ,n}}$, it is clear that $\left({\mathscr {F}}_{W}^{\alpha ,d,n}\left(f_{m}\right) \right) _{m\in {\mathbb {N}}}$ is a Cauchy sequence of $L^{2}({\mathbb {R}}_{+} ^{d+1},\,C_{\alpha +2n,d}^{2}(1+\Vert x\Vert ^{2})^{s}d\mu _{\alpha +2n,d}\left(x\right))$.

Since $L^{2}({\mathbb {R}}_{+}^{d+1},\,C_{\alpha +2n,d} ^{2}(1+\Vert x\Vert ^{2})^{s}d\mu _{\alpha +2n,d}\left(x\right))$ is complete, there exists a function $f\in L^{2}({\mathbb {R}}_{+}^{d+1},\,C_{\alpha +2n,d}^{2}(1+\Vert x\Vert ^{2})^{s}d\mu _{\alpha +2n,d}\left(x\right))$ such that

$$\begin{aligned} \underset{m\rightarrow +\infty }{\lim }\Vert {\mathscr {F}}_{W}^{\alpha ,d,n}\left(f_{m}\right) -f\Vert _{L^{2}({\mathbb {R}}_{+}^{d+1},\,C_{\alpha +2n,d} ^{2}(1+\Vert x\Vert ^{2})^{s}d\mu _{\alpha +2n,d}\left(x\right))}=0. \end{aligned}$$

(3.4)

Then $f\in {\mathscr {S}}_{*}^{\prime }$ and $h=\left({\mathscr {F}}_{W} ^{\alpha ,d,n}\right) ^{-1}\left(f\right) \in {\mathscr {S}}_{n,*}^{\prime }.$

So, ${\mathscr {F}}_{W}^{\alpha ,d,n}(h)=f\in L^{2}({\mathbb {R}}_{+} ^{d+1},\,C_{\alpha +2n,d}^{2}(1+\Vert x\Vert ^{2})^{s}d\mu _{\alpha +2n,d}\left(x\right))$, which proves that $h\in {\mathscr {H}}^{s,\alpha ,n}({\mathbb {R}} _{+}^{d+1})$ and we have

$$\begin{aligned} \Vert f_{m}-h\Vert _{{\mathscr {H}}^{s,\alpha ,n}}=\Vert {\mathscr {F}}_{W}^{\alpha ,d,n}\left(f_{m}\right) -f\Vert _{L^{2}({\mathbb {R}}_{+}^{d+1},C_{\alpha +2n,d}^{2}(1+\Vert x\Vert ^{2})^{s}d\mu _{\alpha +2n,d}\left(x\right))}\underset{m\rightarrow +\infty }{\rightarrow }0. \end{aligned}$$

Hence, ${\mathscr {H}}^{s,\alpha ,n}({\mathbb {R}}_{+}^{d+1})$ is complete. $\square$

Proposition 3.3

Let $s,t\in {\mathbb {R}}$. The operator ${\mathscr {O}}_{t}$ defined by:

$$\begin{aligned} \forall x\in {\mathbb {R}}_{+}^{d+1},\,{\mathscr {O}}_{t}u\left(x\right) =C_{\alpha +2n,d}^{2}\int _{{\mathbb {R}}_{+}^{d+1}}(1+\Vert \xi \Vert)^{t} \Lambda _{\alpha ,d,n}(-x,\xi){\mathscr {F}}_{W}^{\alpha ,d,n}(u)\left(\xi \right) d\mu _{\alpha +2n,d}(\xi) \end{aligned}$$

is a isomorphism from ${\mathscr {H}}^{s,\alpha ,n}({\mathbb {R}}_{+}^{d+1})$ onto ${\mathscr {H}}^{s-t,\alpha ,n}({\mathbb {R}}_{+}^{d+1})$.

Proof

Let $s,t\in {\mathbb {R}}$ and $u\in {\mathscr {H}}^{s,\alpha ,n}({\mathbb {R}}_{+} ^{d+1}).$ The function:

$\xi \mapsto (1+\Vert \xi \Vert)^{t}(1+\Vert \xi \Vert ^{2})^{\frac{s-t}{2}}{\mathscr {F}}_{W}^{\alpha ,d,n}(u)\left(\xi \right)$ belongs to $L_{\alpha +2n}^{2}({\mathbb {R}}_{+}^{d+1})$ and have

$$\begin{aligned} \forall \xi \in {\mathbb {R}}_{+}^{d+1},\,{\mathscr {F}}_{W}^{\alpha ,d,n}\left({\mathscr {O}}_{t}u\right) \left(\xi \right) =(1+\Vert \xi \Vert)^{t}\mathscr {F}_{W}^{\alpha ,d,n}\left(u\right) \left(\xi \right) . \end{aligned}$$

Thus

$$\begin{aligned}&\int _{{\mathbb {R}}_{+}^{d+1}}(1+\Vert \xi \Vert ^{2})^{s-t}\left| \mathscr {F}_{W}^{\alpha ,d,n}({\mathscr {O}}_{t}u)\left(\xi \right) \right| ^{2} d\mu _{\alpha +2n,d}\left(\xi \right) \\&\le \kappa _{t}\int _{{\mathbb {R}}_{+}^{d+1}}(1+\Vert \xi \Vert ^{2})^{s}\left| {\mathscr {F}}_{W}^{\alpha ,d,n}(u)\left(\xi \right) \right| ^{2}d\mu _{\alpha +2n,d}\left(\xi \right) \end{aligned}$$

where

$$\begin{aligned} \kappa _{t}=\underset{x\in {\mathbb {R}}_{+}^{d+1}}{\sup }\left[ \frac{(1+\Vert x\Vert)^{2t}}{(1+\Vert x\Vert ^{2})^{t}}\right] \le 2^{\left| t\right| }. \end{aligned}$$

Then, ${\mathscr {O}}_{t}u\in {\mathscr {H}}^{s-t,\alpha ,n}({\mathbb {R}}_{+}^{d+1})$ and we have

$$\begin{aligned} \left\| {\mathscr {O}}_{t}u\right\| _{{\mathscr {H}}^{s-t,\alpha ,n}}\le 2^{\frac{\left| t\right| }{2}}\left\| u\right\| _{\mathscr {H}^{s,\alpha ,n}}. \end{aligned}$$

Now, let $v\in {\mathscr {H}}^{s-t,\alpha ,n}({\mathbb {R}}_{+}^{d+1})$ and put

$$\begin{aligned} u=\left({\mathscr {F}}_{W}^{\alpha ,d,n}\right) ^{-1}\left((1+\Vert \xi \Vert)^{-t}{\mathscr {F}}_{W}^{\alpha ,d,n}(v)\right) . \end{aligned}$$

From the definition of the operator ${\mathscr {O}}_{t}$, we have $\mathscr {O}_{t}u=v$ and we get

$$\begin{aligned}&\int _{{\mathbb {R}}_{+}^{d+1}}(1+\Vert \xi \Vert ^{2})^{s}\left| \mathscr {F}_{W}^{\alpha ,d,n}\left(u\right) \left(\xi \right) \right| ^{2} d\mu _{\alpha +2n,d}\left(\xi \right) \\&\quad =\int _{{\mathbb {R}}_{+}^{d+1}}(1+\Vert \xi \Vert ^{2})^{s}(1+\Vert \xi \Vert)^{-2t}\left| {\mathscr {F}}_{W}^{\alpha ,d,n}\left(v\right) \left(\xi \right) \right| ^{2}d\mu _{\alpha +2n,d}\left(\xi \right) \\& \quad \le 2^{\left| t\right| }\int _{{\mathbb {R}}_{+}^{d+1}}(1+\Vert \xi \Vert ^{2})^{s-t}\left| {\mathscr {F}}_{W}^{\alpha ,d,n}(v)\left(\xi \right) \right| ^{2}d\mu _{\alpha +2n,d}\left(\xi \right) . \end{aligned}$$

Hence, $u\in {\mathscr {H}}^{s,\alpha ,n}({\mathbb {R}}_{+}^{d+1})$ and we obtain

$$\begin{aligned} \left\| u\right\| _{{\mathscr {H}}^{s,\alpha ,n}}\le 2^{\frac{\left| t\right| }{2}}\left\| {\mathscr {O}}_{t}u\right\| _{\mathscr {H}^{s-t,\alpha ,n}}. \end{aligned}$$

Which completes the proof. $\square$

Remark 3.1

The dual of ${\mathscr {H}}^{s,\alpha ,n}({\mathbb {R}}_{+}^{d+1})\,$can be identified with ${\mathscr {H}}^{-s,\alpha ,n}({\mathbb {R}}_{+}^{d+1})$. The relation of the identification is as follows:

$$\begin{aligned} \langle u,v\rangle _{0,\alpha ,n}=C_{\alpha +2n,d}^{2}\int _{{\mathbb {R}}_{+}^{d+1} }{\mathscr {F}}_{W}^{\alpha ,d,n}(u)(\xi)\overline{{\mathscr {F}}_{W}^{\alpha ,d,n}(v)(\xi)}d\mu _{\alpha +2n,d}(\xi), \end{aligned}$$

(3.5)

with $u\in {\mathscr {H}}^{s,\alpha ,n}({\mathbb {R}}_{+}^{d+1})$ and $v\in \mathscr {H}^{-s,\alpha ,n}({\mathbb {R}}_{+}^{d+1})$.

Proposition 3.4

Let $s_{1},s,s_{2}\in {\mathbb {R}}$, satisfying $s_{1}<s<s_{2}$. Then, for all $\varepsilon >0,$ there exists a nonnegative constant $C_{\varepsilon }$ such that for all $u\in {\mathscr {H}}^{s_{2},\alpha ,n}({\mathbb {R}}_{+}^{d+1})$, we have

$$\begin{aligned} \Vert u\Vert _{{\mathscr {H}}^{s,\alpha ,n}}\le C_{\varepsilon }\Vert u\Vert _{{\mathscr {H}}^{s_{1},\alpha ,n}}+\varepsilon \Vert u\Vert _{{\mathscr {H}}^{s_{2},\alpha ,n}}. \end{aligned}$$

(3.6)

Proof

Let $s_{1},s_{2}\in {\mathbb {R}}$ and $s=\left(1-t\right) s_{1}+ts_{2},$ $t\in \left] 0,\,1\right[$. Let $u\in {\mathscr {H}}^{s_{2},\alpha ,n} ({\mathbb {R}}_{+}^{d+1}).$ We put $t=\displaystyle \frac{1}{p}$ and $1-t=\displaystyle \frac{1}{q}$, applying the Hölder’s inequality, we get

$$\begin{aligned} \Vert u\Vert _{{\mathscr {H}}^{s,\alpha ,n}}&\le \Vert u\Vert _{{\mathscr {H}}^{s_{1} ,\alpha ,n}}^{1-t}\times \Vert u\Vert _{{\mathscr {H}}^{s_{2},\alpha ,n}}^{t}\\&\le \left(\varepsilon ^{\frac{-t}{1-t}}\Vert u\Vert _{{\mathscr {H}}^{s_{1} ,\alpha ,n}}\right) ^{1-t}\times \left(\varepsilon \Vert u\Vert _{{\mathscr {H}}^{s_{2},\alpha ,n}}\right) ^{t}\\&\le \varepsilon ^{\frac{s-s_{1}}{s-s_{2}}}\Vert u\Vert _{{\mathscr {H}}^{s_{1} ,\alpha ,n}}+\varepsilon \Vert u\Vert _{{\mathscr {H}}^{s_{2},\alpha ,n}}. \end{aligned}$$

Then the relation (3.6) is proved. $\square$

Proposition 3.5

Let $s\in {\mathbb {R}}$ and $m\in {\mathbb {N}}$. Then for all $\varepsilon >2m,\,$ the operator $\left(\Delta _{W}^{\alpha ,d,n}\right) ^{m}$ is continuous from ${\mathscr {H}}^{s,\alpha ,n}({\mathbb {R}}_{+}^{d+1})$ into ${\mathscr {H}}^{s-\varepsilon ,\alpha ,n}({\mathbb {R}}_{+}^{d+1})$.

Proof

Let $m\in {\mathbb {N}},\,\varepsilon >2m$, $s\in {\mathbb {R}}$ and $u\in \mathscr {H}^{s,\alpha ,n}({\mathbb {R}}_{+}^{d+1})$.

Using the relation (2.29), we can see that $\left(\Delta _{W}^{\alpha ,d}\right) ^{m} u\in {\mathscr {H}}^{s-\varepsilon ,\alpha ,n}({\mathbb {R}}_{+}^{d+1})$ and we have

$$\begin{aligned} \Vert \left(\Delta _{W}^{\alpha ,d}\right) ^{m}u\Vert _{\mathscr {H}^{s-\varepsilon ,\alpha ,n}}\le \Vert u\Vert _{{\mathscr {H}}^{s,\alpha ,n}}. \end{aligned}$$

Thus the proof is finished. $\square$

4 Pseudo-differential operators

NotationsWe need the following notations

• For $r\ge 0,$ we designate by ${\mathcal {S}}^{r},$ the space of $C^{\infty } -$function a on ${\mathbb {R}}^{d+1}\times {\mathbb {R}}^{d+1}$ such that for each compact set $K\subset {\mathbb {R}}^{d+1}$ and each $\beta ,\gamma \in {\mathbb {N}}$, there exists a constant $C=C\left(K,\beta ,\gamma \right)$ satisfying:

  $$\begin{aligned} \forall \left(x,\ \xi \right) \in K\times {\mathbb {R}}^{d+1},\ \left| D_{\xi }^{\beta }D_{x}^{\gamma }a\left(x,\ \xi \right) \right| \le C(1+\Vert \xi \Vert ^{2})^{\frac{r}{2}}. \end{aligned}$$

  (4.1)

• For $r,l\in {\mathbb {R}}$ with $l>0,$ we denote by ${\mathcal {S}}^{r,l},$ the space consits of all $C^{\infty }-$function a on ${\mathbb {R}}^{d+1} \times {\mathbb {R}}^{d+1}$ such that for each $\beta ,\gamma \in {\mathbb {N}}$, there exist a positive constant $C=C\left(r,l,\beta ,\gamma \right)$ satisfying the relation:

  $$\begin{aligned} \forall \left(x,\ \xi \right) \in {\mathbb {R}}^{d+1}\times {\mathbb {R}} ^{d+1},\ \left| D_{\xi }^{\beta }D_{x}^{\gamma }a\left(x,\ \xi \right) \right| \le C(1+\Vert \xi \Vert ^{2})^{\frac{r}{2}}(1+\Vert x\Vert ^{2})^{-\frac{l}{2}}. \end{aligned}$$

  (4.2)

Definition 4.1

The pseudo-differential operator $A\left(a,\Delta _{W} ^{\alpha ,d,n}\right)$ associated with $a\left(x,\xi \right) \in {\mathcal {S}}^{r}$ is defined for $u\in {\mathscr {S}}_{n,*}({\mathbb {R}}^{d+1})$ by:

$$\begin{aligned} \left[ A\left(a,\Delta _{W}^{\alpha ,d,n}\right) u\right] \left(x\right) =\int _{{\mathbb {R}}_{+}^{d+1}}\Lambda _{\alpha ,d,n}(-x,y)a\left(x,y\right) {\mathscr {F}}_{W}^{\alpha ,d,n}(u)\left(y\right) d\mu _{\alpha +2n,d}\left(y\right) . \end{aligned}$$

(4.3)

Theorem 4.1

If $a\left(x,\ \xi \right) \in {\mathcal {S}}^{r}$, then its associated pseudo-differential operator $A\left(a,\ \Delta _{W}^{\alpha ,d}\right)$ is a well-defined mapping from ${\mathscr {S}}_{n,*} ({\mathbb {R}}^{d+1})$ into $C^{\infty }\left({\mathbb {R}}^{d+1}\right) .$

Proof

Let $a\left(x,\ \xi \right) \in {\mathcal {S}}^{r}$ and $s>r+\displaystyle \frac{d}{2}+\alpha +2n+1$. From the relation (4.1), we have for any compact set $K\subset {\mathbb {R}}^{d+1}$ and any $\gamma \in {\mathbb {N}}$,

$$\begin{aligned} \forall \left(x,\ \xi \right) \in K\times {\mathbb {R}}^{d+1},\ \left| D_{x}^{\gamma }a\left(x,\ \xi \right) \right| \le C(1+\Vert \xi \Vert ^{2})^{\frac{r}{2}}. \end{aligned}$$

(4.4)

Let $u\in {\mathscr {S}}_{n,*}({\mathbb {R}}^{d+1})$ and $x\in K$, using the relations (4.4), (2.7) and the Cauchy-Schwartz inequality, we obtain

$$\begin{aligned}&\int _{{\mathbb {R}}_{+}^{d+1}}\left| a\left(x,\ \xi \right) \Lambda _{\alpha ,d,n}(-x,\xi){\mathscr {F}}_{W}^{\alpha ,d,n}(u)\left(\xi \right) \right| d\mu _{\alpha +2n,d}\left(\xi \right) \\&\le Cx_{d+1}^{2n}\int _{{\mathbb {R}}_{+}^{d+1}}(1+\Vert \xi \Vert ^{2})^{\frac{r}{2}}\left| {\mathscr {F}}_{W}^{\alpha ,d,n}(u)\left(\xi \right) \right| d\mu _{\alpha +2n,d}\left(\xi \right) \\&\le \frac{C}{C_{\alpha +2n,d}}x_{d+1}^{2n}\left(\int _{{\mathbb {R}}_{+} ^{d+1}}(1+\Vert \xi \Vert ^{2})^{r-s}d\mu _{\alpha +2n,d}\left(\xi \right) \right) ^{\frac{1}{2}}\left\| u\right\| _{{\mathscr {H}}^{s,\alpha ,n}}. \end{aligned}$$

This relation proves that $A\left(a,\ \Delta _{W}^{\alpha ,d}\right) \left(u\right)$ is well-defined and continuous on ${\mathbb {R}}_{+}^{d+1}$.

By the same argument, we can prove

$$\begin{aligned} \int _{{\mathbb {R}}_{+}^{d+1}}\left| D_{x}^{\gamma }a\left(x,\xi \right) \Lambda _{\alpha ,d,n}(-x,\xi){\mathscr {F}}_{W}^{\alpha ,d,n}(u)\left(\xi \right) \right| d\mu _{\alpha ,d}\left(\xi \right) \le C^{\prime }\left\| u\right\| _{{\mathscr {H}}^{s,\alpha ,n}} \end{aligned}$$

where $C^{\prime }$ is a positive constant.

Consequently, in vertue of Leibniz formula, we obtain the result. $\square$

The next lemma plays an important role in this section.

Lemma 4.1

Let $t\ge 0$ and $l>2\alpha +4n+d+2.$ Then, for all $a\left(x,\ \xi \right) \in {\mathcal {S}}^{r,l},$we have:

$$\begin{aligned} \left| {\mathscr {F}}_{W}^{\alpha ,d,n}\left({\mathscr {M}}_{n}a\left(.,y\right) \right) \left(\xi \right) \right| \le C(1+\Vert y\Vert ^{2})^{\frac{r}{2}}(1+\Vert \xi \Vert ^{2})^{-\frac{t}{2}}, \end{aligned}$$

(4.5)

where C is a constant depending on $r,t,\alpha ,d,n$ and l.

Proof

Let $k\in {\mathbb {N}}$. By invoking (2.7), (2.13) and (4.2), we obtain

$$\begin{aligned} \Vert \xi \Vert ^{2k}\left| {\mathscr {F}}_{W}^{\alpha ,d,n}\left({\mathscr {M}}_{n} a\left(.,y\right) \right) \left(\xi \right) \right|&=\left| {\mathscr {F}}_{W}^{\alpha ,d,n}\left[ \left(\triangle _{W}^{\alpha ,d,n}\right) ^{k}\left({\mathscr {M}}_{n}a\left(.,y\right) \right) \left(\xi \right) \right] \right| \\&\le \int _{{\mathbb {R}}_{+}^{d+1}}\left| \left(\triangle _{W,x} ^{\alpha ,d,n}\right) ^{k}\left(x_{d+1}^{2n}a\left(x,y\right) \right) \right| \left| \Lambda _{\alpha ,d,n}(x,\xi)\right| d\mu _{\alpha ,d}(x)\\&\le \int _{{\mathbb {R}}_{+}^{d+1}}\left| \left(\triangle _{W,x} ^{\alpha ,d,n}\right) ^{k}\left(x_{d+1}^{2n}a\left(x,y\right) \right) \right| x_{d+1}^{2n}d\mu _{\alpha ,d}(x)\\&\le C_{1}(1+\Vert y\Vert ^{2})^{\frac{r}{2}}\int _{{\mathbb {R}}_{+}^{d+1} }(1+\Vert x\Vert ^{2})^{-\frac{l}{2}}\left(1+x_{d+1}^{4n}\right) d\mu _{\alpha ,d}(x)\\&\le C_{2}(1+\Vert y\Vert ^{2})^{\frac{r}{2}} \end{aligned}$$

where $l>2\alpha +4n+d+2$ and

$$\begin{aligned} C_{2}=C_{1}\int _{{\mathbb {R}}_{+}^{d+1}}(1+\Vert x\Vert ^{2})^{-\frac{l}{2} }\left(1+x_{d+1}^{4n}\right) d\mu _{\alpha ,d}(x)=C_{2}\left(r,k,\alpha ,d,n\right) . \end{aligned}$$

We put $m=\left[ \frac{t}{2}\right] +1,\,t\ge 0,$ where $\left[ \frac{t}{2}\right]$ is the integer part of $\frac{t}{2}$. We get

$$\begin{aligned} (1+\Vert \xi \Vert ^{2})^{m}\left| {\mathscr {F}}_{W}^{\alpha ,d,n}\left({\mathscr {M}}_{n}a\left(.,y\right) \right) \left(\xi \right) \right|&=\sum _{k=0}^{m}C_{m}^{k}\Vert \xi \Vert ^{2k}\left| {\mathscr {F}}_{W} ^{\alpha ,d,n}\left({\mathscr {M}}_{n}a\left(.,y\right) \right) \left(\xi \right) \right| \\&\le \sum _{k=0}^{m}C_{m}^{k}C_{2}\left(r,k,\alpha ,d,n\right) (1+\Vert y\Vert ^{2})^{\frac{r}{2}}\\&\le C(1+\Vert y\Vert ^{2})^{\frac{r}{2}} \end{aligned}$$

where C is a constant depending on $r,t,\alpha ,d,n$ and l.

Hence, we obtain

$$\begin{aligned} \left| {\mathscr {F}}_{W}^{\alpha ,d,n}\left({\mathscr {M}}_{n}a\left(.,y\right) \right) \left(\xi \right) \right|&\le C(1+\Vert y\Vert ^{2})^{\frac{r}{2}}(1+\Vert \xi \Vert ^{2})^{-m}\\&\le C(1+\Vert y\Vert ^{2})^{\frac{r}{2}}(1+\Vert \xi \Vert ^{2})^{-\frac{t}{2}}. \end{aligned}$$

$\square$

The following theorem gives an alternative form of $A\left(a,\ \Delta _{W}^{\alpha ,d}\right)$ which will be useful in the sequel.

Theorem 4.2

Let $l>2\alpha +4n+d+2$, $a\left(x,\ \xi \right) \in {\mathcal {S}}^{r,l}$ and $u\in {\mathscr {S}}_{n,*}({\mathbb {R}}^{d+1})$. Then, the pseudo-differential operator $A\left(a,\ \Delta _{W}^{\alpha ,d}\right)$ admits the following representation:

$$\begin{aligned} \left[ A\left(a,\Delta _{W}^{\alpha ,d}\right) u\right] \left(x\right) =C_{\alpha +2n,d}^{2}\int _{{\mathbb {R}}_{+}^{d+1}}\Lambda _{\alpha ,d,n} (-x,\xi)\times \end{aligned}$$

(4.6)

$$\begin{aligned} \left[ \int _{{\mathbb {R}}_{+}^{d+1}}{\mathscr {M}}_{n,\xi }^{-1}\mathscr {M}_{n,y}^{-1}T_{-y}^{\alpha ,d,n}{\mathscr {M}}_{n}{\mathscr {F}}_{W}^{\alpha ,d,n}\left({\mathscr {M}}_{n}a\left(.,y\right) \right) \left(\xi \right) {\mathscr {F}}_{W}^{\alpha ,d,n}(u)\left(y\right) d\mu _{\alpha +2n,d}\left(y\right) \right] d\mu _{\alpha +2n,d}\left(\xi \right) \end{aligned}$$

where all involved integrals are absolutely convergent.

Proof

We put

$$\begin{aligned} g_{x}\left(y,\xi \right) =\Lambda _{\alpha ,d,n}(-x,\xi){\mathscr {M}}_{n,\xi }^{-1}{\mathscr {M}}_{n,y}^{-1}T_{-y}^{\alpha ,d,n}{\mathscr {M}}_{n}\mathscr {F}_{W}^{\alpha ,d,n}\left({\mathscr {M}}_{n}a\left(.,y\right) \right) \left(\xi \right) {\mathscr {F}}_{W}^{\alpha ,d,n}(u)\left(y\right) . \end{aligned}$$

We shall prove that $g_{x}$ belongs to $L^{1}\left({\mathbb {R}}_{+} ^{d+1}\times {\mathbb {R}}_{+}^{d+1},\ d\mu _{\alpha +2n,d}\left(y\right) d\mu _{\alpha +2n,d}\left(\xi \right) \right) .$

Let $t>l$ and $\gamma >\frac{r}{2}-\frac{t}{2}+\frac{d}{2}+\alpha +2n+1.$ Using the relations (2.19), (2.7) and (4.5), we obtain

$$\begin{aligned}&\left| {\mathscr {M}}_{n,\xi }^{-1}{\mathscr {M}}_{n,y}^{-1}T_{-y}^{\alpha ,d,n}{\mathscr {M}}_{n}{\mathscr {F}}_{W}^{\alpha ,d,n}\left({\mathscr {M}}_{n}a\left(.,\,y\right) \right) \left(\xi \right) \right| \\&\quad\le C_{1}(1+\Vert y\Vert ^{2})^{\frac{r}{2}}\left| T_{-y}^{\alpha +2n,d}\left[ (1+\Vert x\Vert ^{2})^{-\frac{t}{2}}\right] \left(\xi \right) \right| . \end{aligned}$$

Hence from (2.21), we get

$$\begin{aligned} \left| {\mathscr {M}}_{n,\xi }^{-1}{\mathscr {M}}_{n,y}^{-1}T_{-y}^{\alpha ,d,n}{\mathscr {M}}_{n}{\mathscr {F}}_{W}^{\alpha ,d,n}\left({\mathscr {M}}_{n}a\left(.,\,y\right) \right) \left(\xi \right) \right| \le C_{2}(1+\Vert y\Vert ^{2})^{\frac{r-t}{2}}(1+\Vert \xi \Vert ^{2})^{-\frac{t}{2}} \end{aligned}$$

(4.7)

where $C_{2}$ is a constant depending on $r,t,\alpha ,d,n$ and l. On the other hand since $u\in {\mathscr {S}}_{n,*}({\mathbb {R}}^{d+1})$, then there exist $C_{3}>0$ such that

$$\begin{aligned} \forall y\in {\mathbb {R}}_{+}^{d+1},\,\left| {\mathscr {F}}_{W}^{\alpha ,d,n}(u)\left(y\right) \right| \le C_{3}(1+\Vert y\Vert ^{2})^{-\gamma }. \end{aligned}$$

Hence, we get

$$\begin{aligned} \left| g_{x}\left(y,\xi \right) \right| \le Cx_{d+1}^{2n}(1+\Vert y\Vert ^{2})^{\frac{r}{2}-\frac{t}{2}-\gamma }(1+\Vert \xi \Vert ^{2})^{-\frac{t}{2}}. \end{aligned}$$

Since $t>l>2\alpha +4n+d+2$ and $\gamma >\frac{r}{2}-\frac{t}{2}+\frac{d}{2}+\alpha +2n+1,$ the function $g_{x}$ belongs to $L^{1}\left({\mathbb {R}} _{+}^{d+1}\times {\mathbb {R}}_{+}^{d+1},d\mu _{\alpha +2n,d}\left(y\right) d\mu _{\alpha +2n,d}\left(\xi \right) \right)$. So, the result follows by applying the inverse theorem and using the relation (2.25). $\square$

Now, we are in a situation to establish the fundamental result of this section given by the following result.

Theorem 4.3

Let $s,\ \frac{l}{2}>\alpha +2n+\frac{d}{2}+1,\,a\left(x,\,\xi \right) \in {\mathcal {S}}^{r,l}$ and $A\left(x,\ \Delta _{W} ^{\alpha ,d,n}\right)$ be the associated pseudo-differential operator. Then $A\left(a,\ \Delta _{W}^{\alpha ,d,n}\right)$ maps continuously from ${\mathscr {H}}^{s+r,\alpha ,n}({\mathbb {R}}_{+}^{d+1})$ to ${\mathscr {H}}^{s,\alpha ,n}({\mathbb {R}}_{+}^{d+1})$. Moreover, we have

$$\begin{aligned} \forall u\in {\mathscr {S}}_{n,*}({\mathbb {R}}^{d+1})\text {, }\left\| A\left(a,\ \Delta _{W}^{\alpha ,d,n}\right) u\right\| _{{\mathscr {H}}^{s,\alpha ,n} }\le k\left\| u\right\| _{{\mathscr {H}}^{s+r,\alpha ,n}} \end{aligned}$$

(4.8)

where k is a constant depending on $s,r,\alpha ,d,n$ and l.

Proof

Let $s,\frac{l}{2}>\alpha +2n+\frac{d}{2}+1$. We put

$$\begin{aligned} \varphi _{s}\left(\xi \right) =\int _{{\mathbb {R}}_{+}^{d+1}}{\mathscr {M}}_{n,\xi }^{-1}{\mathscr {M}}_{n,y}^{-1}T_{y}^{\alpha ,d,n}{\mathscr {M}}_{n}\mathscr {F}_{W}^{\alpha ,d,n}\left({\mathscr {M}}_{n}a\left(.,y\right) \right) \left(-\xi \right) {\mathscr {F}}_{W}^{\alpha ,d,n}(u)\left(y\right) d\mu _{\alpha +2n,d}\left(y\right) \end{aligned}$$

From the relations (4.7), we have

$$\begin{aligned} \left| \varphi _{s}\left(\xi \right) \right| \le C_{2}(1+\Vert \xi \Vert ^{2})^{-\frac{t}{2}}\int _{{\mathbb {R}}_{+}^{d+1}}(1+\Vert y\Vert ^{2})^{\frac{r-t}{2}}\left| {\mathscr {F}}_{W}^{\alpha ,d,n}(u)\left(y\right) \right| d\mu _{\alpha +2n,d}\left(y\right) . \end{aligned}$$

Hence using the Cauchy-Schwartz inequality, we obtain

$$\begin{aligned} C_{\alpha +2n,d}(1+\Vert \xi \Vert ^{2})^{\frac{s}{2}}\left| \varphi _{s}\left(\xi \right) \right| \le C_{3}(1+\Vert \xi \Vert ^{2})^{\frac{s}{2}-\frac{t}{2}}\left\| u\right\| _{{\mathscr {H}}^{s+r,\alpha ,n}} \end{aligned}$$

where

$$\begin{aligned} C_{3}=C_{2}\left(\int _{{\mathbb {R}}_{+}^{d+1}}(1+\Vert y\Vert ^{2})^{-s-t} d\mu _{\alpha +2n,d}\left(y\right) \right) ^{\frac{1}{2}}. \end{aligned}$$

Then

$$\begin{aligned} \left\| A\left(a,\ \Delta _{W}^{\alpha ,d,n}\right) u\right\| _{{\mathscr {H}}^{s,\alpha ,n}}=C_{\alpha +2n,d}\left\| (1+\Vert \xi \Vert ^{2})^{\frac{s}{2}}\varphi _{s}\right\| _{\alpha +2n,2}\le k\left\| u\right\| _{{\mathscr {H}}^{s+r,\alpha ,n}} \end{aligned}$$

where $t>\left| s\right| +\alpha +2n+\displaystyle \frac{d}{2}+1$ and

$$\begin{aligned} k=C_{3}\left(\int _{{\mathbb {R}}_{+}^{d+1}}(1+\Vert \xi \Vert ^{2})^{s-t} d\mu _{\alpha +2n,d}\left(\xi \right) \right) ^{\frac{1}{2}}. \end{aligned}$$

$\square$