Skip to content

Advertisement

  • Research
  • Open Access

Characterization of \(W^{p}\)-type of spaces involving fractional Fourier transform

Journal of Inequalities and Applications20152015:31

https://doi.org/10.1186/s13660-014-0544-9

  • Received: 25 November 2014
  • Accepted: 23 December 2014
  • Published:

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.

Keywords

  • fractional Fourier transform
  • convex functions
  • Gel’fand and Shilov spaces of type W
  • \(L_{p}\)-space

MSC

  • 46F12
  • 46E15

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 [24], 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. □

Declarations

Acknowledgements

The first author is thankful to DST-CIMS, Banaras Hindu University, Varanasi, India for providing the research facilities and the second author is also thankful to DST-CIMS, Banaras Hindu University, Varanasi, India for awarding the Junior Research Fellowship since December 2012.

Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

Authors’ Affiliations

(1)
DST-Centre for Interdisciplinary Mathematical Sciences, Banaras Hindu University, Varanasi, 221005, India
(2)
Department of Mathematical Sciences, Indian Institute of Technology (BHU), Varanasi, 221005, India

References

  1. Gurevich, BL: New types of test function spaces and spaces of generalized functions and the Cauchy problem for operator equations. Dissertation, Kharkov (1956) Google Scholar
  2. Gel’fand, IM, Shilov, GE: Generalized Functions, vol. 3. Academic Press, New York (1967) MATHGoogle Scholar
  3. Friedman, A: Generalized Functions and Partial Differential Equations. Prentice Hall, New York (1963) MATHGoogle Scholar
  4. Pathak, RS, Upadhyay, SK: \(W^{p}\)-Space and Fourier transformation. Proc. Am. Math. Soc. 121(3), 733-738 (1994) MATHMathSciNetGoogle Scholar
  5. Betancor, JJ, Rodriguez-Mesa, L: Characterization of W-type spaces. Proc. Am. Math. Soc. 126(5), 1371-1379 (1988) View ArticleMathSciNetGoogle Scholar
  6. Upadhyay, SK: W-Spaces and pseudo-differential operators. Appl. Anal. 82, 381-397 (2003) View ArticleMATHMathSciNetGoogle Scholar
  7. De Bie, H, De Schepper, N: Fractional Fourier transforms of hyper complex signals. Signal Image Video Process. 6, 381-388 (2012) View ArticleGoogle Scholar
  8. Upadhyay, SK, Kumar, A, Dubey, JK: Characterization of spaces of type W and pseudo-differential operators of infinite order involving fractional Fourier transform. J. Pseud.-Differ. Oper. Appl. 5(2), 215-230 (2014) View ArticleMathSciNetGoogle Scholar
  9. Prasad, A, Mahato, A: The fractional wavelet transform on spaces of type W. Integral Transforms Spec. Funct. 24(3), 239-250 (2012) View ArticleMathSciNetGoogle Scholar
  10. Pathak, RS: On Hankel transformable spaces and a Cauchy problem. Can. J. Math. 34(1), 84-106 (1985) View ArticleGoogle Scholar

Copyright

© Upadhyay and Kumar; licensee Springer 2015

Advertisement