Characterization of $W^{p}$-type of spaces involving fractional Fourier transform

Abstract

The characterizations of $W^{p}$-type of spaces and mapping relations between W- and $W^{p}$-type of spaces are discussed by using the fractional Fourier transform. The uniqueness of the Cauchy problems is also investigated by using the same transform.

1 Introduction

The spaces of W-type were studied by Gurevich [1], Gel’fand and Shilov [2] and Friedman [3]. They investigated the behavior of the Fourier transformation on W-type spaces. The spaces of W-types are applied to the theory of partial differential equations. Pathak and Upadhyay [4] investigated the spaces $W_{M}^{p}$, $W_{M,a}^{p}$, $W^{\Omega,b,p}$, $W^{\Omega,p}$, $W^{\Omega,p}_{M}$, $W^{\Omega,b,p}_{M,a}$ in terms of $L^{p}$ norms. Here M, Ω are certain continuous increasing convex functions and a, b are positive constants and $p\geq1$. It was shown that the Fourier transformation F is to be a continuous linear mapping as follows: $F: W^{p}_{M,a}\rightarrow W^{\Omega,\frac{1}{a},r}$, $F: W^{\Omega ,b,p}\rightarrow W^{r}_{M,\frac{1}{b}}$, $F: W^{\Omega ,b,p}_{M,a}\rightarrow W^{\Omega,\frac{1}{a},r}_{M, \frac{1}{a}}$. Using the theory of the Hankel transform, Betancor and Rodriguez-Mesa [5] gave a new characterization of the space of $We^{p}_{\mu}$-type and established the results $We^{p}_{M,a}= We_{M,a}$, $We^{p,\Omega,b}= We^{\Omega,b}$, $We^{p,\Omega,b}_{M,a}= We^{\Omega,b}_{M,a}$. Upadhyay [6] established the results of the following types: $W^{p}_{M,a}= W_{M,a}$, $W^{p,\Omega,b}= W^{\Omega,b}$, $W^{p,\Omega,b}_{M,a}= W^{\Omega,b}_{M,a}$ by exploiting the theory of Fourier transformations. Motivated by the work of Pathak and Upadhyay [4] and Upadhyay [6] we shall extend a similar type of results in n dimensions by using the theory of the fractional Fourier transformations. Let $\mathbb{R}^{n}$ be the usual Euclidean space given by

$$\mathbb{R}^{n} = \bigl\{ (x_{1},\ldots,x_{n}) \mbox{: } x_{j}\mbox{'s are real numbers}\bigr\} . $$

Assume $x = ( x_{1},\ldots,x_{n} )$ and $y = ( y_{1},\ldots,y_{n} )$. Then the inner product of x and y is defined by

$$ \langle x ,y \rangle=x\cdot y = {\sum_{j=1}^{n} x_{j}\cdot y_{j}} $$

(1.1)

and the norm of x is defined by

$$ \vert x\vert = \Biggl( {\sum_{j=1}^{n} x^{2}_{j}} \Biggr)^{\frac{1}{2}}= \bigl(x^{2}_{1}+ \cdots+x^{2}_{n} \bigr)^{\frac{1}{2}}. $$

(1.2)

The $L^{p}$ norm of a function f in $L^{p}(\mathbb{R}^{n})$, $1\leq p\leq\infty$, is denoted by $\Vert f\Vert _{p}$ and defined as

$$ \Vert f\Vert _{p} = \biggl(\int_{\mathbb{R}^{n}}{ \bigl\vert f(x)\bigr\vert ^{p} \, dx} \biggr)^{\frac{1}{p}}. $$

(1.3)

The n-dimensional fractional Fourier transform (FrFT) with parameter α of $f(x)$ on $x \in\mathbb{R}^{n}$ is denoted by $(F_{\alpha }f)(\xi)$ [7, 8] and defined as

$$ \hat{f}_{\alpha}(\xi) = (F_{\alpha}f) (\xi) = \int _{\mathbb{R}^{n}} K_{\alpha}(x ,\xi) f(x)\, dx, \quad \xi\in \mathbb{R}^{n}, $$

(1.4)

where

$$ K_{\alpha}(x ,\xi)= \begin{cases} C_{\alpha}e^{\frac{i(\vert x\vert ^{2}+{\vert \xi \vert }^{2})\cot {\alpha}}{2} - i \langle x , {\xi} \rangle\csc{\alpha}} & \text{if } {\alpha}\ne n\pi, \\ \frac{1}{(2\pi)^{\frac{n}{2}}}e^{-i \langle x , {\xi} \rangle} &\text{if }\alpha= \frac{\pi}{2}, \end{cases} \quad \forall n\in \mathbb{Z} $$

and

$$ C_{\alpha} =(2\pi i \sin\alpha)^{\frac{-n}{2}}e^{\frac{in\alpha}{2}} = \frac{1}{[\pi(1 - e^{-2i\alpha})]^{\frac{n}{2}}}. $$

(1.5)

The corresponding inversion formula is given by

$$ {f}(x) = \frac{1}{(2\pi)^{\frac{n}{2}}} \int_{\mathbb{R}^{n}} \overline {K_{\alpha}(x ,\xi)} \hat{f}_{\alpha}(\xi) \, d\xi, \quad x\in \mathbb{R}^{n}, $$

(1.6)

where the kernel

$$ \overline{K_{\alpha}(x ,\xi)}= C_{\alpha} e^{-\frac{i(\vert x\vert ^{2}+{\vert \xi \vert }^{2})\cot {\alpha}}{2}+i \langle x,{\xi} \rangle\csc{\alpha}}, $$

and $C_{\alpha}$ is defined by (1.5).

Now from the technique of [9, p.2], (1.1) can be written as

$$\begin{aligned} F_{\alpha}\bigl[f(x)\bigr](\xi) =& C_{\alpha}e^{\frac{{i \vert \xi \vert ^{2} \cot \alpha}}{2}} \int_{\mathbb{R}^{n}}{e^{-i \langle x,\xi \rangle \csc\alpha} \bigl[f(x)e^{\frac{{i \vert x\vert ^{2} cot\alpha}}{2}} \bigr]\, dx} \\ =& (2\pi)^{\frac{n}{2}}C_{\alpha}e^{\frac{{i \vert \xi \vert ^{2} \cot \alpha}}{2}} \bigl[f(x)e^{\frac{{i \vert x \vert ^{2} \cot\alpha}}{2}}\hat{ \bigr]}(\xi\csc\alpha). \end{aligned}$$

(1.7)

Replacing $f(x)= e^{-\frac{i\vert x\vert ^{2} \cot\alpha }{2}} \phi(x)$ in (1.3), we obtain

$$ F_{\alpha}\bigl[e^{-\frac{i\vert x\vert ^{2}cot\alpha}{2}}\phi(x)\bigr](\xi)= (2 \pi)^{\frac{n}{2}}C_{\alpha}e^{\frac{{i \vert \xi \vert ^{2} \cot\alpha }}{2}}\bigl[\phi(x)\hat{\bigr]}(\xi\csc\alpha). $$

(1.8)

Now substituting $\xi= w \sin\alpha$, where $w\in \mathbb{R}^{n}$ in (1.4), we obtain

$$ F_{\alpha}\bigl[e^{-\frac{i\vert x\vert ^{2}\cot\alpha}{2}}\phi(x)\bigr](w \sin \alpha)= (2 \pi)^{\frac{n}{2}}C_{\alpha}e^{\frac{i \vert w\sin\alpha \vert ^{2} \cot\alpha}{2}}\bigl[\phi(x)\hat{\bigr]}(w). $$

(1.9)

Let $\psi = F_{\alpha} [e^{-\frac{i\vert x\vert ^{2}\cot\alpha }{2}}\phi(x) ]$, then (1.6) can be written as

$$ \psi(w \sin\alpha)=(2\pi)^{\frac{n}{2}}C_{\alpha}e^{\frac{i \vert w\sin \alpha \vert ^{2} \cot\alpha}{2}}\bigl[\phi(x)\hat{ \bigr]}(w). $$

(1.10)

Now we recall the definitions of W- and $W^{p}$-type of spaces from [2–4], which are given below. Let $\mu_{j}$ and $w_{j}$, $j=1,\ldots,n$, be continuous and increasing functions on $[0, \infty)$ with $\mu_{j}(0)= w_{j}(0)=0$ and $\mu_{j}(\infty)=w_{j}(\infty)=\infty$.

We define

$$\begin{aligned}& M_{j}(x_{j})=\int_{0}^{x_{j}}{ \mu_{j}(\xi_{j})} \, d\xi_{j} \quad (x_{j}\geq0), \end{aligned}$$

(1.11)

$$\begin{aligned}& \Omega_{j}(y_{j})= \int_{0}^{y_{j}}{w_{j}( \eta_{j})}\, d\eta_{j}\quad (y_{j}\geq0), \end{aligned}$$

(1.12)

where $j= 1,\ldots,n$. The functions $M_{j}(x_{j})$ and $\Omega _{j}(y_{j})$ are continuous, increasing, and convex with $M_{j}(0)= \Omega _{j}(0)= 0$ and $M_{j}(\infty)= \Omega_{j}(\infty)= \infty$, we have

$$\begin{aligned}& M_{j}(-x_{j})=M_{j}(x_{j}), \qquad M_{j}(x_{j})+ M_{j} \bigl(x'_{j}\bigr)\leq M_{j} \bigl(x_{j}+x'_{j}\bigr), \end{aligned}$$

(1.13)

$$\begin{aligned}& \Omega_{j}(-y_{j})=\Omega_{j}(y_{j}), \qquad \Omega_{j}(y_{j})+ \Omega_{j} \bigl(y'_{j}\bigr)\leq\Omega_{j} \bigl(y_{j}+y'_{j}\bigr). \end{aligned}$$

(1.14)

We define

$$\begin{aligned}& \mu(\xi) = \bigl(\mu_{1}(\xi_{1})\bigr),\ldots,\bigl( \mu_{n}(\xi_{n})\bigr), \\& w(\eta)= \bigl(w_{1}(\eta_{1})\bigr),\ldots, \bigl(w_{n}(\eta_{n})\bigr). \end{aligned}$$

The space $W_{M,a}(\mathbb{R}^{n})$ consists of all $C^{\infty}$-complex valued functions $\phi(x)$ on $x\in\mathbb{R}^{n}$, which for any $\delta\in\mathbb{R}^{n}_{+}$ satisfy the inequality

$$ \bigl\vert D^{k}_{x}\phi(x)\bigr\vert \leq C_{k,\delta} \exp\bigl[-M\bigl[(a-\delta)x\bigr]\bigr], $$

(1.15)

and the space $W_{M,a}^{p}(\mathbb{R}^{n})$ consists of all infinitely differentiable functions $\phi(x)$ on $x\in\mathbb{R}^{n}$, which for any $\delta\in\mathbb{R}^{n}_{+}$ satisfy the inequality

$$ \biggl(\int_{\mathbb{R}^{n}}{\bigl\vert \exp \bigl[-M\bigl[(a- \delta)x\bigr] \bigr]D^{k}_{x}\phi(x)\bigr\vert ^{p}}\, dx \biggr)^{\frac{1}{p}}\leq C_{k,\delta ,p}, \quad p\geq1 $$

(1.16)

for each $k\in\mathbb{Z}^{n}_{+}$ where $D^{k}_{x}=D^{k_{1}}_{x_{1}}\cdots D^{k_{n}}_{x_{n}}$,

$$ \exp\bigl[-M\bigl[(a-\delta)x\bigr]\bigr]=\exp\bigl[-M_{1} \bigl[(a_{1}-\delta_{1})x_{1}\bigr]-\cdots -M_{n}\bigl[(a_{n}-\delta_{n})x_{n} \bigr]\bigr] $$

and $a_{1},\ldots,a_{n}$, $C_{k,\delta,p}$, $C_{k,\delta}$ are positive constants depending on the function $\phi(x)$.

The space $W^{\Omega,b}(\mathbb{C}^{n})$ consists of all entire analytic functions $\phi(z)$, where $z=x+iy$ and $x,y \in \mathbb{R}^{n}$, which for any $\rho\in\mathbb{R}^{n}_{+}$ satisfy the inequality

$$ \bigl\vert z^{k}\phi(z)\bigr\vert \leq C_{k,\rho} \exp \bigl[\Omega\bigl[(b + \rho)y\bigr]\bigr], \quad k\in\mathbb{Z}^{n}_{+}, $$

(1.17)

where

$$ z^{k}=z_{1}^{k_{1}}\cdots z_{n}^{k_{n}}, $$

and $b_{1},\ldots,b_{n}$, $C_{k,\rho}$ are positive constants depending on the function $\phi(x)$ and the space $W^{\Omega,b,p}$ consists of all entire analytic functions $\phi(z)$ such that for $k\in\mathbb{Z}^{n}_{+}$, $\rho\in\mathbb{R}^{n}_{+}$, there exists a constant $C_{k,\rho,p} > 0$ such that

$$ \biggl(\int_{\mathbb{R}^{n}}\bigl\vert \exp \bigl[\Omega\bigl[(b+ \rho)y\bigr] \bigr] z^{k}\phi(z)\bigr\vert ^{p}\,dx \biggr)^{\frac{1}{p}}\leq C_{k,\rho,p}, $$

(1.18)

where

$$ \exp\bigl[\Omega\bigl[(b+\rho)\bigr]y\bigr] = \exp\bigl[\Omega_{1} \bigl[ (b_{1}+\rho_{1})y_{1}\bigr]+\cdots+ \Omega_{n}\bigl[ (b_{n}+\rho_{n})y_{n} \bigr]\bigr]. $$

The space $W^{\Omega,b}_{M,a}(\mathbb{C}^{n})$ consists of all entire analytic functions $\phi(z)$ such that there exist constants $\rho, \delta\in\mathbb{R}^{n}_{+}$ and $C_{\delta,\rho } > 0$ such that

$$ \bigl\vert \phi(z)\bigr\vert \leq C_{\delta,\rho} \exp\bigl[ -M\bigl[(a - \delta) x\bigr] + \Omega\bigl[(b + \rho)y\bigr]\bigr] , $$

(1.19)

and the space $W^{\Omega,b,p}_{M,a}(\mathbb{C}^{n})$ consists of all entire analytic functions $\phi(z)$ such that for $\rho,\delta \in\mathbb{R}^{n}_{+}$ and $C_{\rho,\delta,p} > 0$,

$$ \biggl(\int_{\mathbb{R}^{n}}\bigl\vert \exp \bigl[ M\bigl[(a-\delta)x \bigr]-\Omega\bigl[(b+\rho )y\bigr] \bigr]\phi(z)\bigr\vert ^{p}\,dx \biggr)^{\frac{1}{p}} \leq C_{\rho,\delta ,p}, $$

(1.20)

where $\exp [ M[(a-\delta)x] ]$ and $\exp [-\Omega[(b+\rho)y] ]$ have the usual meaning like (1.16) and (1.18), and the constants $C_{\rho,\delta,p}$, a, b, and ρ, δ depend only on the function $\phi(z)$.

Let $M_{j}(x_{j})$ and $\Omega_{j}(y_{j})$ be the functions defined by (1.11) and (1.12), respectively, the functions $\mu_{j}(\xi_{j})$ and $w_{j}(\eta_{j})$ which occur in these equations are mutually inverse, that is, $\mu_{j}(w_{j}(\eta_{j}))=\eta_{j}$ and $w_{j}(\mu_{j}(\xi_{j}))=\xi_{j}$, then the corresponding functions $M_{j}(x_{j})$ and $\Omega_{j}(y_{j})$ are said to be the dual in the sense of Young. In this case, the Young inequality,

$$ x_{j}y_{j}\leq M_{j}(x_{j})+ \Omega_{j}(y_{j}), $$

(1.21)

holds for any $x_{j}\geq$, $y_{j}\geq0$.

2 Characterization of $W^{p}$-type of spaces

In this section we study the characterization of $W^{p}$-type of spaces by using the fractional Fourier transformation.

Theorem 2.1

Let $M(x)$ and $\Omega(y)$ be the pair of functions which are dual in the sense of Young. Then

$$ F_{\alpha} \bigl[ W^{p}_{M,a} \bigr]\subset W^{\Omega,\frac{1}{a},r} \quad \textit{for any } 1\leq p, r <\infty. $$

(2.1)

Proof

Let $e^{-\frac{i\vert x\vert ^{2}\cot\alpha }{2}}\phi(x)\in W^{p}_{M,a}(\mathbb{R}^{n})$ and $\sigma=w+i\tau$. Then for any p and r, using the technique of [3, pp.20-21] and (1.10), we have

$$ \bigl\Vert (\sigma\sin\alpha)^{k}\psi(\sigma\sin\alpha)\bigr\Vert _{r}= \biggl[\int_{\mathbb{R}^{n}} \bigl\vert (\sigma\sin \alpha)^{k}\psi(\sigma\sin \alpha)\bigr\vert ^{r}\, dw \biggr]^{\frac{1}{r}}. $$

Now using the inequality $\vert \sigma \vert ^{|k|}\leq\frac{ \vert \sigma \vert ^{|k+2|} + \vert \sigma \vert ^{|k|}}{\vert w\vert ^{2} +1}$, we have

$$\begin{aligned}& {\bigl\Vert (\sigma\sin\alpha)^{k}\psi(\sigma\sin\alpha)\bigr\Vert _{r}} \\& \quad \leq \biggl[\vert \sin\alpha \vert ^{r|k|}\int _{\mathbb{R}^{n}} \biggl(\frac{\vert \sigma \vert ^{|k+2|} + \vert \sigma \vert ^{|k|}}{ \vert w\vert ^{2} +1} \biggr)^{r} \bigl\vert \psi(\sigma\sin\alpha)\bigr\vert ^{r} \,dw \biggr]^{\frac{1}{r}} \\& \quad \leq \biggl[\vert \sin\alpha \vert ^{r|k|}\int _{\mathbb{R}^{n}} \biggl(\frac{\vert \sigma \vert ^{|k+2|}}{\vert w\vert ^{2} +1} \biggr)^{r} \bigl\vert \psi(\sigma\sin\alpha)\bigr\vert ^{r} \,dw \biggr]^{\frac{1}{r}} \\& \qquad {}+ \biggl[\vert \sin\alpha \vert ^{r|k|}\int _{\mathbb{R}^{n}} \biggl(\frac{ \vert \sigma \vert ^{|k|}}{\vert w\vert ^{2} +1} \biggr)^{r} \bigl\vert \psi(\sigma\sin\alpha)\bigr\vert ^{r} \,dw \biggr]^{\frac{1}{r}}. \end{aligned}$$

Using (1.7), we get

$$\begin{aligned}& {\bigl\Vert (\sigma\sin\alpha)^{|k|}\psi(\sigma\sin\alpha)\bigr\Vert _{r}} \\& \quad \leq C_{\alpha,k+2} \biggl[\int_{\mathbb{R}^{n}}{ \frac{d\sigma}{( \vert w\vert ^{2}+1)^{r}} \biggl(\int_{\mathbb{R}^{n}}\bigl\vert \exp \bigl[ \langle x,\tau\rangle \bigr] \bigr\vert \bigl\vert D^{k+2}\phi(x) \bigr\vert \,dx \biggr)^{r}} \biggr]^{\frac{1}{r}} \\& \qquad {}+ C_{\alpha,k} \biggl[\int_{\mathbb{R}^{n}}{ \frac {dw}{(\vert w\vert ^{2}+1)^{r}} \biggl(\int_{\mathbb{R}^{n}}\bigl\vert \exp \bigl[ \langle x,\tau\rangle \bigr] \bigr\vert \bigl\vert D^{k}\phi(x) \bigr\vert \,dx \biggr)^{r}} \biggr]^{\frac{1}{r}} \\& \quad \leq \biggl[C_{\alpha,k+2} \biggl\{ \int_{\mathbb{R}^{n}}{ \frac {dw}{(\vert w\vert ^{2}+1)^{r}} \biggl(\int_{\mathbb{R}^{n}}\bigl\vert \exp \bigl[ \langle x,\tau\rangle \bigr] \bigr\vert \bigl\vert D^{k+2}\phi(x) \bigr\vert \,dx \biggr)^{r}} \biggr\} ^{\frac{1}{r}} \\& \qquad {}+C_{\alpha,k} \biggl\{ \int_{\mathbb{R}^{n}}{ \frac {dw}{(\vert w\vert ^{2}+1)^{r}} \biggl(\int_{\mathbb{R}^{n}}\bigl\vert \exp \bigl[ \langle x,\tau\rangle \bigr] \bigr\vert \bigl\vert D^{k}\phi(x) \bigr\vert \,dx \biggr)^{r}} \biggr\} ^{\frac{1}{r}} \biggr] \\& \quad \leq C_{\alpha,k+2} \biggl[ \int_{\mathbb{R}^{n}}{ \frac{dw}{( \vert w\vert ^{2}+1)^{r}} \biggl( \biggl(\int_{\mathbb{R}^{n}}\bigl\vert \exp \bigl[M\bigl[(a-\delta)x\bigr]\bigr]D^{k+2}\phi(x)\bigr\vert ^{p} \,dx \biggr)^{\frac{1}{p}}} \\& \qquad {}\times \biggl(\int_{\mathbb{R}^{n}}{\bigl\vert \exp \bigl[|x||\tau| -M\bigl[(a-\delta)x\bigr]\bigr]\bigr\vert ^{p'} \,dx} \biggr)^{\frac{1}{p'}} \biggr)^{r} \biggr]^{\frac{1}{r}} \\& \qquad {}+ C_{\alpha,k} \biggl[ \int_{\mathbb{R}^{n}}{ \frac{dw}{(\vert w \vert ^{2}+1)^{r}} \biggl( \biggl(\int_{\mathbb{R}^{n}}\bigl\vert \exp \bigl[M\bigl[(a-\delta )x\bigr]\bigr]D^{k}\phi(x)\bigr\vert ^{p} \,dx \biggr)^{\frac{1}{p}}} \\& \qquad {}\times \biggl(\int_{\mathbb{R}^{n}}{\bigl\vert \exp \bigl[|x||\tau| -M\bigl[(a-\delta)x\bigr]\bigr]\bigr\vert ^{p'} \,dx} \biggr)^{\frac{1}{p'}} \biggr)^{r} \biggr]^{\frac{1}{r}}. \end{aligned}$$

Now using the Young inequality (1.21) and the arguments of [3, p.23], we get

$$ \bigl\Vert (\sigma\sin\alpha)^{k}\psi(\sigma\sin\alpha)\bigr\Vert _{r}\leq D'_{k+2,\rho,\alpha,r} e^{\Omega[\frac{\tau}{\gamma}]}+ D'_{k,\rho ,\alpha,r} e^{\Omega[\frac{\tau}{\gamma}]}. $$

In the above expression, we set $\frac{1}{\gamma}= ( \frac{1}{a} +\rho )$, since $\gamma= a-\delta$ and ρ is arbitrarily small together with δ. Therefore, we have

$$ \bigl\Vert (\sigma\sin\alpha)^{k}\psi(\sigma\sin\alpha)\bigr\Vert _{r}\leq C^{r}_{k,\rho, \vert \sin\alpha \vert } e^{\Omega[\frac{\tau}{\gamma}]}. $$

 □

Theorem 2.2

Let $M(x)$ and $\Omega(y)$ be the pair of functions which are dual in the sense of Young. Then

$$ F_{\alpha} \bigl[ W^{\Omega,b,p} \bigr]\subset W^{r}_{M,\frac{1}{b}} \quad \textit{for any } 1\leq p, r <\infty. $$

Proof

Let $e^{-\frac{i\vert z\vert ^{2}cot\alpha }{2}}\phi(z)\in W^{\Omega,b,p}(\mathbb{C}^{n})$ and $\sigma=w+i\tau$. Then from the arguments of [8, Theorem 2.2], we have

$$\begin{aligned} \bigl\vert D^{k}_{w}\psi(w\sin\alpha)\bigr\vert \leq& C_{k,r,\eta,s,\rho} \exp\bigl[- \langle y,w \rangle+ \Omega (b+\rho)y \bigr]\\ \leq& C_{k,r,\eta,s,\rho} \exp\bigl[-y_{1}w_{1}+ \Omega_{1}\bigl[(b_{1}+\rho _{1})y_{1} \bigr] \\ &{}-\cdots -y_{n}w_{n}+\Omega_{n} \bigl[(b_{n}+\rho_{n})y_{n}\bigr] \bigr]. \end{aligned}$$

From the arguments of [4, p.737] we have

$$ \bigl\vert D^{k}_{w}\psi(w\sin\alpha)\bigr\vert \leq C_{k,\rho,r,\eta,s} \exp \biggl[ -M \biggl[ \biggl( \frac{1}{b}-\delta \biggr)w \biggr] -M \biggl[ \frac {\rho^{2}w}{b^{3}} \biggr] \biggr]. $$

Hence,

$$ \biggl\lVert \exp\biggl[ M\biggl[\biggl(\frac{1}{b}-\delta\biggr)w\biggr]\biggr] D^{k}_{w}\phi(w \sin\alpha) \biggr\rVert _{r} \leq C^{r}_{k,\rho,r,\eta,s}. $$

This implies that

$$ \psi(w\sin\alpha)\in W^{r}_{M,\frac{1}{b}}. $$

 □

Theorem 2.3

Let $M_{0}(x)$ and $\Omega _{0}(y)$ be the functions which are dual in the sense of Young to the functions $M(x)$ and $\Omega(y)$, respectively. Then

$$ F_{\alpha} \bigl[ W^{\Omega,b,p}_{M,a,} \bigr]\subset W^{\Omega_{0},\frac {1}{a},r}_{M_{0}, \frac{1}{b}}\quad \textit{for any } 1\leq p, r <\infty. $$

Proof

Let $e^{-\frac{i\vert x\vert ^{2}cot\alpha }{2}}\phi(x)\in W^{\Omega,b,p}_{M,a}(\mathbb{R}^{n})$ and $\sigma= w + i\tau$. Then by the technique of [2, pp.23-24] and (1.10), we have

$$\begin{aligned} \bigl\vert \psi(\sigma\sin\alpha)\bigr\vert \leq&\bigl\vert (2\pi i \sin \alpha )^{\frac{-n}{2}}e^{\frac{in\alpha}{2}}\bigr\vert \int_{\mathbb{R}^{n}} \bigl\vert e^{\overline{i \langle\sigma,z \rangle}}\bigr\vert \bigl\vert \phi (z)\bigr\vert \,dx \\ \leq& \bigl\vert (2\pi i \sin\alpha)^{\frac{-n}{2}}e^{\frac{in\alpha }{2}}\bigr\vert \int_{\mathbb{R}^{n}}\bigl\vert e^{- \langle w,y \rangle - \langle\tau,x \rangle}\bigr\vert \bigl\vert \phi(z)\bigr\vert \,dx \\ \leq& D C_{\rho,\delta}\bigl\vert e^{- \langle w,y \rangle} \bigr\vert \int _{\mathbb{R}^{n}}\bigl\vert e^{- \langle\tau,x \rangle}\bigr\vert \exp \bigl[ -M \bigl[(a-\delta)x\bigr]+ \Omega\bigl[(b+\rho)y\bigr] \bigr]\,dx \\ \leq& C_{\rho,\delta,\alpha}e^{- \langle w,y \rangle}\int_{\mathbb{R}^{n}} \exp \bigl[ -M\bigl[(a-\delta)x\bigr]+ \Omega\bigl[(b+\rho)y\bigr] \bigr] \exp \bigl[ \vert \tau \vert \vert x\vert \bigr]\,dx \\ =&C_{\rho,\delta,\alpha} \exp \bigl[- \langle w,y \rangle+ \Omega\bigl[(b+\rho)y \bigr] \bigr] \int_{\mathbb{R}^{n}}\exp \bigl[ -M\bigl[(a-\delta )x \bigr]+ \vert \tau \vert \vert x\vert \bigr] \,dx. \end{aligned}$$

Now using the arguments of [4, p.738], we have

$$ \bigl\vert \psi(\sigma\sin\alpha)\bigr\vert \leq C_{\rho,\delta,\alpha} \exp \biggl[ -M_{0} \biggl[ \biggl(\frac{1}{b}-\delta_{0} \biggr)w \biggr] + \Omega_{0} \biggl[ \biggl(\frac {1}{a} + \rho_{0}\biggr)\tau \biggr]- M_{0} \biggl[ \frac{\rho_{0}^{2} w}{b^{3}} \biggr] \biggr]. $$

Hence,

$$ \biggl\Vert \exp \biggl[M_{0} \biggl[ \biggl(\frac{1}{b}- \delta_{0} \biggr)w \biggr]-\Omega_{0} \biggl[ \biggl( \frac{1}{a}+\rho_{0} \biggr)\tau \biggr] \biggr]\psi(\sigma\sin \alpha)\biggr\Vert _{r}\leq C^{\prime r}_{\alpha,\delta,\rho} . $$

 □

3 Relation between W- and $W^{p}$-types of spaces

In this section the mapping relations between W- and $W^{p}$-types of spaces are discussed.

Theorem 3.1

Let $M(x)$, $\Omega(y)$ be the pair of functions which are dual in the sense of Young. Then

$$W^{p}_{M,a}= W_{M,a},\quad 1 \leq p <\infty. $$

Proof

Now, for showing the above theorem we shall prove the following lemma. □

Lemma 3.2

Let $1 \leq p <\infty$. Then $W^{p}_{M,a} \subset W_{M,a}$.

Proof

Let $e^{-\frac{|x|^{2}\cot\alpha}{2}}\phi(x)\in W^{p}_{M,a}(\mathbb{R}^{n})$ and $\sigma= w + i\tau$. Then from the arguments of Theorem 2.1, we get

$$ F_{\alpha}\bigl(W^{p}_{M,a}\bigr)\subset W^{\Omega,\frac{1}{a}}. $$

(3.1)

From the inverse property of the fractional Fourier transform, we have

$$ W^{p}_{M,a}\subset F^{-1}_{\alpha} \bigl(W^{\Omega,\frac{1}{a}}\bigr). $$

(3.2)

Now, let $\hat{\phi}_{\alpha}(\sigma)\in W^{\Omega,\frac{1}{a}}$. Then by the technique of [2, pp.21-22] and (1.6), we have

$$ \phi(x)= C_{\alpha}\int_{\mathbb{R}^{n}} e^{-\frac{i(|x|^{2} + |\sigma |^{2})\cot\alpha}{2}+ i\langle x, \sigma\rangle\csc\alpha} \hat{ \phi }_{\alpha}(\sigma) \,dw. $$

Therefore,

$$\begin{aligned} \phi^{(k)}(x) =& C_{\alpha}\int_{\mathbb{R}^{n}} D^{k}_{x}\bigl(e^{-\frac {i|x|^{2}\cot\alpha}{2}+ i\langle x, \sigma\rangle\csc\alpha}\bigr)e^{-\frac {i|\sigma|^{2}\cot\alpha}{2}}\hat{ \phi}_{\alpha}(\sigma) \,dw \\ =& C_{\alpha}\int_{\mathbb{R}^{n}}\sum _{r\leq k}\binom{k}{r}\bigl(D^{r}_{x} e^{-\frac{i|x|^{2}\cot\alpha}{2}}\bigr) \bigl(D^{k-r}_{x} e^{i\langle x, \sigma \rangle\csc\alpha} \bigr) e^{-\frac{i|\sigma|^{2}\cot\alpha}{2}}\hat{\phi}_{\alpha}(\sigma) \,dw \\ =& C_{\alpha}\sum_{r\leq k}\binom{k}{r} \int_{\mathbb{R}^{n}}\sum_{\eta \leq r}C_{\eta}( \cot\alpha) x^{\eta}e^{-\frac{i|x|^{2}\cot\alpha}{2}} (i\sigma\csc\alpha)^{k-r} \\ &{} \times e^{i\langle x, \sigma\rangle\csc\alpha}e^{-\frac{i|\sigma |^{2}\cot\alpha}{2}}\hat{\phi}_{\alpha}(\sigma) \,dw \\ =& C_{\alpha}\sum_{r\leq k}\binom{k}{r} \sum_{\eta\leq r}C_{\eta}(\cot \alpha)\int _{\mathbb{R}^{n}}e^{-\frac{i|x|^{2}\cot\alpha}{2}}(i\sigma\csc \alpha)^{k-r} \\ &{} \times\bigl(D^{\eta}_{\sigma}e^{i\langle x, \sigma\rangle\csc\alpha } \bigr)e^{-\frac{i|\sigma|^{2}\cot\alpha}{2}}\hat{\phi}_{\alpha}(\sigma) \,dw \\ =&C_{\alpha}\sum_{r\leq k}\binom{k}{r}\sum _{\eta\leq r}C_{\eta}(\cot \alpha) (i\csc \alpha)^{-|\eta|+|k-r|} (-1)^{|\eta|}\int_{\mathbb{R}^{n}}e^{-\frac{i|x|^{2}\cot\alpha}{2}}e^{i\langle x, \sigma\rangle\csc \alpha} \\ &{} \times D^{\eta}_{\sigma}\bigl(\sigma^{k-r}e^{-\frac{i|\sigma|^{2}\cot \alpha}{2}} \hat{\phi}_{\alpha}(\sigma)\bigr) \,dw \\ =& C_{\alpha}\sum_{r\leq k}\binom{k}{r} \sum_{\eta\leq r}C_{\eta}(\cot \alpha) (i\csc \alpha)^{-|\eta|+|k-r|} (-1)^{|\eta|}\int_{\mathbb{R}^{n}}e^{-\frac{i|x|^{2}\cot\alpha}{2}}e^{i\langle x, \sigma\rangle\csc \alpha} \\ &{} \times\sum_{\beta\leq\eta}\binom{\eta}{\beta} \bigl(D^{\eta}_{\sigma }e^{-\frac{i|\sigma|^{2}\cot\alpha}{2}}\bigr) \bigl(D^{\eta-\beta}_{\sigma} \sigma ^{k-r}\hat{\phi}_{\alpha}(\sigma)\bigr)\,dw \\ =&C_{\alpha}\sum_{r\leq k}\binom{k}{r}\sum _{\eta\leq r}C_{\eta}(\cot \alpha) (i\csc \alpha)^{-|\eta|+|k-r|} (-1)^{|\eta|}\int_{\mathbb{R}^{n}}e^{-\frac{i|x|^{2}\cot\alpha}{2}}e^{i\langle x, \sigma\rangle\csc \alpha} \\ &{} \times\sum_{\beta\leq\eta}\binom{\eta}{\beta}\biggl(\sum _{\lambda\leq\eta }C_{\lambda}(\cot\alpha) \sigma^{\lambda}\biggr)e^{-\frac{i|\sigma|^{2}\cot \alpha}{2}} \\ &{}\times\sum_{m\leq\eta-\beta} \binom{\eta-\beta}{m} \bigl(D^{m}_{\sigma} \sigma^{k-r} \bigr) \bigl(D^{\eta-\beta-m}_{\sigma}\hat{\phi}_{\alpha}(\sigma) \bigr)\,dw. \end{aligned}$$

Hence,

$$\begin{aligned} \bigl\vert \phi^{(k)}(x)\bigr\vert \leq& |C_{\alpha}|\sum _{r\leq k}\binom{k}{r}\sum _{\eta \leq r}\bigl\vert C_{\eta}(\cot\alpha)\bigr\vert \vert \csc\alpha|^{-|\eta|+|k-r|} \\ &{}\times\sum_{\beta\leq \eta} \binom{\eta}{\beta} \frac{(k-r)!}{(k-r-m)!} \sum_{\lambda\leq\beta }\bigl|C_{\lambda}( \cot\alpha)\bigr\vert \\ &{} \times\sum_{m\leq\eta-\beta}\binom{\eta-\beta}{m}\int _{\mathbb{R}^{n}}\bigl\vert e^{i\langle x, \sigma\rangle\csc\alpha}\bigr\vert \bigl\vert \sigma^{k-r-m+\lambda} D^{\eta-\beta-m}_{\sigma}\hat{\phi}_{\alpha}( \sigma)\bigr\vert \,dw. \end{aligned}$$

Therefore,

$$\begin{aligned} \bigl\vert \phi^{(k)}(x)\bigr\vert \leq&\vert C_{\alpha} \vert \sum_{r\leq k}\binom{k}{r}\sum _{\eta \leq r}\bigl\vert C_{\eta}(\cot\alpha)\bigr\vert \vert \csc\alpha \vert ^{-|\eta|+|k-r|} \\ &{}\times\sum_{\beta\leq \eta} \binom{\eta}{\beta} \frac{(k-r)!}{(k-r-m)!} \sum_{\lambda\leq\beta }\bigl\vert C_{\lambda}(\cot\alpha)\bigr\vert \\ &{} \times\sum_{m\leq\eta-\beta}\binom{\eta-\beta}{m}\int _{\mathbb{R}^{n}} e^{-\langle x, \tau\rangle\csc\alpha} \biggl(\frac{|\sigma |^{|k-r-m+\lambda|+2} +|\sigma|^{|k-r-m+\lambda|}}{(|w|^{2} +1)} \biggr) \\ &{} \times\bigl\vert D^{\eta-\beta-m}_{\sigma}\hat{ \phi}_{\alpha}(\sigma )\bigr\vert \,dw \\ \leq& C'_{\alpha,k} \exp\biggl[-\langle x, \tau\rangle\csc \alpha+ \Omega \biggl[\biggl(\frac{1}{a} + \rho\biggr)\tau\biggr]\biggr]. \end{aligned}$$

(3.3)

Now using the arguments of [4, p.23], we get

$$ \bigl\vert \phi^{(k)}(x)\bigr\vert \leq C'_{\alpha,k} \exp\bigl[-M\bigl[(a-\delta)x\csc\alpha\bigr]\bigr] $$

(3.4)

for arbitrarily small δ together with ρ. Hence the above expression gives

$$ F_{\alpha}^{-1}\bigl(W^{\Omega,\frac{1}{a}}\bigr)\subset W_{M,a}. $$

(3.5)

Thus (3.2) and (3.5) imply that

$$ W^{p}_{M,a}\subset W_{M,a}. $$

(3.6)

 □

Lemma 3.3

Let $1\leq p< \infty$. Then $W_{M,a}\subset W^{p}_{M,a}$.

Proof

Let $e^{-\frac{i|x|^{2}\cot\alpha}{2}}\phi(x)\in W_{M,a}(\mathbb{R}^{n})$ and $\sigma= w + i\tau\in\mathbb{C}^{n}$. Then from [8, Theorem 2.1], it follows that

$$ F_{\alpha}(W_{M,a})\subset W^{\Omega,\frac{1}{a}}. $$

(3.7)

Now by the inverse property of the fractional Fourier transform we have

$$ W_{M,a}\subset F_{\alpha}^{-1} \bigl(W^{\Omega,\frac{1}{a}} \bigr). $$

(3.8)

Again let $\hat{\phi}_{\alpha}(\sigma)\in W^{\Omega,\frac{1}{a}}$. Then from (3.3) we have

$$ \bigl\vert \phi^{k}(x)\bigr\vert \leq C'_{\alpha,k} \exp\biggl[-\langle x, \tau\rangle\csc\alpha + \Omega\biggl[\biggl( \frac{1}{a} + \rho\biggr)\tau\biggr]\biggr]. $$

(3.9)

Using (1.21) and [6, p.385] we get

$$ \bigl\vert \phi^{k}(x)\bigr\vert \leq C'_{\alpha,k} \exp\bigl[-M\bigl[(a-\delta)x\csc\alpha \bigr]-M\bigl[a^{3} \rho^{2} x\csc\alpha\bigr]\bigr]. $$

(3.10)

Therefore,

$$ \bigl\Vert \exp\bigl[M\bigl[(a-\delta)x\csc\alpha\bigr]\bigr] \phi^{k}(x)\bigr\Vert _{p}\leq C'_{\alpha, k} \bigl\Vert e^{-M[a^{3}\rho^{2} x\csc\alpha]}\bigr\Vert _{p}. $$

(3.11)

This implies that

$$ F_{\alpha}^{-1}\bigl(W^{\Omega,\frac{1}{a}}\bigr)\subset W^{p}_{M,a}. $$

(3.12)

From (3.8) and (3.12) we find

$$ W_{M,a}\subset W^{p}_{M,a}. $$

(3.13)

Now from (3.6) and (3.13) we get the result

$$ W^{p}_{M,a}= W_{M,a}. $$

(3.14)

 □

Theorem 3.4

Let $M(x)$ and $\Omega(y)$ be the same functions as in Theorem  3.1. Then

$$ W^{\Omega,b,p}= W^{\Omega,b}, \quad 1\leq p<\infty. $$

(3.15)

Proof

Let $e^{-\frac{i|z|^{2}\cot\alpha}{2}}\phi(z)\in W^{\Omega,b,p}$. Then from Theorem 2.2 it follows that

$$ F_{\alpha}\bigl( W^{\Omega,b,p}\bigr)\subset W_{M,\frac{1}{a}}. $$

(3.16)

By the inverse property of the fractional Fourier transform, we have

$$ W^{\Omega,b,p}\subset F_{\alpha}^{-1} (W_{M,\frac{1}{a}}). $$

(3.17)

Now let $\hat{\phi}_{\alpha}(x)\in W_{M,\frac{1}{b}}$. Then from the technique of [2, pp.20-21], we have

$$\begin{aligned}& {(i\sigma\csc\alpha)^{k} \phi(\sigma)} \\& \quad = C_{\alpha}\int_{\mathbb{R}^{n}} e^{-\frac{i(|x|^{2} + |\sigma|^{2})\cot \alpha}{2}} \bigl( D^{k}_{x}e^{i\langle x, \sigma\rangle\csc\alpha } \bigr) \hat{ \phi}_{\alpha}(x) \,dx \\& \quad = C_{\alpha}(-1)^{|k|}\int_{\mathbb{R}^{n}}e^{i\langle x, \sigma\rangle \csc\alpha} \bigl( D^{k}_{x} e^{-\frac{i|x|^{2} \cot\alpha}{2}}\hat{\phi }_{\alpha}(x) \bigr) e^{-\frac{ i|\sigma|^{2}\cot\alpha}{2}} \,dx \\& \quad = C_{\alpha}(-1)^{|k|}\int_{\mathbb{R}^{n}}e^{i\langle x, \sigma\rangle \csc\alpha} \sum_{r\leq k}\binom{k}{r} \bigl(D^{r}_{x} e^{-\frac{i|x|^{2} \cot\alpha}{2}} \bigr) \bigl(D^{k-r}_{x} \hat{ \phi}_{\alpha}(x) \bigr)e^{-\frac{ i|\sigma |^{2}\cot\alpha}{2}} \,dx \\& \quad = C_{\alpha}(-1)^{|k|}\int_{\mathbb{R}^{n}}e^{i\langle x, \sigma\rangle \csc\alpha} \sum_{r\leq k}\binom{k}{r}\sum _{\beta\leq r}C_{\beta}(\cot \alpha) e^{-\frac{i|x|^{2} \cot\alpha}{2}}e^{-\frac{ i|\sigma|^{2}\cot\alpha }{2}} \bigl(x^{\beta}D^{k-r}_{x}\hat{\phi}_{\alpha}(x) \bigr)\,dx. \end{aligned}$$

Therefore,

$$\begin{aligned} \bigl\vert (\sigma\csc\alpha)^{k}\phi(s)\bigr\vert \leq& |C_{\alpha}|\sum_{r\leq k}\binom {k}{r}\sum _{\beta\leq r}\bigl\vert C_{\beta}(\cot\alpha)\bigr\vert \int_{\mathbb{R}^{n}} \bigl\vert e^{-\langle x, \tau\rangle\csc\alpha}\bigr\vert \bigl\vert x^{\beta}D^{k-r}_{x} \hat{\phi }_{\alpha}(x)\bigr\vert \,dx \\ \leq&|C_{\alpha}|\sum_{r\leq k} \binom{k}{r}\sum_{\beta\leq r}\bigl\vert C_{\beta }(\cot\alpha)\bigr\vert \int_{\mathbb{R}^{n}} e^{[|x||\tau\csc\alpha|-M[(\frac{1}{b}-\delta )x]]} \,dx. \end{aligned}$$

Now using the arguments of [2, p.21], we get

$$ \bigl\vert (\sigma\csc\alpha)^{k}\phi(\sigma)\bigr\vert \leq C_{\alpha,k} \exp\bigl[\Omega \bigl[(b+\rho)\tau\csc\alpha\bigr]\bigr], $$

where ρ is arbitrarily small together with δ. Thus we have

$$ F_{\alpha}^{-1}(W_{M,\frac{1}{b}})\subset W^{\Omega,b}. $$

(3.18)

Therefore (3.17) and (3.18) yield

$$ W^{\Omega,b,p} \subset W^{\Omega,b} . $$

(3.19)

Again we take $e^{-\frac{i|z|^{2}\cot\alpha}{2}} \phi(z)\in W^{\Omega ,b}(\mathbb{C}^{n})$. Then from [8, Theorem 2.2], we have

$$ F_{\alpha}\bigl(W^{\Omega,b}\bigr)\subset W_{M,\frac{1}{b}}. $$

(3.20)

By the inverse property of the fractional Fourier transform, we have

$$ W^{\Omega,b}\subset F_{\alpha}^{-1}( W_{M,\frac{1}{b}}). $$

(3.21)

Furthermore, we take $\hat{\phi}(x)\in W_{M,\frac{1}{b}}(\mathbb{R}^{n})$. Then from the arguments of [2, pp.20-21], we have

$$\begin{aligned}& {\bigl\Vert (i\sigma\csc\alpha)^{k}\phi(\sigma)\bigr\Vert _{p}} \\& \quad = \biggl( \int_{\mathbb{R}^{n}}\bigl\vert (i\sigma\csc \alpha)^{k}\phi(\sigma )\bigr\vert ^{p}\,dw \biggr)^{\frac{1}{p}} \\& \quad = \biggl( \int_{\mathbb{R}^{n}} \biggl( \frac{|i\sigma\csc\alpha|^{|k|+2} +|i\sigma\csc\alpha|^{|k|}}{(|w|^{2}+1)} \biggr)^{p} \bigl\vert \phi(\sigma )\bigr\vert ^{p}\,dw \biggr)^{\frac{1}{p}} \\& \quad \leq \biggl( \int_{\mathbb{R}^{n}} \biggl( \frac{|i\sigma\csc\alpha |^{|k|+2} }{(|w|^{2}+1)} \biggr)^{p} \bigl|\phi(\sigma)\bigr|^{p}\,dw \biggr)^{\frac {1}{p}} + \biggl( \int_{\mathbb{R}^{n}} \biggl( \frac{|i\sigma\csc\alpha |^{|k|}}{(|w|^{2}+1)} \biggr)^{p} \bigl\vert \phi(\sigma)\bigr\vert ^{p}\,dw \biggr)^{\frac {1}{p}} \\& \quad \leq \biggl( \int_{\mathbb{R}^{n}}\frac{dw}{(|w|^{2} +1)^{p}} \biggl( C_{\alpha,k}\int_{\mathbb{R}^{n}}\bigl\vert e^{-\langle x, \tau\rangle\csc\alpha} \hat{\phi}_{\alpha}(x)\bigr\vert \,dx \biggr)^{p} \biggr)^{\frac{1}{p}} \\& \qquad {} + \biggl( \int_{\mathbb{R}^{n}}\frac{dw}{(|w|^{2} +1)^{p}} \biggl( C'_{\alpha,k}\int_{\mathbb{R}^{n}}\bigl\vert e^{-\langle x, \tau\rangle\csc\alpha} \hat{\phi}_{\alpha}(x)\bigr\vert \,dx \biggr)^{p} \biggr)^{\frac{1}{p}} \\& \quad \leq \bigl(C_{\alpha,k} C_{k+2,\delta} +C'_{\alpha,k} C_{k,\delta}\bigr) \biggl( \int_{\mathbb{R}^{n}}\frac{dw}{(|w|^{2} +1)^{p}} \\& \qquad {}\times \biggl(\int_{\mathbb{R}^{n}} \biggl\vert \exp\biggl[-\langle x, \tau \rangle\csc\alpha-M\biggl[\biggl(\frac{1}{b}-\delta \biggr)x\biggr]\biggr] \biggr\vert \,dx \biggr)^{p} \biggr)^{\frac{1}{p}} \\& \quad \leq C_{\alpha,k,\delta} \biggl(\int_{\mathbb{R}^{n}}\biggl\vert \exp\biggl[-\langle x, \tau \rangle\csc\alpha-M\biggl[\biggl(\frac{1}{b}- \delta\biggr)x\biggr]\biggr]\biggr\vert \,dx \biggr) \biggl( \int _{\mathbb{R}^{n}}\frac{dw}{(|w|^{2} +1)^{p}} \biggr)^{\frac{1}{p}} \\& \quad \leq C_{\alpha,k,\delta} C_{p} \int_{\mathbb{R}^{n}} \biggl\vert \exp\biggl[-\langle x, \tau\rangle\csc\alpha-M\biggl[\biggl( \frac{1}{b}-\delta\biggr)x\biggr]\biggr]\biggr\vert \,dx. \end{aligned}$$

(3.22)

Now using the Young inequality (1.21) and from the arguments of [2, p.21], we get

$$ \bigl\Vert (i\sigma\csc\alpha)^{k}\phi(\sigma)\bigr\Vert _{p} \leq C_{\alpha ,k,\delta,p} \exp\bigl[\Omega\bigl[(b+ \rho)\tau\bigr] \bigr]. $$

This implies that

$$ F_{\alpha}^{-1}(W_{M,\frac{1}{b}})\subset W^{\Omega,b,p}. $$

(3.23)

Now (3.21) and (3.23) give

$$ W^{\Omega,b}\subset W^{\Omega,b,p}. $$

(3.24)

Finally, (3.19) and (3.24) give

$$ W^{\Omega,b}= W^{\Omega,b,p}. $$

(3.25)

 □

Theorem 3.5

Let $\Omega_{0}(y)$ and $M_{0}(x)$ be the functions which are dual in the sense of Young to the functions $M(x)$ and $\Omega(y)$, respectively. Then

$$ W^{\Omega,b,p}_{M,a}=W^{\Omega,b}_{M,a}, \quad 1\leq p< \infty. $$

(3.26)

Proof

Let $e^{-\frac{i|x|^{2}\cot\alpha}{2}}\phi(x)\in W^{\Omega,b,p}_{M,a}$. Then from Theorem 2.3, it follows that

$$ F_{\alpha}\bigl(W^{\Omega,b,p}_{M,a}\bigr)\subset W^{\Omega_{0},\frac {1}{a}}_{M_{0},\frac{1}{b}}. $$

(3.27)

By the inverse property of the fractional Fourier transform we get

$$ W^{\Omega,b,p}_{M,a}\subset F_{\alpha}^{-1} \bigl(W^{\Omega_{0},\frac {1}{a}}_{M_{0},\frac{1}{b}}\bigr). $$

(3.28)

Now let $\hat{\phi}_{\alpha}(z)\in W_{M_{0},\frac{1}{b}}^{\Omega _{0},\frac{1}{a}}$. Then from the arguments of [2, p.24], we get

$$ \phi(\sigma+ i\tau)= C_{\alpha}\int_{\mathbb{R}^{n}} e^{-\frac{i(|z|^{2} + |\sigma|^{2})\cot\alpha}{2}+i\langle\sigma, z\rangle\csc\alpha} \hat {\phi}_{\alpha}(z) \,dx. $$

Therefore,

$$\begin{aligned}& {\bigl\vert \phi(\sigma+ i\tau)\bigr\vert } \\& \quad \leq |C_{\alpha}|\int_{\mathbb{R}^{n}} \bigl\vert \exp \bigl[-\langle w, y\rangle\csc \alpha-\langle\tau, x\rangle\csc\alpha\bigr]\bigr\vert \bigl\vert \hat{\phi}_{\alpha }(z)\bigr\vert \,dx \\& \quad \leq C_{\alpha} C_{\delta,\rho}\int_{\mathbb{R}^{n}} \bigl\vert \exp\bigl[-\langle w, y\rangle\csc\alpha-\langle\tau, x\rangle\csc \alpha\bigr]\bigr\vert \\& \qquad {}\times\biggl\vert \exp\biggl[-M_{0}\biggl[\biggl( \frac{1}{b}-\delta\biggr)x\biggr]+\Omega_{0}\biggl[\biggl( \frac{1}{a}+\rho \biggr)y\biggr]\biggr]\biggr\vert \,dx \\& \quad \leq C_{\delta,\rho,\alpha} \exp\biggl[\Omega_{0}\biggl[\biggl( \frac{1}{a}+\rho \biggr)y\biggr]-\langle w, y\rangle\csc\alpha\biggr] \\& \qquad {}\times\int _{\mathbb{R}^{n}} \exp\biggl[-M_{0}\biggl[\biggl( \frac {1}{b}-\delta\biggr)x\biggr]+\langle\tau, x\rangle\csc\alpha\biggr] \,dx. \end{aligned}$$

Now using (1.21), we have

$$\begin{aligned} \bigl\vert \phi(\sigma+ i\tau)\bigr\vert \leq& C'_{\delta,\rho,\alpha} \exp \biggl[-M \biggl[\frac{w\csc\alpha}{\frac{1}{a}+ \rho} \biggr] + \Omega \biggl[ \frac{\tau \csc\alpha}{(\frac{1}{b}+ 2\delta)} \biggr] \biggr] \\ \leq&C'_{\delta,\rho,\alpha} \exp \biggl[ -M \biggl[\biggl( \frac{1}{a}-\delta _{0}\biggr)w\csc\alpha \biggr] +\Omega \biggl[\biggl(\frac{1}{b}+ \rho_{0}\biggr)\tau\csc \alpha \biggr] \biggr], \end{aligned}$$

where $\rho_{0}$ and $\delta_{0}$ are arbitrarily small together with ρ and δ, respectively. This shows that

$$ F_{\alpha}^{-1}\bigl(W^{\Omega_{0},b,}_{M_{0},a}\bigr) \subset W^{\Omega,b}_{M,a}. $$

(3.29)

Thus from (3.28) and (3.29), we get

$$ W^{\Omega,b,p}_{M,a}\subset W^{\Omega,b}_{M,a}. $$

(3.30)

Similarly it is easy to show that

$$ W^{\Omega,b}_{M,a}\subset W^{\Omega,b,p}_{M,a}. $$

(3.31)

Finally, (3.30) and (3.31) imply that

$$ W^{\Omega,b,p}_{M,a}=W^{\Omega,b}_{M,a}. $$

(3.32)

 □

4 Uniqueness class of a Cauchy problem

In this section we apply the theory of the fractional Fourier transform which is discussed in (1.4) and (1.6) to establish a uniqueness theorem for the Cauchy problem:

$$\begin{aligned}& \frac{\partial u(x,t)}{\partial t} = P(i\triangle_{x})u(x, t),\quad \forall(x, t)\in \mathbb{R}^{n} \times[0, T], \end{aligned}$$

(4.1)

$$\begin{aligned}& u(x, 0)= u_{0}(x), \end{aligned}$$

(4.2)

where

$$\begin{aligned} \triangle_{x}^{k} =& \triangle_{x_{1}}^{k_{1}} \cdots\triangle _{x_{n}}^{k_{n}} \end{aligned}$$

(4.3)

$$\begin{aligned} =& \biggl( \frac{\partial}{\partial x_{1}}-ix_{1} \cot\alpha \biggr)^{k_{1}}\cdots \biggl( \frac{\partial}{\partial x_{n}}-ix_{n} \cot \alpha \biggr)^{k_{n}} \end{aligned}$$

(4.4)

is a differential operator and $u(x, t)$ is an $N\times1$ column vector. Here P is an $N\times N$ polynomial matrix with constant coefficients of order k. A similar problem has been investigated by Gel’fand and Shilov [2], and Friedman [3] by exploiting the theory of Fourier transforms. Also, Pathak [10] studied the uniqueness of the Cauchy problem by using the theory of the Hankel transform.

Theorem 4.1

The Cauchy problem (4.1) and (4.2) possesses a unique solution $u(x, t)$ in the space $(W^{\Omega_{0},\frac{1}{a-\theta}}_{M_{0},\frac{1}{b-\theta}})'$ for the interval $0 \leq t \leq T$, $T< (2cp_{0})^{-1}(d/2)^{p_{0}}$, $\theta< a$, and for any initial function $u_{0}(x)$ belonging to the same space, where $p_{0}$ is the reduced order of the system (4.1) and (4.2) with $i\triangle_{x}$ replaced by $i\frac{\partial }{\partial x}$ and c being a constant depending on P.

Proof

From the fundamental result [3, p.177], the Cauchy problem (4.1) and (4.2) will have a solution in the space $\Phi_{1}'$ for $0 \leq t\leq T$ if there exists a solution of the adjoint problem,

$$\begin{aligned}& \frac{\partial}{\partial t}\phi(x, t)= \tilde{P}\bigl(i\triangle_{x}^{*} \bigr)\phi (x, t), \end{aligned}$$

(4.5)

$$\begin{aligned}& \phi(x, t_{0})= \phi_{0}(x)\in\Phi, \end{aligned}$$

(4.6)

in the space $\Phi_{1}$ for $0\leq t\leq t_{0}$, where $t_{0}$ is any point in the interval $0\leq t\leq T$, $\tilde{P}$ is the adjoint of P and $\triangle_{x}^{*}$ is the conjugate of $\triangle_{x}$.

Applying the fractional Fourier transform to (4.5) and (4.6), we get

$$\begin{aligned}& \frac{d}{ dt}\varPsi_{\alpha}(\sigma, t)= \tilde{P}(\sigma\csc\alpha )\varPsi(\sigma, t), \end{aligned}$$

(4.7)

$$\begin{aligned}& \varPsi_{\alpha}(\sigma, t_{0})= \varPsi_{\alpha,0}( \sigma), \end{aligned}$$

(4.8)

where $\varPsi_{\alpha}(\sigma, t)= (F_{\alpha}\phi)(x, t)$. A formal solution of (4.7) and (4.8) is given by

$$ \varPsi_{\alpha}(\sigma, t)= \exp\bigl[(t-t_{0})\tilde{P}( \sigma\csc\alpha )\bigr]\varPsi_{\alpha,0}(\sigma). $$

(4.9)

Let us write

$$ Q(\sigma\csc\alpha,t_{0},t)= \exp\bigl[(t-t_{0}) \tilde{P}(\sigma\csc\alpha)\bigr], $$

(4.10)

consisting of entire analytic functions of σ where $\sigma= w +i\tau$. Since $p_{0}$ is the reduced order of the system (4.1) and (4.2), using the inequality

$$\vert \sigma\csc\alpha \vert ^{p_{0}} \leq2^{p_{0}}\bigl( \vert w\csc\alpha \vert ^{p_{0}} + |\tau \csc\alpha|^{p_{0}} \bigr) $$

and the arguments of [2, p.53] in (4.10) we obtain

$$\bigl\vert Q(\sigma\csc\alpha,t_{0},t)\bigr\vert \leq C \exp \bigl[(p_{0})^{-1}\theta ^{p_{0}}\bigl(\vert w\csc \alpha \vert ^{p_{0}} + |\tau\csc\alpha|^{p_{0}} \bigr)\bigr] $$

under the assumptions $t_{0}\leq t\leq t_{0}+T$ and $2^{p_{0}+1}cT< (p_{0})^{-1}\theta ^{p_{0}}$.

If we set

$$M(w\csc\alpha)= {|w\csc\alpha|}^{p_{0}}/{p_{0}},\qquad \Omega(\tau \csc\alpha)= {|\tau\csc\alpha|}^{p_{0}}/{p_{0}}, $$

then

$$\bigl\vert Q(\sigma\csc\alpha,t_{0},t)\bigr\vert \leq C \exp \bigl[M(\theta\cdot w\csc\alpha) + \Omega(\theta\cdot\tau\csc\alpha)\bigr]. $$

Now, let us assume that

$$\phi_{0}(x)\in\Phi= W^{\Omega_{0},\frac{1}{a}}_{M_{0},\frac{1}{b}}. $$

Then

$$\psi_{\alpha,0}(\sigma)= (F_{\alpha}\phi_{0}) (x)\in W^{\Omega,b}_{M,a}. $$

We now apply the theorem [2, p.54] for given a. One can always choose the time interval $0\leq t\leq T$ so small that the inequality $\theta< a$ holds; for such values of T the matrix $Q(\sigma\csc\alpha ,t_{0},t)$ will be a multiplier in the space $W^{\Omega,b}_{M,a}$ which maps this space into the space $W^{\Omega,b+\theta}_{M,a-\theta}$ taking T sufficiently small. Thus the Cauchy problem (4.7) and (4.8) has a unique solution in $W^{\Omega,b+\theta }_{M,a-\theta}$. Also we can show that

$$F^{-1}_{\alpha}\bigl[W^{\Omega,b+\theta}_{M,a-\theta}\bigr]= \Phi_{1}= W^{\Omega _{0},\frac{1}{a-\theta}}_{M_{0},\frac{1}{b+\theta}}, $$

and the Cauchy problem (4.5) and (4.6) has a unique solution in $W^{\Omega_{0},\frac{1}{a-\theta}}_{M_{0},\frac{1}{b+\theta }}$.

Now using the arguments of [3, Theorem 6, p.177], we get the complete proof. □