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.

1 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.

2 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. □

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

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