On function spaces with fractional Fourier transform in weighted Lebesgue spaces

Abstract

Let w and ω be weight functions on $\mathbb{R} ^{d}$. In this work, we define $A_{\alpha,p}^{w,\omega} ( \mathbb{R}^{d} )$ to be the vector space of $f\in L_{w}^{1} ( \mathbb{R}^{d} )$ such that the fractional Fourier transform ${F_{\alpha}}f$ belongs to $L_{\omega}^{p} ( \mathbb{R}^{d} )$ for $1\leq p<\infty$. We endow this space with the sum norm ${\Vert f\Vert _{A_{\alpha,p}^{w,\omega}}}={\Vert f\Vert _{1,w}}+ {\Vert {{F_{\alpha}}f} \Vert _{p,\omega}}$ and show that $A_{\alpha,p}^{w,\omega} ( \mathbb{R}^{d} )$ becomes a Banach space and invariant under time-frequency shifts. Further we show that the mapping $y\rightarrow{T_{y}f}$ is continuous from $\mathbb{R} ^{d}$ into $A_{\alpha,p}^{w,\omega} ( \mathbb{R}^{d} )$, the mapping $z\rightarrow{M_{z}}f$ is continuous from $\mathbb{R}^{d}$ into $A_{\alpha,p}^{w,\omega} ( \mathbb{R}^{d} )$ and $A_{\alpha,p}^{w,\omega} ( \mathbb{R}^{d} )$ is a Banach module over $L_{w}^{1} ( \mathbb{R}^{d} )$ with Θ convolution operation. At the end of this work, we discuss inclusion properties of these spaces.

Introduction

In this work, for any function $f: \mathbb{R} ^{d}\rightarrow \mathbb{C}$, the translation and modulation operator are defined as ${T_{x}}f ( t ) =f ( {t-x} )$ and ${M_{w}}f ( t ) ={e^{iwt}} f ( t )$ for all $y,w\in \mathbb{R}^{d}$, respectively. Also we write the Lebesgue space $( {{L^{p}} ( \mathbb{R}^{d} ) ,{{\Vert \cdot \Vert }_{p}}} )$, for $1\leq p<\infty$. Let w be a weight function on $\mathbb{R} ^{d}$, that is, a measurable and locally bounded function w satisfying $w ( x ) \geq1$ and $w ( {x+y} ) \leq w ( x ) w ( y )$ for all $x,y\in \mathbb{R}^{d}$. We define, for $1\leq p<\infty$,

$$ L_{w}^{p} \bigl( \mathbb{R} ^{d} \bigr) = \bigl\{ f|{fw\in{L^{p}} \bigl( \mathbb{R} ^{d} \bigr) } \bigr\} . $$

It is well known that $L_{w}^{p} ( \mathbb{R} ^{d} )$ is a Banach space under the norm ${\Vert f\Vert _{p,w}}={\Vert {fw}\Vert _{p}}$.

Let ${w_{1}}$ and ${w_{2}}$ are two weight functions. We say that ${w_{1}}\prec{w_{2}}$ if there exists $c>0$, such that ${w_{1}} ( x ) \leq c{w_{2}} ( x )$ for all $x\in \mathbb{R}^{d}$ [1, 2].

The Fourier transform $\hat{f}$ (or $\mathcal{F}f$) of $f\in{L^{1}} ( \mathbb{R} )$ is given by

$$ \hat{f}(w)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}{f(t){e^{-iwt}} \, dt}. $$

The fractional Fourier transform is a generalization of the Fourier transform with a parameter α and can be interpreted as a rotation by an angle α in the time-frequency plane. The fractional Fourier transform with angle α of a function f is defined by

$$ \mathcal{F}_{\alpha}f(u)=\int_{-\infty}^{+\infty}{{K_{\alpha}}(u,t)f(t) \, dt}, $$

where

$$ {K_{\alpha}}(u,t)=\left \{ \begin{array}{l@{\quad}l} \sqrt{\frac{{1-i\cot\alpha}}{{2\pi}}}{e^{i ( {\frac {{{u^{2}}+{t^{2}}} }{2}} ) \cot\alpha-iut\operatorname{cosec}\alpha}},&\text{if } \alpha\text{ is not multiple of }\pi, \\ \delta(t-u),&\text{if }\alpha=2k\pi,k\in \mathbb{Z}, \\ \delta(t+u),&\text{if }\alpha=(2k+1)\pi,k\in \mathbb{Z}, \end{array} \right . $$

and δ is a Dirac delta function. The fractional Fourier transform with $\alpha=\frac{\pi}{2}$ corresponds to the Fourier transform [3–9].

The fractional Fourier transform can be extended to higher dimensions as [9]:

$$\begin{aligned}& ( \mathcal{F} {{_{{\alpha_{1}},\ldots,{\alpha_{n}}}}f} ) ( {{u_{1}}, \ldots,{u_{n}}} ) \\& \quad =\int_{-\infty}^{+\infty}{\cdots\int_{-\infty}^{+\infty}{{K_{{\alpha_{1}},\ldots,{\alpha_{n}}}} ( {{u_{1}},\ldots,{u_{n}};{t_{1}}, \ldots,{t_{n}}} ) }}f ( {{t_{1}}, \ldots,{t_{n}}} ) \, d{t_{1}}\cdots \, d{t_{n}}, \end{aligned}$$

or shortly

$$ \mathcal{F}_{\alpha}f ( u ) =\int_{-\infty}^{+\infty}{\cdots\int_{-\infty}^{+\infty}{{K_{\alpha}} ( {u,t} ) }}f ( t )\, dt, $$

where

$$ {K_{\alpha}} ( {u,t} ) ={K_{{\alpha_{1}},\ldots,{\alpha_{n}}}} ( {{u_{1}},\ldots,{u_{n}};{t_{1}}, \ldots,{t_{n}}} ) ={K_{{\alpha _{1}}}} ( {{u_{1}},{t_{1}}} ) {K_{{\alpha_{2}}}} ( {{u_{2}},{t_{2}}} ) \cdots{K_{{\alpha_{n}}}} ( {{u_{n}},{t_{n}}} ) . $$

In this work we define the function spaces with fractional Fourier transform in weighted Lebesgue spaces and discuss some properties of these spaces.

On function spaces with fractional Fourier transform in weighted Lebesgue spaces

Definition 1

Let w and ω be weight functions on $\mathbb{R}^{d}$ and $1\leq p<\infty$. The space $A_{\alpha,p}^{w,\omega} ( \mathbb{R} ^{d} )$ consist of all ${f\in L_{w}^{1} ( \mathbb{R} ^{d} ) }$ such that $\mathcal{F}{{_{\alpha}f}\in L_{\omega }^{p} ( \mathbb{R} ^{d} ) }$. The norm on the vector space $A_{\alpha,p}^{w,\omega } ( \mathbb{R} ^{d} )$ is

$$ {\Vert f\Vert _{A_{\alpha,p}^{w,\omega}}}={\Vert f\Vert _{1,w}}+{\Vert \mathcal{F} {_{\alpha}f} \Vert _{p,\omega}}. $$

Theorem 2

$( {A_{\alpha,p}^{w,\omega} ( \mathbb{R} ^{d} ) ,{{\Vert \cdot \Vert }_{A_{\alpha,p}^{w,\omega}}}} )$ is a Banach space for $1\leq p<\infty$.

Proof

Let $( f_{n} ) _{n\in \mathbb{N}}$ is a Cauchy sequence in ${A_{\alpha,p}^{w,\omega} ( \mathbb{R} ^{d} ) }$. Thus $( f_{n} ) _{n\in \mathbb{N}}$ and $( \mathcal{F}{_{\alpha}}f_{n} ) _{n\in \mathbb{N}}$ are Cauchy sequences in ${L_{w}^{1} ( \mathbb{R} ^{d} ) }$ and ${L_{\omega}^{p} ( \mathbb{R} ^{d} ) }$, respectively. Since ${L_{w}^{1} ( \mathbb{R} ^{d} ) }$ and ${L_{\omega}^{p} ( \mathbb{R} ^{d} ) }$ are Banach spaces, there exist $f\in{L_{w}^{1} ( \mathbb{R} ^{d} ) }$ and $g\in{L_{\omega}^{p} ( \mathbb{R} ^{d} ) }$ such that ${\Vert f_{n}-f\Vert _{1,w}\rightarrow0}$, ${\Vert \mathcal{F}{_{\alpha}}f_{n}-g\Vert _{p,\omega }\rightarrow0}$ and hence ${\Vert f_{n}-f\Vert _{1}\rightarrow0}$ and ${\Vert \mathcal{F}{_{\alpha}}f_{n}-g\Vert _{p}\rightarrow0}$. Then $( \mathcal{F}{_{\alpha}}f_{n} ) _{n\in \mathbb{N} }$ has a subsequence $( \mathcal{F}{_{\alpha}}f_{n_{k}} ) _{n_{k}\in \mathbb{N}}$ that converges pointwise to g almost everywhere. Also it is easy to see that ${\Vert f_{n_{k}}-f\Vert _{1}\rightarrow0}$. Then we have

$$\begin{aligned} \bigl\vert \mathcal{F}_{\alpha}f ( u ) -g ( u ) \bigr\vert \leq&\bigl\vert \mathcal{F}_{\alpha} ( {{f_{{n_{k}}}}-f} ) ( u ) \bigr\vert +\bigl\vert \mathcal{F}_{\alpha }{f_{{n_{k}}}} ( u ) -g ( u ) \bigr\vert \\ \leq&\prod_{j=1}^{d}{\biggl\vert \sqrt{\frac{{1-i\cot{\alpha_{j}}}}{{2\pi}}}\biggr\vert } \\ &{}\times \int_{\mathbb{R} ^{d}}{\bigl\vert { ( {{f_{{n_{k}}}}-f} ) (t)}\bigr\vert \bigl\vert {{e^{\sum_{j=1}^{d}{ ( {\frac{i}{2} ( {{u_{j}}^{2}+{t_{j}}^{2}} ) \cot{\alpha_{j}}-i{u_{j}}{t_{j}}\operatorname{cosec}{\alpha_{j}}} ) }}}}\bigr\vert \, dt} \\ &{}+\bigl\vert \mathcal{F} {{_{\alpha }} {f_{{n_{k}}}} ( u ) -g ( u ) }\bigr\vert \\ =&\prod_{j=1}^{d}{\biggl\vert \sqrt{\frac{{1-i\cot{\alpha_{j}}}}{{2\pi}}}\biggr\vert } {\Vert {{f_{{n_{k}}}}-f} \Vert _{1}}+\bigl\vert \mathcal{F} {{_{\alpha}} {f_{{n_{k}}}} ( u ) -g ( u ) }\bigr\vert . \end{aligned}$$

From this inequality, we obtain $\mathcal{F}_{\alpha}f=g$ almost everywhere. Thus ${{{\Vert f_{n}-f\Vert }_{A_{\alpha ,p}^{w,\omega}}\rightarrow0}}$ and $f\in{A_{\alpha,p}^{w,\omega } ( \mathbb{R} ^{d} ) }$. Hence $( {A_{\alpha,p}^{w,\omega} ( \mathbb{R} ^{d} ) ,{{\Vert \cdot \Vert }_{A_{\alpha,p}^{w,\omega}}}} )$ is a Banach space. □

The following proposition is generalization of the one-dimensional and two-dimensional versions. The proof of this proposition is very similar to the proofs of one-dimensional and two-dimensional versions in [3, 5, 10, 11], and we omit the details.

Proposition 3

Let $\alpha= ( {{\alpha_{1}},{\alpha_{2}},\ldots,{\alpha _{d}}} )$, where ${\alpha_{i}}\neq k\pi$ for each index i with $1\leq i\leq d$ and $k\in \mathbb{Z}$. Then

$$ (1) \quad \mathcal{F}_{\alpha} ( {{T_{y}}f} ) (u)={e^{\sum_{j=1}^{d}{ ( {\frac{i}{2}y_{j}^{2}\sin{\alpha_{j}}\cos{\alpha_{j}}-i{u_{j}}{y_{j}}\sin{\alpha_{j}}} ) }}}\mathcal{F} {_{\alpha}}f({u_{1}}-{y_{1}} \cos{\alpha_{1}},\ldots,{u_{d}}-{y_{d}}\cos{ \alpha_{d}}) $$

(1)

for all $f\in{L^{1}} ( \mathbb{R}^{d} )$ and $y\in \mathbb{R}^{d}$;

$$ (2) \quad \mathcal{F}_{\alpha} ( {{M_{v}}f} ) (u)={e^{\sum_{j=1}^{d}{ ( {-\frac{i}{2}v_{j}^{2}\sin{\alpha_{j}}\cos {\alpha_{j}+}i{u_{j}}{v_{j}}\cos{\alpha_{j}}} ) }}}\mathcal{F} {_{\alpha}}f({u_{1}}-{v_{1}} \sin{\alpha_{1}},\ldots,{u_{d}}-{v_{d}}\sin{ \alpha_{d}}) $$

for all $f\in{L^{1}} ( \mathbb{R} ^{d} )$ and $v\in \mathbb{R} ^{d}$.

Theorem 4

Let $\alpha= ( {{\alpha_{1}},{\alpha_{2}},\ldots,{\alpha _{d}}} )$, where ${\alpha_{i}}\neq k\pi$ for each index i with $1\leq i\leq d$ and $k\in \mathbb{Z}$.

• (1) Let $1\leq p<\infty$, w and ω be weight functions on $\mathbb{R} ^{d}$. Then the space $A_{\alpha,p}^{w,\omega} ( \mathbb{R} ^{d} )$ is translation invariant.

• (2) Let ω be a bounded weight function on $\mathbb{R} ^{d}$. Then the mapping $y\rightarrow{T_{y}}f$ of $\mathbb{R}^{d}$ into $A_{\alpha,p}^{w,\omega} ( \mathbb{R} ^{d} )$ is continuous.

Proof

(1) Let $f\in{A_{\alpha,p}^{w,\omega} ( \mathbb{R} ^{d} ) }$. Then $f\in{L_{w}^{1} ( \mathbb{R} ^{d} ) }$ and $\mathcal{F}{_{\alpha}}f\in{L_{\omega}^{p} ( \mathbb{R} ^{d} ) }$. It is well known that the space ${L_{w}^{1} ( \mathbb{R} ^{d} ) }$ is translation invariant and holds ${\Vert {{T_{y}}f} \Vert _{1,w}}\leq w ( y ) {\Vert f\Vert _{1,w}}$ for all $y\in \mathbb{R} ^{d}$ [12]. Let $b= ( {{y_{1}}\cos{\alpha _{1}},\ldots,{y_{d}}\cos{\alpha_{d}}} )$. By using the equality (1), we get

$$\begin{aligned} {\bigl\Vert \mathcal{F} {_{\alpha}} ( {{T_{y}}f} ) \bigr\Vert _{p,\omega}} =&{ \biggl( {\int_{\mathbb{R} ^{d}}{{{\bigl\vert \mathcal{F} {_{\alpha}} ( {{T_{y}}f} ) { ( u ) }\bigr\vert }^{p}} {\omega^{p}} ( u ) \, du}} \biggr) ^{\frac{1}{p}}} \\ =&\biggl( {\int_{\mathbb{R} ^{d}}{{{\bigl\vert \mathcal{F} {_{\alpha}f ( {{u_{1}}-{y_{1}}\cos { \alpha_{1}},\ldots,{u_{d}}-{y_{d}}\cos{ \alpha_{d}}} ) }\bigr\vert }^{p}}}} \\ &\biggl.\biggl.{}\times {{{{\bigl\vert {{e^{\sum_{j=1}^{d}{ ( {\frac{i}{2}{y_{j}}^{2}\sin{\alpha_{j}}\cos{\alpha_{j}}-i{u_{j}}{y_{j}}\sin{\alpha_{j}}} ) }}}} \bigr\vert }^{p}} {\omega^{p}} ( u )\, du}}\biggr) \biggr.^{\frac{1}{p}} \\ \leq&{ \biggl( {\int_{\mathbb{R} ^{d}}{{{\bigl\vert \mathcal{F} {_{\alpha}f ( {u-b} ) } \bigr\vert }^{p}} { \omega^{p}} ( {u-b} ) {\omega^{p}} ( b ) \,du}} \biggr) ^{\frac{1}{p}}} \\ =&\omega ( b ) {\Vert \mathcal{F} {_{\alpha}f} \Vert _{p,\omega}} \end{aligned}$$

for all $y\in \mathbb{R} ^{d}$. Hence, we have

$$ {\Vert {{T_{y}}f}\Vert _{A_{\alpha,p}^{w,\omega}}}\leq w ( y ) {\Vert f \Vert _{1,w}}+\omega ( b ) { \Vert \mathcal{F} {_{\alpha}f} \Vert _{p,\omega}}< \infty. $$

This means that $A_{\alpha,p}^{w,\omega} ( \mathbb{R}^{d} )$ is translation invariant.

(2) Let f $\in{A_{\alpha,p}^{w,\omega} ( \mathbb{R} ^{d} ) }$. We will show that if $\lim_{n\rightarrow\infty }{y_{n}}=0$ for any sequence ${ ( {{y_{n}}} ) _{n\in \mathbb{N} }}\subset \mathbb{R} ^{d}$, then $\lim_{n\rightarrow\infty}{{T_{{y_{n}}}}f}=f$, which will complete the proof. It is well known that the mapping $y\rightarrow{T_{y}}f$ is continuous from $\mathbb{R}^{d}$ into ${L_{w}^{1} ( \mathbb{R} ^{d} ) }$ (see [12]). Thus, we have

$$ {\Vert {{T_{{y_{n}}}}f-f}\Vert _{1,w}}\rightarrow0 $$

(2)

as $n\rightarrow\infty$. Also,

$$\begin{aligned} {\bigl\Vert \mathcal{F} {_{\alpha} ( {{T_{{y_{n}}}}f-f} ) }\bigr\Vert _{p,\omega}} =&{\bigl\Vert \mathcal{F} _{\alpha} ( {{T_{{y_{n}}}}f} ) -\mathcal{F} {_{\alpha}f}\bigr\Vert _{p,\omega}} \\ =&{\bigl\Vert {e^{\sum_{j=1}^{d}{ ( {\frac{i}{2}{{ ( {y_{n}^{j}} ) }^{2}}\sin{\alpha_{j}}\cos{\alpha_{j}}-i{u_{j}}y_{n}^{j}\sin{\alpha _{j}}} ) }}} {T_{ ( {y_{n}^{1}\cos{\alpha_{1}},\ldots,y_{n}^{d}\cos {\alpha _{d}}} ) }} ( \mathcal{F} {_{\alpha}f} ) -\mathcal{F} {{ _{\alpha}}f}\bigr\Vert _{p,\omega}} \\ \leq&{\bigl\Vert { \bigl( {T_{ ( {y_{n}^{1}\cos{\alpha_{1}},\ldots,y_{n}^{d}\cos {\alpha _{d}}} ) }} ( \mathcal{F} {_{\alpha}f} ) -\mathcal{F} {{ _{\alpha}}f} \bigr) }\bigr\Vert _{p,\omega}} \\ &{} +{\bigl\Vert { \bigl( {e^{\sum_{j=1}^{d}{ ( {\frac{i}{2}{{ ( {y_{n}^{j}} ) }^{2}}\sin{\alpha_{j}}\cos{\alpha _{j}}-i{u_{j}}y_{n}^{j}\sin{\alpha_{j}}} ) }}}-1 \bigr) } \mathcal{F} {_{\alpha}f} \bigr\Vert _{p,\omega}}. \end{aligned}$$

Since $\mathcal{F}{_{\alpha}}f\in{L_{\omega}^{p} ( \mathbb{R} ^{d} ) }$, the mapping $y\rightarrow{T_{y}} ( \mathcal{F}{{_{\alpha}}}f )$ is continuous from $\mathbb{R}^{d}$ into ${L_{\omega}^{p} ( \mathbb{R}^{d} ) }$ for all $y\in \mathbb{R}^{d}$ [12]. Then we obtain ${\Vert {T_{ ( {y_{n}^{1}\cos{\alpha_{1}},\ldots,y_{n}^{d}\cos{\alpha_{d}}} ) }} ( \mathcal{F}{_{\alpha}f} ) -\mathcal{F}{_{\alpha}f} \Vert _{p,\omega}}\rightarrow0$ as $n\rightarrow\infty$. Now let ${h_{{y_{n}}}} ( u ) =\vert {{e^{\sum_{j=1}^{d}{ ( {\frac{i}{2}{{ ( {y_{n}^{j}} ) }^{2}}\sin{\alpha_{j}}\cos{\alpha _{j}}-i{u_{j}}y_{n}^{j}\sin{\alpha_{j}}} ) }}}-1}\vert \vert \mathcal{F}{_{\alpha}f ( u ) }\vert$. Since $\lim_{n\rightarrow\infty}{y_{n}}=0$ and ω is a bounded weight function on $\mathbb{R}^{d}$, we see that $\lim_{n\rightarrow\infty}h_{{y_{n}}}^{p} ( u ) {\omega^{p}} ( u ) =0$ for all $u\in \mathbb{R}^{d}$. Also, since

$$ {h_{{y_{n}}}} ( u ) =\bigl\vert {{e^{\sum_{j=1}^{d}{ ( {\frac{i}{2}{{ ( {y_{n}^{j}} ) }^{2}}\sin{\alpha_{j}}\cos {\alpha _{j}}-i{u_{j}}y_{n}^{j}\sin{\alpha_{j}}} ) }}}-1}\bigr\vert \bigl\vert \mathcal{F} {_{\alpha}f ( u ) }\bigr\vert \leq 2\bigl\vert \mathcal{F} {_{\alpha}f ( u ) }\bigr\vert $$

and $\mathcal{F}_{\alpha}f\in L_{\omega}^{p} ( \mathbb{R}^{d} )$, we can write $h_{{y_{n}}}^{p} ( u ) {\omega ^{p}} ( u ) \leq{2^{p}}{\vert \mathcal{F}{_{\alpha}f ( u ) }\vert ^{p}}{\omega^{p}} ( u )$. Thus, by the Lebesgue dominated convergence theorem,

$$ {\bigl\Vert { \bigl( {{e^{\sum_{j=1}^{d}{ ( {\frac{i}{2}{{ ( {y_{n}^{j}} ) }^{2}}\sin{\alpha_{j}}\cos{\alpha_{j}}-i{u_{j}}y_{n}^{j}\sin{\alpha_{j}}} ) }}}-1} \bigr) }\mathcal {F} {_{\alpha}f}\bigr\Vert _{p,\omega}}\rightarrow0 $$

as $\lim_{n\rightarrow\infty}{y_{n}}=0$. Hence,

$$ {\Vert {{T_{{y_{n}}}}f-f}\Vert _{A_{\alpha,p}^{w,\omega }}\rightarrow0} $$

(3)

as $n\rightarrow\infty$. Combining (2) and (3),

$$ {\Vert {{T_{{y_{n}}}}f-f}\Vert _{A_{\alpha,p}^{w,\omega }}}={\Vert {{T_{{y_{n}}}}f-f}\Vert _{1,w}}+{\bigl\Vert \mathcal {F} {{_{\alpha}} ( {{T_{{y_{n}}}}f-f} ) }\bigr\Vert _{p,\omega }\rightarrow0} $$

as $n\rightarrow\infty$. This is the desired result. □

Theorem 5

Let $\alpha= ( {{\alpha_{1}},{\alpha_{2}},\ldots,{\alpha _{d}}} )$, where ${\alpha_{i}}\neq k\pi$ for each index i with $1\leq i\leq d$ and $k\in \mathbb{Z}$.

• (1) Let $1\leq p<\infty$, w and ω be weight functions on $\mathbb{R}^{d}$. Then $A_{\alpha,p}^{w,\omega} ( \mathbb{R}^{d} )$ is invariant under modulations.

• (2) Let ω be a bounded weight function on $\mathbb{R}^{d}$. Then the mapping $z\rightarrow{M_{z}}f$ is continuous from $\mathbb{R}^{d}$ into $A_{\alpha,p}^{w,\omega} ( \mathbb{R}^{d} )$.

Proof

(1) Let f $\in{A_{\alpha,p}^{w,\omega} ( \mathbb{R} ^{d} ) }$. Then $f\in{L_{w}^{1} ( \mathbb{R} ^{d} ) }$ and $\mathcal{F}{_{\alpha}}f\in{L_{\omega}^{p} ( \mathbb{R} ^{d} ) }$. It is easy to see that ${\Vert {{M_{\eta}}f} \Vert _{1,w}}={\Vert f\Vert _{1,w}}$ and ${{M_{\eta }}f\in L_{w}^{1} ( \mathbb{R} ^{d} ) }$. Let $c= ( {{\eta_{1}}\sin{\alpha_{1}},\ldots,{\eta _{d}}\sin{\alpha_{d}}} ) \in \mathbb{R} ^{d}$. Thus by Proposition 3, we have

$$\begin{aligned} {\bigl\Vert \mathcal{F} {_{\alpha}} ( {{M_{\eta}}f} ) \bigr\Vert _{p,\omega}} =&{ \biggl( {\int_{\mathbb{R} ^{d}}{{{\bigl\vert \mathcal{F} {_{\alpha}} ( {{M_{\eta}}f} ) { ( u ) }\bigr\vert }^{p}} {\omega^{p}} ( u ) \, du}} \biggr) ^{\frac{1}{p}}} \\ =&\biggl( {{{\int_{\mathbb{R} ^{d}}{\bigl\vert \mathcal{F} {_{\alpha}f ( {{u_{1}}-{\eta _{1}}\sin{\alpha_{1}},\ldots,{u_{d}}-{\eta_{d}}\sin{ \alpha_{d}}} ) } \bigr\vert }}^{p}}} \\ &\biggl.\biggl.{}\times{{{{\bigl\vert {{e^{\sum_{j=1}^{d}{ ( {-\frac{i}{2}\eta_{j}^{2}\sin{\alpha_{j}}\cos{\alpha_{j}}+i{u_{j}}{\eta_{j}}\cos{\alpha_{j}}} ) }}}}\bigr\vert }^{p}} {\omega^{p}} ( u )\, du}}\biggr) \biggr.^{\frac {1}{p}} \\ \leq&{ \biggl( {\int_{\mathbb{R} ^{d}}{{{\bigl\vert \mathcal{F} {_{\alpha}f ( {u-c} ) } \bigr\vert }^{p}} { \omega^{p}} ( {u-c} ) {\omega^{p}} ( c )\, du}} \biggr) ^{\frac{1}{p}}} \\ =&\omega ( c ) {\Vert \mathcal{F} {_{\alpha}f} \Vert _{p,\omega}} \end{aligned}$$

for all ${\eta}\in \mathbb{R} ^{d}$. Hence, we get

$$ {\Vert {{M_{\eta}}f} \Vert _{A_{\alpha,p}^{w,\omega}}}\leq {\Vert f\Vert _{1,w}}+\omega ( c ) {\Vert \mathcal{F} {_{\alpha}f} \Vert _{p,\omega}}< \infty. $$

(2) The proof technique of this part is the same as that of Theorem 4(2). So, for the sake of brevity, we will not prove it. □

The following definition is an extension of the convolution in [13, 14] of two functions to n dimensions.

Definition 6

Let $\alpha= ( {{\alpha_{1}},{\alpha_{2}},\ldots,{\alpha _{d}}} )$, where ${\alpha_{i}}\neq k\pi$ for each index i with $1\leq i\leq d$ and $k\in \mathbb{Z}$. Then the convolution of two functions $f,g\in{L^{1}} ( \mathbb{R} ^{d} )$ is the function $f\Theta g$ defined by

$$ ( {f\Theta g} ) ( x ) =\int_{\mathbb{R}^{d}}{f ( y ) g ( {x-y} ) {e^{\sum_{j=1}^{d}{i{y_{j}} ( {{y_{j}}-{x_{j}}} ) \cot{\alpha_{j}}}}}\, dy}. $$

It is easy to see that $f\Theta g$ belongs to ${L^{1}} ( \mathbb{R} ^{d} )$ by Fubini’s theorem.

Theorem 7

Let $\alpha= ( {{\alpha_{1}},{\alpha_{2}},\ldots,{\alpha _{d}}} )$, where ${\alpha_{i}}\neq k\pi$ for each index i with $1\leq i\leq d$ and $k\in \mathbb{Z}$, and $f,g\in{L^{1}} ( \mathbb{R} ^{d} )$. Then

$$ \mathcal{F}_{\alpha} ( {f\Theta g} ) ( u ) = \Biggl[ {\prod _{j=1}^{d}\sqrt{\frac{{2\pi}}{{1-i\cot{\alpha_{j}}}}}} \Biggr] {e^{\sum_{j=1}^{d}{-\frac{i}{2}u_{j}^{2}\cot{\alpha_{j}}}}}\mathcal{F}_{\alpha}f ( u ) \mathcal{F}_{\alpha}g ( u ), $$

where $\mathcal{F}_{\alpha}f$ and $\mathcal{F}_{\alpha}g$ are the fractional Fourier transforms of functions f and g, respectively.

Proof

Let $\alpha= ( {{\alpha_{1}},{\alpha_{2}},\ldots,{\alpha _{d}}} )$, where ${\alpha_{i}}\neq k\pi$ for each index i with $1\leq i\leq d$ and $k\in \mathbb{Z}$, and $f,g\in{L^{1}} ( \mathbb{R} ^{d} )$. We can write from the definition of the fractional Fourier transform

$$\begin{aligned} \mathcal{F}_{\alpha} ( {f\Theta g} ) ( u ) =& \Biggl[ {\prod _{j=1}^{d}\sqrt{\frac{{1-i\cot{\alpha_{j}}}}{{2\pi}}}} \Biggr] \int_{ \mathbb{R} ^{d}}{ ( {f\Theta g} ) ( t ) {e^{\sum_{j=1}^{d}{ ( {\frac{i}{2} ( {u_{j}^{2}+t_{j}^{2}} ) \cot{\alpha _{j}}-i{u_{j}}{t_{j}}\operatorname{cosec}{\alpha_{j}}} ) }}}\, dt} \\ =& \Biggl[ {\prod_{j=1}^{d} \sqrt{\frac{{1-i\cot{\alpha_{j}}}}{{2\pi}}}} \Biggr] \int_{\mathbb{R} ^{d}}{\int _{ \mathbb{R} ^{d}}{f ( y ) g ( {t-y} ) }} {e^{\sum_{j=1}^{d}{i{y_{j}} ( {{y_{j}}-{t_{j}}} ) \cot{\alpha_{j}}}}} \\ &{}\times {e^{\sum_{j=1}^{d}{ ( {\frac{i}{2} ( {u_{j}^{2}+t_{j}^{2}} ) \cot{\alpha _{j}}-i{u_{j}}{t_{j}}\operatorname{cosec}{\alpha_{j}}} ) }}}\, dt\, dy. \end{aligned}$$

We make the substitution $t-y=k$ and obtain

$$\begin{aligned} \mathcal{F}_{\alpha} ( {f\Theta g} ) ( u ) =& \Biggl[ {\prod _{j=1}^{d}\sqrt{\frac{{1-i\cot{\alpha_{j}}}}{{2\pi}}}} \Biggr] \int_{\mathbb{R} ^{d}}{ \biggl( {\int_{\mathbb{R} ^{d}}{f ( y ) {e^{\sum_{j=1}^{d}{ ( {\frac{i}{2} ( {u_{j}^{2}+y_{j}^{2}} ) \cot{\alpha_{j}}-i{u_{j}}{y_{j}}\operatorname {cosec}{\alpha_{j}}} ) }}}\, dy}} \biggr) } \\ &{}\times g ( k ) {e^{\sum_{j=1}^{d}{ ( {\frac{i}{2}k_{j}^{2}\cot{\alpha_{j}}-i{u_{j}}{k_{j}}\operatorname{cosec}{\alpha_{j}}} ) }}}\, dk \\ =& \Biggl[ {\prod_{j=1}^{d}\sqrt{ \frac{{2\pi}}{{1-i\cot{\alpha_{j}}}}}} \Biggr] {e^{\sum_{j=1}^{d}{-\frac{i}{2}u_{j}^{2}\cot{\alpha_{j}}}}} { \Biggl[ {\prod _{j=1}^{d}\sqrt{\frac{{1-i\cot{\alpha_{j}}}}{{2\pi}}}} \Biggr] ^{2}} \\ &{}\times \int_{ \mathbb{R} ^{d}}{ \biggl( {\int_{ \mathbb{R} ^{d}}{f ( y ) {e^{\sum_{j=1}^{d}{ ( {\frac{i}{2} ( {u_{j}^{2}+y_{j}^{2}} ) \cot{\alpha_{j}}-i{u_{j}}{y_{j}}\operatorname {cosec}{\alpha_{j}}} ) }}}\, dy}} \biggr) } \\ &{}\times g ( k ) {e^{\sum_{j=1}^{d}{ ( {\frac{i}{2} ( {k_{j}^{2}+u_{j}^{2}} ) \cot{\alpha_{j}}-i{u_{j}}{k_{j}}\operatorname{cosec}{\alpha_{j}}} ) }}}\, dk \\ =& \Biggl[ {\prod_{j=1}^{d}\sqrt{ \frac{{2\pi}}{{1-i\cot{\alpha_{j}}}}}} \Biggr] {e^{\sum_{j=1}^{d}{-\frac{i}{2}u_{j}^{2}\cot{\alpha_{j}}}}} \Biggl[ {\prod _{j=1}^{d}\sqrt{\frac{{1-i\cot{\alpha_{j}}}}{{2\pi}}}} \Biggr] \\ &{}\times \int_{ \mathbb{R} ^{d}}\mathcal{F} {_{\alpha}f ( u ) g ( k ) { e^{\sum_{j=1}^{d}{ ( {\frac{i}{2} ( {k_{j}^{2}+u_{j}^{2}} ) \cot{\alpha_{j}}-i{u_{j}}{k_{j}}\operatorname{cosec}{\alpha _{j}}} ) }}}\, dk} \\ =& \Biggl[ {\prod_{j=1}^{d}\sqrt{ \frac{{2\pi}}{{1-i\cot{\alpha_{j}}}}}} \Biggr] {e^{\sum_{j=1}^{d}{-\frac{i}{2}u_{j}^{2}\cot{\alpha_{j}}}}}\mathcal{F}_{\alpha}f ( u ) \mathcal {F} {_{\alpha}}g ( u ) . \end{aligned}$$

 □

Theorem 8

Let $\alpha= ( {{\alpha_{1}},{\alpha_{2}},\ldots,{\alpha _{d}}} )$, where ${\alpha_{i}}\neq k\pi$ for each index i with $1\leq i\leq d$ and $k\in \mathbb{Z}$. $L_{w}^{1} ( \mathbb{R} ^{d} )$ is a Banach algebra under Θ convolution.

Proof

It is well known that $L_{w}^{1} ( \mathbb{R} ^{d} )$ is a Banach space [2]. Let $f,g\in L_{w}^{1} ( \mathbb{R} ^{d} )$, then we have

$$\begin{aligned} {\Vert {f\Theta g}\Vert _{1,w}} =&\int_{\mathbb{R} ^{d}}{ \vert {f\Theta g}\vert w ( x ) \, dy} \\ =&\int_{ \mathbb{R} ^{d}}{\biggl\vert {\int_{ \mathbb{R} ^{d}}{f ( y ) }g ( {x-y} ) {e^{\sum_{j=1}^{d}{i{y_{j}} ( {{y_{j}}-{x_{j}}} ) \cot{\alpha_{j}}}}}\, dy} \biggr\vert w ( x ) \, dx} \\ \leq&\int_{ \mathbb{R} ^{d}}{ \biggl( {\int_{ \mathbb{R} ^{d}}{ \bigl\vert {g ( {x-y} ) }\bigr\vert w ( {x-y} ) \, dx}} \biggr) \bigl\vert {f ( y ) }\bigr\vert }w ( y ) \, dy \\ =&{\Vert g\Vert _{1,w}}\int_{ \mathbb{R} ^{d}}{ \bigl\vert {f ( y ) }\bigr\vert }w ( y )\, dy \\ =&{\Vert g\Vert _{1,w}} {\Vert f\Vert _{1,w}}. \end{aligned}$$

(4)

It is easy to show that the other conditions of the Banach algebra are satisfied. □

Theorem 9

Let $\alpha= ( {{\alpha_{1}},{\alpha_{2}},\ldots,{\alpha _{d}}} )$, where ${\alpha_{i}}\neq k\pi$ for each index i with $1\leq i\leq d$ and $k\in \mathbb{Z}$. $A_{\alpha,p}^{w,\omega} ( \mathbb{R}^{d} )$ is a Banach Θ-convolution module over $L_{w}^{1} ( \mathbb{R} ^{d} )$.

Proof

It is well known that $A_{\alpha,p}^{w,\omega} ( \mathbb{R} ^{d} )$ is a Banach space by Theorem 2. Let $f\in A_{\alpha ,p}^{w,\omega} ( \mathbb{R} ^{d} )$ and $g\in L_{w}^{1} ( \mathbb{R} ^{d} )$. By using the inequality (4), we get

$$\begin{aligned} {\bigl\Vert \mathcal{F} {_{\alpha} ( {f\Theta g} ) } \bigr\Vert _{p,\omega}} =&{\Biggl\Vert { \Biggl[ {\prod_{j=1}^{d} \sqrt{\frac{{2\pi}}{{1-i\cot{\alpha_{j}}}}}} \Biggr] {e^{\sum_{j=1}^{d}{-\frac{i}{2}u_{j}^{2}\cot{\alpha_{j}}}}}}\mathcal{F} {_{\alpha}f ( u ) }\mathcal{F} {_{\alpha}g ( u ) }\Biggr\Vert _{p,\omega}} \\ =&\biggl\vert {\prod _{j=1}^{d}\sqrt{ \frac{{2\pi}}{{1-i\cot{\alpha_{j}}}}}}\biggr\vert { \biggl( {\int_{\mathbb{R} ^{d}}{{{ \bigl\vert \mathcal{F} {_{\alpha}f ( u ) } \bigr\vert }^{p}} {{\bigl\vert \mathcal{F} {_{\alpha}g ( u ) } \bigr\vert }^{p}} {\omega^{p}} ( u )\, du}} \biggr) ^{\frac{1}{p}}} \\ =&\biggl\vert {\prod _{j=1}^{d}\sqrt{ \frac{{2\pi}}{{1-i\cot{\alpha_{j}}}}}}\biggr\vert \Biggl( \int_{\mathbb{R} ^{d}}{{\bigl\vert \mathcal{F} {_{\alpha}f ( u ) } \bigr\vert } ^{p}} {{\Biggl\vert {\prod _{j=1}^{d}\sqrt{\frac{{1-i\cot{\alpha_{j}}}}{{2\pi}}}}\Biggr\vert }^{p}} \\ &\biggl.\biggl.{}\times\biggl\vert {\int_{\mathbb{R} ^{d}}{g ( t ) } {e^{\sum_{j=1}^{d}{ ( {\frac{i}{2} ( {u_{j}^{2}+t_{j}^{2}} ) \cot{\alpha_{j}}-i{u_{j}}{t_{j}}\operatorname {cosec}{ \alpha_{j}}} ) }}}\, dt}\biggr\vert ^{p} {\omega^{p}} ( u )\, du \Biggr) \biggr.^{\frac{1}{p}} \\ \leq&{ \biggl( {\int_{\mathbb{R} ^{d}}{{{\bigl\vert \mathcal{F} {_{\alpha}f ( u ) } \bigr\vert }^{p}} {{ \biggl( { \int_{\mathbb{R} ^{d}}{\bigl\vert {g ( t ) }\bigr\vert }\, dt} \biggr) }^{p}} {\omega ^{p}} ( u ) \,du}} \biggr) ^{\frac{1}{p}}} \\ =&{\Vert g\Vert _{1}} { \biggl( {\int _{\mathbb{R} ^{d}}{{{\bigl\vert \mathcal{F} {_{\alpha}f ( u ) } \bigr\vert }^{p}} {\omega^{p}} ( u )\, du}} \biggr) ^{\frac{1}{p}}} \\ \leq&{\Vert g\Vert _{1,w}} {\Vert \mathcal{F} {_{\alpha}f} \Vert _{p,\omega }}. \end{aligned}$$

(5)

Combining (4) and (5), we obtain

$$\begin{aligned} {\Vert {f\Theta g}\Vert _{A_{\alpha,p}^{w,\omega}}} =&{ \Vert {f\Theta g}\Vert _{1,w}}+{\bigl\Vert \mathcal{F} {_{\alpha} ( {f\Theta g} ) }\bigr\Vert _{p,\omega}} \\ \leq&{\Vert g\Vert _{1,w}} {\Vert f\Vert _{1,w}}+{\Vert g\Vert _{1,w}} { \Vert \mathcal{F} {_{\alpha}f} \Vert _{p,\omega}} \\ =&{\Vert f\Vert _{A_{\alpha ,p}^{w,\omega}} \Vert g\Vert _{1,w}}. \end{aligned}$$

This is the desired result. It is easy to see that the other conditions of the module are satisfied. □

Inclusion properties of the space $A_{\alpha ,p}^{w,\omega} (\mathbb{R}^{d} )$

Proposition 10

For every $0\neq f\in A_{\alpha,p}^{w,1} ( \mathbb{R} ^{d} )$ there exists $c ( f ) >0$ such that

$$ c ( f ) w ( x ) \leq{ \Vert {{T_{x}}f}\Vert _{A_{\alpha,p}^{w,1}}}\leq w ( x ) {\Vert f\Vert _{A_{\alpha,p}^{w,1}}}. $$

Proof

Let $0\neq f\in A_{\alpha,p}^{w,1} ( \mathbb{R}^{d} )$. By [12], there exists $c ( f ) >0$ such that

$$ c ( f ) w ( x ) \leq{ \Vert {{T_{x}}f}\Vert _{1,w}}\leq w ( x ) {\Vert f\Vert _{1,w}}. $$

(6)

By using (6) and the equality ${\Vert \mathcal {F}{_{\alpha} ( {{T_{x}}f} ) }\Vert _{p}}={\Vert \mathcal{F}{{_{\alpha}}f} \Vert _{p}}$, we obtain

$$\begin{aligned} c ( f ) w ( x ) &\leq{ \Vert {{T_{x}}f}\Vert _{1,w}} \leq{ \Vert {{T_{x}}f}\Vert _{1,w}}+{\bigl\Vert \mathcal{F} {_{\alpha} ( {{T_{x}}f} ) }\bigr\Vert _{p}} \\ &\leq w ( x ) {\Vert f\Vert _{1,w}}+{\Vert \mathcal {F} {_{\alpha}f} \Vert _{p}} \\ &\leq w ( x ) {\Vert f\Vert _{1,w}}+w ( x ) {\Vert \mathcal{F} {{_{\alpha }}f}\Vert _{p}} \\ &=w ( x ) {\Vert f\Vert _{A_{\alpha,p}^{w,1}}} \end{aligned}$$

for all $f\in A_{\alpha,p}^{w,1} ( \mathbb{R} ^{d} )$. □

Lemma 11

Let ${w_{1}}$, ${w_{2}}$, ${\omega_{1}}$ and ${\omega_{2}}$ be weight functions on $\mathbb{R}^{d}$. If $A_{\alpha,p}^{{w_{1}},{\omega_{1}}} ( \mathbb{R} ^{d} ) \subset A_{\alpha,p}^{{w_{2}},{\omega_{2}}} ( \mathbb{R} ^{d} )$, then $A_{\alpha,p}^{{w_{1}},{\omega_{1}}} ( \mathbb{R} ^{d} )$ is a Banach space under the norm $|\!|\!|f|\!|\!|={\Vert f\Vert _{A_{\alpha ,p}^{{w_{1}}, {\omega_{1}}}}}+{\Vert f\Vert _{A_{\alpha ,p}^{{w_{2}},{\omega _{2}}}}}$.

Proof

Let $( f_{n} ) _{n\in \mathbb{N}}$ is a Cauchy sequence in $( A_{\alpha,p}^{{w_{1}},{\omega_{1}}} ( \mathbb{R} ^{d} ) ,|\!|\!|\cdot|\!|\!|)$. Then $( f_{n} ) _{n\in \mathbb{N}}$ is a Cauchy sequence in $( A_{\alpha,p}^{{w_{1}},{\omega_{1}}} ( \mathbb{R} ^{d} ) ,{\Vert \cdot \Vert _{A_{\alpha,p}^{{w_{1}},{\omega _{1}}}}} )$ and $( A_{\alpha,p}^{{w_{2}},{\omega_{2}}} ( \mathbb{R} ^{d} ) ,{\Vert \cdot \Vert _{A_{\alpha,p}^{{w_{2}},{\omega _{2}}}}} )$. As these spaces are Banach spaces, there exist $f\in A_{\alpha ,p}^{{w_{1}},{\omega_{1}}} ( \mathbb{R} ^{d} )$ and $g\in A_{\alpha,p}^{{w_{2}},{\omega_{2}}} ( \mathbb{R} ^{d} )$ such that ${\Vert {{f_{n}}-f}\Vert _{A_{\alpha ,p}^{{w_{1}},{\omega_{1}}}}}\rightarrow0$, ${\Vert {{f_{n}}-g} \Vert _{A_{\alpha,p}^{{w_{2}},{\omega_{2}}}}}\rightarrow0$. Using the inequalities ${\Vert \cdot \Vert _{1}}\leq{ \Vert \cdot \Vert _{1,{w_{1}}}}\leq{ \Vert \cdot \Vert _{A_{\alpha ,p}^{{w_{1}},{\omega _{1}}}}}$ and ${\Vert \cdot \Vert _{1}}\leq{ \Vert \cdot \Vert _{1,{w_{2}}}}\leq{ \Vert \cdot \Vert _{A_{\alpha ,p}^{{w_{2}},{\omega _{2}}}}}$, we obtain ${\Vert {{f_{n}}-f}\Vert _{1}}\rightarrow0$ and ${\Vert {{f_{n}}-g}\Vert _{1}}\rightarrow0$. Also ${\Vert {f-g}\Vert _{1}}\leq{ \Vert {{f_{n}}-f} \Vert _{1}}+{\Vert {{f_{n}}-g}\Vert _{1}}$, we have $f=g$. Hence $|\!|\!|{{f_{n}}-f}|\!|\!|\rightarrow0$ and $f\in A_{\alpha,p}^{{w_{1}},{\omega_{1}}} ( \mathbb{R} ^{d} )$. That means $( A_{\alpha,p}^{{w_{1}},{\omega_{1}}} (\mathbb{R}^{d} ) , |\!|\!|\cdot|\!|\!|)$ is a Banach space. □

Theorem 12

Let ${w_{1}}$ and ${w_{2}}$ be weight functions on $\mathbb{R} ^{d}$. Then $A_{\alpha,p}^{{w_{1}},1} ( \mathbb{R} ^{d} ) \subset A_{\alpha,p}^{{w_{2}},1} ( \mathbb{R} ^{d} )$ if and only if ${w_{2}}\prec{w_{1}}$.

Proof

Suppose that ${w_{2}}\prec{w_{1}}$. Thus there exists ${c_{1}}>0$ such that ${w_{2}} ( x ) \leq{c}_{1}{w_{1}} ( x )$ for all $x\in \mathbb{R}^{d}$. Also let $f\in A_{\alpha,p}^{{w_{1}},1} ( \mathbb{R} ^{d} )$. Then we write

$$ {\Vert f\Vert _{1,{w_{2}}}}\leq{c_{1}} {\Vert f \Vert _{1,{w_{1}}}}< \infty. $$

Hence we have

$$ {\Vert f\Vert _{A_{\alpha,p}^{{w_{2}},1}}=\Vert f\Vert _{1,{w_{2}}}+\Vert \mathcal{F} {_{\alpha}f} \Vert _{p}\leq c_{1}\Vert f \Vert _{1,{w_{1}}}+c_{1}\Vert \mathcal{F} {_{\alpha}f} \Vert _{p}=c_{1}\Vert f\Vert _{A_{\alpha ,p}^{{w_{1}},1}}}. $$

Therefore, $A_{\alpha,p}^{{w_{1}},1} ( \mathbb{R} ^{d} ) \subset A_{\alpha,p}^{{w_{2}},1} ( \mathbb{R} ^{d} )$.

Conversely, suppose that $A_{\alpha,p}^{{w_{1}},1} ( \mathbb{R} ^{d} ) \subset A_{\alpha,p}^{{w_{2}},1} ( \mathbb{R} ^{d} )$. For every $f\in A_{\alpha,p}^{{w_{1}},1} ( \mathbb{R} ^{d} )$, we have $f\in A_{\alpha,p}^{{w_{2}},1} ( \mathbb{R} ^{d} )$. By Proposition 10, there are constants ${c_{1}},{c_{2}},{c_{3}},{c_{4}}>0$ such that

$$ {c_{1}} {w_{1}} ( x ) \leq{ \Vert {{T_{x}}f} \Vert _{A_{\alpha,p}^{{w_{1}},1}}}\leq{c_{2}} {w_{1}} ( x ) $$

(7)

and

$$ {c_{3}} {w_{2}} ( x ) \leq{ \Vert {{T_{x}}f} \Vert _{A_{\alpha,p}^{{w_{2}},1}}}\leq{c_{4}} {w_{2}} ( x ) $$

(8)

for all $x\in \mathbb{R} ^{d}$. It is well known from Lemma 11 that the space ${A_{\alpha ,p}^{{w_{1}} ,1} ( \mathbb{R} ^{d} ) }$ is a Banach space under the norm ${|\!|\!|f|\!|\!|}$, $f\in A_{\alpha,p}^{{w_{1}},1} ( \mathbb{R} ^{d} )$. Then by the closed graph theorem the norms ${\Vert \cdot \Vert _{A_{\alpha,p}^{{w_{1}},1}}}$ and ${\Vert \cdot \Vert _{A_{\alpha,p}^{{w_{2}},1}}}$ are equivalent on $A_{\alpha,p}^{{w_{1}} ,1} ( \mathbb{R} ^{d} )$. So, there exists ${c}>0$ such that ${\Vert f \Vert _{A_{\alpha,p}^{{w_{2}},1}}\leq \Vert f\Vert _{A_{\alpha ,p}^{{w_{1}},1}}}$ for all $f\in A_{\alpha,p}^{{w_{1}},1} ( \mathbb{R} ^{d} )$. Moreover, as ${T_{x}}f\in A_{\alpha,p}^{{w_{2}},1} ( \mathbb{R} ^{d} )$, we have

$$ {\Vert {{T_{x}}f}\Vert _{A_{\alpha,p}^{{w_{2}},1}}}\leq c{\Vert {{T_{x}}f}\Vert _{A_{\alpha,p}^{{w_{1}},1}}}. $$

(9)

Then, combining (7), (8), and (9), we obtain

$$ {c_{3}} {w_{2}} ( x ) \leq{ \Vert {{T_{x}}f} \Vert _{A_{\alpha,p}^{{w_{2}},1}}}\leq c{\Vert {{T_{x}}f}\Vert _{A_{\alpha,p}^{{w_{1}},1}}}\leq c{c_{2}} {w_{1}} ( x ) . $$

Thus, ${w_{2}} ( x ) \leq\frac {{c{c_{2}}}}{{{c_{3}}}}{w_{1}} ( x )$. Let $\frac{{c{c_{2}}}}{{{c_{3}}}}=k$. Then we find ${w_{2}} ( x ) \leq k{w_{1}} ( x )$ for all $x\in \mathbb{R} ^{d}$. □

Proposition 13

Let ${w_{1}}$, ${w_{2}}$, ${\omega_{1}}$ and ${\omega_{2}}$ be weight functions on $\mathbb{R} ^{d}$. If ${w_{2}}\prec{w_{1}}$ and ${\omega_{2}}\prec{\omega_{1}}$, then $A_{\alpha,p}^{{w_{1}},{\omega_{1}}} ( \mathbb{R} ^{d} ) \subset A_{\alpha,p}^{{w_{2}},{\omega_{2}}} ( \mathbb{R} ^{d} )$.

Proof

Assume that ${w_{2}}\prec{w_{1 }}$ and ${\omega_{2}}\prec {\omega _{1}}$. Then there exist ${c_{1}},{c_{2}}>0$ such that ${w_{2}} ( x ) \leq{c_{1}}{w_{1}} ( x )$ and ${\omega_{2}} ( x ) \leq{c_{2}}{\omega_{1}} ( x )$ for all $x\in \mathbb{R} ^{d}$. Let $f\in A_{\alpha,p}^{{w_{1}},{\omega_{1}}} ( \mathbb{R} ^{d} )$. As $f\in L_{{w_{1}}}^{1} ( \mathbb{R} ^{d} )$ and $\mathcal{F}_{\alpha}f\in L_{{\omega _{1}}}^{p} ( \mathbb{R} ^{d} )$, we have ${\Vert f\Vert _{1,{w_{2}}}}\leq {c_{1}}{\Vert f\Vert _{1,{w_{1}}}}<\infty$ and ${\Vert \mathcal {F}{{_{\alpha}}f} \Vert _{p,{\omega_{2}}}}\leq{c_{2}}{\Vert \mathcal{F}{_{\alpha}f} \Vert _{p,{\omega_{1}}}}<\infty$. Hence, we obtain $f\in A_{\alpha,p}^{{w_{2}},{\omega_{2}}} ( \mathbb{R} ^{d} )$, and then $A_{\alpha,p}^{{w_{1}},{\omega_{1}}} ( \mathbb{R} ^{d} ) \subset A_{\alpha,p}^{{w_{2}},{\omega_{2}}} ( \mathbb{R} ^{d} )$. □

Duality

Let the mapping $\Phi:A_{\alpha,p}^{w,\omega} ( \mathbb{R} ^{d} ) \rightarrow L_{w}^{1} ( \mathbb{R} ^{d} ) \times L_{\omega}^{p} ( \mathbb{R} ^{d} )$ be defined by $\Phi ( f ) = ( {f},\mathcal {F}{{_{\alpha}}f} )$ for $1\leq p<\infty$ and let $H=\Phi ( {A_{\alpha,p}^{w,\omega} ( \mathbb{R} ^{d} ) } )$. Then

$$ \bigl\Vert {\Phi ( f ) }\bigr\Vert =\bigl\Vert { ( {f},\mathcal{F} {_{\alpha}f} ) }\bigr\Vert ={\Vert f\Vert _{1,w}}+{\Vert \mathcal{F} {_{\alpha}f} \Vert _{p,\omega}} $$

is a norm on H for all $f\in A_{\alpha,p}^{w,\omega} ( \mathbb{R} ^{d} )$. Moreover, we define a set K as

$$\begin{aligned} K =&\biggl\{ ( {\varphi,\psi} ) : \bigl( { ( {\varphi,\psi} ) \in L_{{w^{-1}}}^{\infty} \bigl( \mathbb{R} ^{d} \bigr) \times L_{{\omega^{-1}}}^{p^{\prime}} \bigl( \mathbb{R} ^{d} \bigr) } \bigr) , \\ &\int_{ \mathbb{R} ^{d}}{f ( x ) \varphi ( x )\, dx}+\int _{ \mathbb{R} ^{d}}\mathcal{F} {{_{\alpha}f ( y ) \psi ( y ) \, dy}=0} \text{ for all } { ( {f},\mathcal{F} {_{\alpha}f} ) \in H}\biggr\} , \end{aligned}$$

where $\frac{1}{p}+\frac{1}{{p^{\prime}}}=1$.

The following proposition is proved by the duality theorem, Theorem 1.7 in [15].

Proposition 14

Let $1\leq p<\infty$, and w and ω be weight functions on $\mathbb{R}^{d}$. The dual space of $A_{\alpha,p}^{w,\omega} ( \mathbb{R} ^{d} )$ is isomorphic to ${{L_{{w^{-1}}}^{\infty} ( \mathbb{R} ^{d} ) \times L_{{\omega^{-1}}}^{p^{\prime}} ( \mathbb{R} ^{d} ) }}/ {K}$ where $\frac{1}{p}+\frac {1}{{p^{\prime}}} =1$.