- Research
- Open Access
Characterization of \(W^{p}\)-type of spaces involving fractional Fourier transform
- Santosh Kumar Upadhyay1, 2Email author and
- Anuj Kumar1
Journal of Inequalities and Applications20152015:31
https://doi.org/10.1186/s13660-014-0544-9
© Upadhyay and Kumar; licensee Springer 2015
- Received: 25 November 2014
- Accepted: 23 December 2014
- Published: 28 January 2015
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
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
and the norm of x is defined by
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
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
where
and
The corresponding inversion formula is given by
where the kernel
and \(C_{\alpha}\) is defined by (1.5).
$$\mathbb{R}^{n} = \bigl\{ (x_{1},\ldots,x_{n}) \mbox{: } x_{j}\mbox{'s are real numbers}\bigr\} . $$
$$ \langle x ,y \rangle=x\cdot y = {\sum_{j=1}^{n} x_{j}\cdot y_{j}} $$
(1.1)
$$ \vert x\vert = \Biggl( {\sum_{j=1}^{n} x^{2}_{j}} \Biggr)^{\frac{1}{2}}= \bigl(x^{2}_{1}+ \cdots+x^{2}_{n} \bigr)^{\frac{1}{2}}. $$
(1.2)
$$ \Vert f\Vert _{p} = \biggl(\int_{\mathbb{R}^{n}}{ \bigl\vert f(x)\bigr\vert ^{p} \, dx} \biggr)^{\frac{1}{p}}. $$
(1.3)
$$ \hat{f}_{\alpha}(\xi) = (F_{\alpha}f) (\xi) = \int _{\mathbb{R}^{n}} K_{\alpha}(x ,\xi) f(x)\, dx, \quad \xi\in \mathbb{R}^{n}, $$
(1.4)
$$ K_{\alpha}(x ,\xi)= \begin{cases} C_{\alpha}e^{\frac{i(\vert x\vert ^{2}+{\vert \xi \vert }^{2})\cot {\alpha}}{2} - i \langle x , {\xi} \rangle\csc{\alpha}} & \text{if } {\alpha}\ne n\pi, \\ \frac{1}{(2\pi)^{\frac{n}{2}}}e^{-i \langle x , {\xi} \rangle} &\text{if }\alpha= \frac{\pi}{2}, \end{cases} \quad \forall n\in \mathbb{Z} $$
$$ C_{\alpha} =(2\pi i \sin\alpha)^{\frac{-n}{2}}e^{\frac{in\alpha}{2}} = \frac{1}{[\pi(1 - e^{-2i\alpha})]^{\frac{n}{2}}}. $$
(1.5)
$$ {f}(x) = \frac{1}{(2\pi)^{\frac{n}{2}}} \int_{\mathbb{R}^{n}} \overline {K_{\alpha}(x ,\xi)} \hat{f}_{\alpha}(\xi) \, d\xi, \quad x\in \mathbb{R}^{n}, $$
(1.6)
$$ \overline{K_{\alpha}(x ,\xi)}= C_{\alpha} e^{-\frac{i(\vert x\vert ^{2}+{\vert \xi \vert }^{2})\cot {\alpha}}{2}+i \langle x,{\xi} \rangle\csc{\alpha}}, $$
Now from the technique of [9, p.2], (1.1) can be written as
Replacing \(f(x)= e^{-\frac{i\vert x\vert ^{2} \cot\alpha }{2}} \phi(x)\) in (1.3), we obtain
Now substituting \(\xi= w \sin\alpha\), where \(w\in \mathbb{R}^{n}\) in (1.4), we obtain
Let \(\psi = F_{\alpha} [e^{-\frac{i\vert x\vert ^{2}\cot\alpha }{2}}\phi(x) ]\), then (1.6) can be written as
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 \(\mu_{j}(\infty)=w_{j}(\infty)=\infty \).
$$\begin{aligned} F_{\alpha}\bigl[f(x)\bigr](\xi) =& C_{\alpha}e^{\frac{{i \vert \xi \vert ^{2} \cot \alpha}}{2}} \int_{\mathbb{R}^{n}}{e^{-i \langle x,\xi \rangle \csc\alpha} \bigl[f(x)e^{\frac{{i \vert x\vert ^{2} cot\alpha}}{2}} \bigr]\, dx} \\ =& (2\pi)^{\frac{n}{2}}C_{\alpha}e^{\frac{{i \vert \xi \vert ^{2} \cot \alpha}}{2}} \bigl[f(x)e^{\frac{{i \vert x \vert ^{2} \cot\alpha}}{2}}\hat{ \bigr]}(\xi\csc\alpha). \end{aligned}$$
(1.7)
$$ F_{\alpha}\bigl[e^{-\frac{i\vert x\vert ^{2}cot\alpha}{2}}\phi(x)\bigr](\xi)= (2 \pi)^{\frac{n}{2}}C_{\alpha}e^{\frac{{i \vert \xi \vert ^{2} \cot\alpha }}{2}}\bigl[\phi(x)\hat{\bigr]}(\xi\csc\alpha). $$
(1.8)
$$ F_{\alpha}\bigl[e^{-\frac{i\vert x\vert ^{2}\cot\alpha}{2}}\phi(x)\bigr](w \sin \alpha)= (2 \pi)^{\frac{n}{2}}C_{\alpha}e^{\frac{i \vert w\sin\alpha \vert ^{2} \cot\alpha}{2}}\bigl[\phi(x)\hat{\bigr]}(w). $$
(1.9)
$$ \psi(w \sin\alpha)=(2\pi)^{\frac{n}{2}}C_{\alpha}e^{\frac{i \vert w\sin \alpha \vert ^{2} \cot\alpha}{2}}\bigl[\phi(x)\hat{ \bigr]}(w). $$
(1.10)
We define
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
We define
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
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
for each \(k\in\mathbb{Z}^{n}_{+}\) where \(D^{k}_{x}=D^{k_{1}}_{x_{1}}\cdots D^{k_{n}}_{x_{n}}\),
and \(a_{1},\ldots,a_{n}\), \(C_{k,\delta,p}\), \(C_{k,\delta}\) are positive constants depending on the function \(\phi(x)\).
$$\begin{aligned}& M_{j}(x_{j})=\int_{0}^{x_{j}}{ \mu_{j}(\xi_{j})} \, d\xi_{j} \quad (x_{j}\geq0), \end{aligned}$$
(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)
$$\begin{aligned}& M_{j}(-x_{j})=M_{j}(x_{j}), \qquad M_{j}(x_{j})+ M_{j} \bigl(x'_{j}\bigr)\leq M_{j} \bigl(x_{j}+x'_{j}\bigr), \end{aligned}$$
(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)
$$\begin{aligned}& \mu(\xi) = \bigl(\mu_{1}(\xi_{1})\bigr),\ldots,\bigl( \mu_{n}(\xi_{n})\bigr), \\& w(\eta)= \bigl(w_{1}(\eta_{1})\bigr),\ldots, \bigl(w_{n}(\eta_{n})\bigr). \end{aligned}$$
$$ \bigl\vert D^{k}_{x}\phi(x)\bigr\vert \leq C_{k,\delta} \exp\bigl[-M\bigl[(a-\delta)x\bigr]\bigr], $$
(1.15)
$$ \biggl(\int_{\mathbb{R}^{n}}{\bigl\vert \exp \bigl[-M\bigl[(a- \delta)x\bigr] \bigr]D^{k}_{x}\phi(x)\bigr\vert ^{p}}\, dx \biggr)^{\frac{1}{p}}\leq C_{k,\delta ,p}, \quad p\geq1 $$
(1.16)
$$ \exp\bigl[-M\bigl[(a-\delta)x\bigr]\bigr]=\exp\bigl[-M_{1} \bigl[(a_{1}-\delta_{1})x_{1}\bigr]-\cdots -M_{n}\bigl[(a_{n}-\delta_{n})x_{n} \bigr]\bigr] $$
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
where
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
where
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
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 \(C_{\rho,\delta,p} > 0\),
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 \(\phi(z)\).
$$ \bigl\vert z^{k}\phi(z)\bigr\vert \leq C_{k,\rho} \exp \bigl[\Omega\bigl[(b + \rho)y\bigr]\bigr], \quad k\in\mathbb{Z}^{n}_{+}, $$
(1.17)
$$ z^{k}=z_{1}^{k_{1}}\cdots z_{n}^{k_{n}}, $$
$$ \biggl(\int_{\mathbb{R}^{n}}\bigl\vert \exp \bigl[\Omega\bigl[(b+ \rho)y\bigr] \bigr] z^{k}\phi(z)\bigr\vert ^{p}\,dx \biggr)^{\frac{1}{p}}\leq C_{k,\rho,p}, $$
(1.18)
$$ \exp\bigl[\Omega\bigl[(b+\rho)\bigr]y\bigr] = \exp\bigl[\Omega_{1} \bigl[ (b_{1}+\rho_{1})y_{1}\bigr]+\cdots+ \Omega_{n}\bigl[ (b_{n}+\rho_{n})y_{n} \bigr]\bigr]. $$
$$ \bigl\vert \phi(z)\bigr\vert \leq C_{\delta,\rho} \exp\bigl[ -M\bigl[(a - \delta) x\bigr] + \Omega\bigl[(b + \rho)y\bigr]\bigr] , $$
(1.19)
$$ \biggl(\int_{\mathbb{R}^{n}}\bigl\vert \exp \bigl[ M\bigl[(a-\delta)x \bigr]-\Omega\bigl[(b+\rho )y\bigr] \bigr]\phi(z)\bigr\vert ^{p}\,dx \biggr)^{\frac{1}{p}} \leq C_{\rho,\delta ,p}, $$
(1.20)
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,
holds for any \(x_{j}\geq\), \(y_{j}\geq0\).
$$ x_{j}y_{j}\leq M_{j}(x_{j})+ \Omega_{j}(y_{j}), $$
(1.21)
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
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
Using (1.7), we get
Now using the Young inequality (1.21) and the arguments of [3, p.23], we get
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
□
$$ \bigl\Vert (\sigma\sin\alpha)^{k}\psi(\sigma\sin\alpha)\bigr\Vert _{r}= \biggl[\int_{\mathbb{R}^{n}} \bigl\vert (\sigma\sin \alpha)^{k}\psi(\sigma\sin \alpha)\bigr\vert ^{r}\, dw \biggr]^{\frac{1}{r}}. $$
$$\begin{aligned}& {\bigl\Vert (\sigma\sin\alpha)^{k}\psi(\sigma\sin\alpha)\bigr\Vert _{r}} \\& \quad \leq \biggl[\vert \sin\alpha \vert ^{r|k|}\int _{\mathbb{R}^{n}} \biggl(\frac{\vert \sigma \vert ^{|k+2|} + \vert \sigma \vert ^{|k|}}{ \vert w\vert ^{2} +1} \biggr)^{r} \bigl\vert \psi(\sigma\sin\alpha)\bigr\vert ^{r} \,dw \biggr]^{\frac{1}{r}} \\& \quad \leq \biggl[\vert \sin\alpha \vert ^{r|k|}\int _{\mathbb{R}^{n}} \biggl(\frac{\vert \sigma \vert ^{|k+2|}}{\vert w\vert ^{2} +1} \biggr)^{r} \bigl\vert \psi(\sigma\sin\alpha)\bigr\vert ^{r} \,dw \biggr]^{\frac{1}{r}} \\& \qquad {}+ \biggl[\vert \sin\alpha \vert ^{r|k|}\int _{\mathbb{R}^{n}} \biggl(\frac{ \vert \sigma \vert ^{|k|}}{\vert w\vert ^{2} +1} \biggr)^{r} \bigl\vert \psi(\sigma\sin\alpha)\bigr\vert ^{r} \,dw \biggr]^{\frac{1}{r}}. \end{aligned}$$
$$\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}$$
$$ \bigl\Vert (\sigma\sin\alpha)^{k}\psi(\sigma\sin\alpha)\bigr\Vert _{r}\leq D'_{k+2,\rho,\alpha,r} e^{\Omega[\frac{\tau}{\gamma}]}+ D'_{k,\rho ,\alpha,r} e^{\Omega[\frac{\tau}{\gamma}]}. $$
$$ \bigl\Vert (\sigma\sin\alpha)^{k}\psi(\sigma\sin\alpha)\bigr\Vert _{r}\leq C^{r}_{k,\rho, \vert \sin\alpha \vert } e^{\Omega[\frac{\tau}{\gamma}]}. $$
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
From the arguments of [4, p.737] we have
Hence,
$$\begin{aligned} \bigl\vert D^{k}_{w}\psi(w\sin\alpha)\bigr\vert \leq& C_{k,r,\eta,s,\rho} \exp\bigl[- \langle y,w \rangle+ \Omega (b+\rho)y \bigr]\\ \leq& C_{k,r,\eta,s,\rho} \exp\bigl[-y_{1}w_{1}+ \Omega_{1}\bigl[(b_{1}+\rho _{1})y_{1} \bigr] \\ &{}-\cdots -y_{n}w_{n}+\Omega_{n} \bigl[(b_{n}+\rho_{n})y_{n}\bigr] \bigr]. \end{aligned}$$
$$ \bigl\vert D^{k}_{w}\psi(w\sin\alpha)\bigr\vert \leq C_{k,\rho,r,\eta,s} \exp \biggl[ -M \biggl[ \biggl( \frac{1}{b}-\delta \biggr)w \biggr] -M \biggl[ \frac {\rho^{2}w}{b^{3}} \biggr] \biggr]. $$
$$ \biggl\lVert \exp\biggl[ M\biggl[\biggl(\frac{1}{b}-\delta\biggr)w\biggr]\biggr] D^{k}_{w}\phi(w \sin\alpha) \biggr\rVert _{r} \leq C^{r}_{k,\rho,r,\eta,s}. $$
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
Now using the arguments of [4, p.738], we have
Hence,
□
$$\begin{aligned} \bigl\vert \psi(\sigma\sin\alpha)\bigr\vert \leq&\bigl\vert (2\pi i \sin \alpha )^{\frac{-n}{2}}e^{\frac{in\alpha}{2}}\bigr\vert \int_{\mathbb{R}^{n}} \bigl\vert e^{\overline{i \langle\sigma,z \rangle}}\bigr\vert \bigl\vert \phi (z)\bigr\vert \,dx \\ \leq& \bigl\vert (2\pi i \sin\alpha)^{\frac{-n}{2}}e^{\frac{in\alpha }{2}}\bigr\vert \int_{\mathbb{R}^{n}}\bigl\vert e^{- \langle w,y \rangle - \langle\tau,x \rangle}\bigr\vert \bigl\vert \phi(z)\bigr\vert \,dx \\ \leq& D C_{\rho,\delta}\bigl\vert e^{- \langle w,y \rangle} \bigr\vert \int _{\mathbb{R}^{n}}\bigl\vert e^{- \langle\tau,x \rangle}\bigr\vert \exp \bigl[ -M \bigl[(a-\delta)x\bigr]+ \Omega\bigl[(b+\rho)y\bigr] \bigr]\,dx \\ \leq& C_{\rho,\delta,\alpha}e^{- \langle w,y \rangle}\int_{\mathbb{R}^{n}} \exp \bigl[ -M\bigl[(a-\delta)x\bigr]+ \Omega\bigl[(b+\rho)y\bigr] \bigr] \exp \bigl[ \vert \tau \vert \vert x\vert \bigr]\,dx \\ =&C_{\rho,\delta,\alpha} \exp \bigl[- \langle w,y \rangle+ \Omega\bigl[(b+\rho)y \bigr] \bigr] \int_{\mathbb{R}^{n}}\exp \bigl[ -M\bigl[(a-\delta )x \bigr]+ \vert \tau \vert \vert x\vert \bigr] \,dx. \end{aligned}$$
$$ \bigl\vert \psi(\sigma\sin\alpha)\bigr\vert \leq C_{\rho,\delta,\alpha} \exp \biggl[ -M_{0} \biggl[ \biggl(\frac{1}{b}-\delta_{0} \biggr)w \biggr] + \Omega_{0} \biggl[ \biggl(\frac {1}{a} + \rho_{0}\biggr)\tau \biggr]- M_{0} \biggl[ \frac{\rho_{0}^{2} w}{b^{3}} \biggr] \biggr]. $$
$$ \biggl\Vert \exp \biggl[M_{0} \biggl[ \biggl(\frac{1}{b}- \delta_{0} \biggr)w \biggr]-\Omega_{0} \biggl[ \biggl( \frac{1}{a}+\rho_{0} \biggr)\tau \biggr] \biggr]\psi(\sigma\sin \alpha)\biggr\Vert _{r}\leq C^{\prime r}_{\alpha,\delta,\rho} . $$
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
From the inverse property of the fractional Fourier transform, we have
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
Therefore,
Hence,
Therefore,
Now using the arguments of [4, p.23], we get
for arbitrarily small δ together with ρ. Hence the above expression gives
Thus (3.2) and (3.5) imply that
□
$$ F_{\alpha}\bigl(W^{p}_{M,a}\bigr)\subset W^{\Omega,\frac{1}{a}}. $$
(3.1)
$$ W^{p}_{M,a}\subset F^{-1}_{\alpha} \bigl(W^{\Omega,\frac{1}{a}}\bigr). $$
(3.2)
$$ \phi(x)= C_{\alpha}\int_{\mathbb{R}^{n}} e^{-\frac{i(|x|^{2} + |\sigma |^{2})\cot\alpha}{2}+ i\langle x, \sigma\rangle\csc\alpha} \hat{ \phi }_{\alpha}(\sigma) \,dw. $$
$$\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}$$
$$\begin{aligned} \bigl\vert \phi^{(k)}(x)\bigr\vert \leq& |C_{\alpha}|\sum _{r\leq k}\binom{k}{r}\sum _{\eta \leq r}\bigl\vert C_{\eta}(\cot\alpha)\bigr\vert \vert \csc\alpha|^{-|\eta|+|k-r|} \\ &{}\times\sum_{\beta\leq \eta} \binom{\eta}{\beta} \frac{(k-r)!}{(k-r-m)!} \sum_{\lambda\leq\beta }\bigl|C_{\lambda}( \cot\alpha)\bigr\vert \\ &{} \times\sum_{m\leq\eta-\beta}\binom{\eta-\beta}{m}\int _{\mathbb{R}^{n}}\bigl\vert e^{i\langle x, \sigma\rangle\csc\alpha}\bigr\vert \bigl\vert \sigma^{k-r-m+\lambda} D^{\eta-\beta-m}_{\sigma}\hat{\phi}_{\alpha}( \sigma)\bigr\vert \,dw. \end{aligned}$$
$$\begin{aligned} \bigl\vert \phi^{(k)}(x)\bigr\vert \leq&\vert C_{\alpha} \vert \sum_{r\leq k}\binom{k}{r}\sum _{\eta \leq r}\bigl\vert C_{\eta}(\cot\alpha)\bigr\vert \vert \csc\alpha \vert ^{-|\eta|+|k-r|} \\ &{}\times\sum_{\beta\leq \eta} \binom{\eta}{\beta} \frac{(k-r)!}{(k-r-m)!} \sum_{\lambda\leq\beta }\bigl\vert C_{\lambda}(\cot\alpha)\bigr\vert \\ &{} \times\sum_{m\leq\eta-\beta}\binom{\eta-\beta}{m}\int _{\mathbb{R}^{n}} e^{-\langle x, \tau\rangle\csc\alpha} \biggl(\frac{|\sigma |^{|k-r-m+\lambda|+2} +|\sigma|^{|k-r-m+\lambda|}}{(|w|^{2} +1)} \biggr) \\ &{} \times\bigl\vert D^{\eta-\beta-m}_{\sigma}\hat{ \phi}_{\alpha}(\sigma )\bigr\vert \,dw \\ \leq& C'_{\alpha,k} \exp\biggl[-\langle x, \tau\rangle\csc \alpha+ \Omega \biggl[\biggl(\frac{1}{a} + \rho\biggr)\tau\biggr]\biggr]. \end{aligned}$$
(3.3)
$$ \bigl\vert \phi^{(k)}(x)\bigr\vert \leq C'_{\alpha,k} \exp\bigl[-M\bigl[(a-\delta)x\csc\alpha\bigr]\bigr] $$
(3.4)
$$ F_{\alpha}^{-1}\bigl(W^{\Omega,\frac{1}{a}}\bigr)\subset W_{M,a}. $$
(3.5)
$$ W^{p}_{M,a}\subset W_{M,a}. $$
(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
Now by the inverse property of the fractional Fourier transform we have
Again let \(\hat{\phi}_{\alpha}(\sigma)\in W^{\Omega,\frac{1}{a}}\). Then from (3.3) we have
Using (1.21) and [6, p.385] we get
Therefore,
This implies that
From (3.8) and (3.12) we find
Now from (3.6) and (3.13) we get the result
□
$$ F_{\alpha}(W_{M,a})\subset W^{\Omega,\frac{1}{a}}. $$
(3.7)
$$ W_{M,a}\subset F_{\alpha}^{-1} \bigl(W^{\Omega,\frac{1}{a}} \bigr). $$
(3.8)
$$ \bigl\vert \phi^{k}(x)\bigr\vert \leq C'_{\alpha,k} \exp\biggl[-\langle x, \tau\rangle\csc\alpha + \Omega\biggl[\biggl( \frac{1}{a} + \rho\biggr)\tau\biggr]\biggr]. $$
(3.9)
$$ \bigl\vert \phi^{k}(x)\bigr\vert \leq C'_{\alpha,k} \exp\bigl[-M\bigl[(a-\delta)x\csc\alpha \bigr]-M\bigl[a^{3} \rho^{2} x\csc\alpha\bigr]\bigr]. $$
(3.10)
$$ \bigl\Vert \exp\bigl[M\bigl[(a-\delta)x\csc\alpha\bigr]\bigr] \phi^{k}(x)\bigr\Vert _{p}\leq C'_{\alpha, k} \bigl\Vert e^{-M[a^{3}\rho^{2} x\csc\alpha]}\bigr\Vert _{p}. $$
(3.11)
$$ F_{\alpha}^{-1}\bigl(W^{\Omega,\frac{1}{a}}\bigr)\subset W^{p}_{M,a}. $$
(3.12)
$$ W_{M,a}\subset W^{p}_{M,a}. $$
(3.13)
$$ W^{p}_{M,a}= W_{M,a}. $$
(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
By the inverse property of the fractional Fourier transform, we have
Now let \(\hat{\phi}_{\alpha}(x)\in W_{M,\frac{1}{b}}\). Then from the technique of [2, pp.20-21], we have
Therefore,
Now using the arguments of [2, p.21], we get
where ρ is arbitrarily small together with δ. Thus we have
Therefore (3.17) and (3.18) yield
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
By the inverse property of the fractional Fourier transform, we have
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
Now using the Young inequality (1.21) and from the arguments of [2, p.21], we get
This implies that
Now (3.21) and (3.23) give
Finally, (3.19) and (3.24) give
□
$$ F_{\alpha}\bigl( W^{\Omega,b,p}\bigr)\subset W_{M,\frac{1}{a}}. $$
(3.16)
$$ W^{\Omega,b,p}\subset F_{\alpha}^{-1} (W_{M,\frac{1}{a}}). $$
(3.17)
$$\begin{aligned}& {(i\sigma\csc\alpha)^{k} \phi(\sigma)} \\& \quad = C_{\alpha}\int_{\mathbb{R}^{n}} e^{-\frac{i(|x|^{2} + |\sigma|^{2})\cot \alpha}{2}} \bigl( D^{k}_{x}e^{i\langle x, \sigma\rangle\csc\alpha } \bigr) \hat{ \phi}_{\alpha}(x) \,dx \\& \quad = C_{\alpha}(-1)^{|k|}\int_{\mathbb{R}^{n}}e^{i\langle x, \sigma\rangle \csc\alpha} \bigl( D^{k}_{x} e^{-\frac{i|x|^{2} \cot\alpha}{2}}\hat{\phi }_{\alpha}(x) \bigr) e^{-\frac{ i|\sigma|^{2}\cot\alpha}{2}} \,dx \\& \quad = C_{\alpha}(-1)^{|k|}\int_{\mathbb{R}^{n}}e^{i\langle x, \sigma\rangle \csc\alpha} \sum_{r\leq k}\binom{k}{r} \bigl(D^{r}_{x} e^{-\frac{i|x|^{2} \cot\alpha}{2}} \bigr) \bigl(D^{k-r}_{x} \hat{ \phi}_{\alpha}(x) \bigr)e^{-\frac{ i|\sigma |^{2}\cot\alpha}{2}} \,dx \\& \quad = C_{\alpha}(-1)^{|k|}\int_{\mathbb{R}^{n}}e^{i\langle x, \sigma\rangle \csc\alpha} \sum_{r\leq k}\binom{k}{r}\sum _{\beta\leq r}C_{\beta}(\cot \alpha) e^{-\frac{i|x|^{2} \cot\alpha}{2}}e^{-\frac{ i|\sigma|^{2}\cot\alpha }{2}} \bigl(x^{\beta}D^{k-r}_{x}\hat{\phi}_{\alpha}(x) \bigr)\,dx. \end{aligned}$$
$$\begin{aligned} \bigl\vert (\sigma\csc\alpha)^{k}\phi(s)\bigr\vert \leq& |C_{\alpha}|\sum_{r\leq k}\binom {k}{r}\sum _{\beta\leq r}\bigl\vert C_{\beta}(\cot\alpha)\bigr\vert \int_{\mathbb{R}^{n}} \bigl\vert e^{-\langle x, \tau\rangle\csc\alpha}\bigr\vert \bigl\vert x^{\beta}D^{k-r}_{x} \hat{\phi }_{\alpha}(x)\bigr\vert \,dx \\ \leq&|C_{\alpha}|\sum_{r\leq k} \binom{k}{r}\sum_{\beta\leq r}\bigl\vert C_{\beta }(\cot\alpha)\bigr\vert \int_{\mathbb{R}^{n}} e^{[|x||\tau\csc\alpha|-M[(\frac{1}{b}-\delta )x]]} \,dx. \end{aligned}$$
$$ \bigl\vert (\sigma\csc\alpha)^{k}\phi(\sigma)\bigr\vert \leq C_{\alpha,k} \exp\bigl[\Omega \bigl[(b+\rho)\tau\csc\alpha\bigr]\bigr], $$
$$ F_{\alpha}^{-1}(W_{M,\frac{1}{b}})\subset W^{\Omega,b}. $$
(3.18)
$$ W^{\Omega,b,p} \subset W^{\Omega,b} . $$
(3.19)
$$ F_{\alpha}\bigl(W^{\Omega,b}\bigr)\subset W_{M,\frac{1}{b}}. $$
(3.20)
$$ W^{\Omega,b}\subset F_{\alpha}^{-1}( W_{M,\frac{1}{b}}). $$
(3.21)
$$\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)
$$ \bigl\Vert (i\sigma\csc\alpha)^{k}\phi(\sigma)\bigr\Vert _{p} \leq C_{\alpha ,k,\delta,p} \exp\bigl[\Omega\bigl[(b+ \rho)\tau\bigr] \bigr]. $$
$$ F_{\alpha}^{-1}(W_{M,\frac{1}{b}})\subset W^{\Omega,b,p}. $$
(3.23)
$$ W^{\Omega,b}\subset W^{\Omega,b,p}. $$
(3.24)
$$ W^{\Omega,b}= W^{\Omega,b,p}. $$
(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
By the inverse property of the fractional Fourier transform we get
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
Therefore,
Now using (1.21), we have
where \(\rho_{0}\) and \(\delta_{0}\) are arbitrarily small together with ρ and δ, respectively. This shows that
Thus from (3.28) and (3.29), we get
Similarly it is easy to show that
Finally, (3.30) and (3.31) imply 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)
$$ W^{\Omega,b,p}_{M,a}\subset F_{\alpha}^{-1} \bigl(W^{\Omega_{0},\frac {1}{a}}_{M_{0},\frac{1}{b}}\bigr). $$
(3.28)
$$ \phi(\sigma+ i\tau)= C_{\alpha}\int_{\mathbb{R}^{n}} e^{-\frac{i(|z|^{2} + |\sigma|^{2})\cot\alpha}{2}+i\langle\sigma, z\rangle\csc\alpha} \hat {\phi}_{\alpha}(z) \,dx. $$
$$\begin{aligned}& {\bigl\vert \phi(\sigma+ i\tau)\bigr\vert } \\& \quad \leq |C_{\alpha}|\int_{\mathbb{R}^{n}} \bigl\vert \exp \bigl[-\langle w, y\rangle\csc \alpha-\langle\tau, x\rangle\csc\alpha\bigr]\bigr\vert \bigl\vert \hat{\phi}_{\alpha }(z)\bigr\vert \,dx \\& \quad \leq C_{\alpha} C_{\delta,\rho}\int_{\mathbb{R}^{n}} \bigl\vert \exp\bigl[-\langle w, y\rangle\csc\alpha-\langle\tau, x\rangle\csc \alpha\bigr]\bigr\vert \\& \qquad {}\times\biggl\vert \exp\biggl[-M_{0}\biggl[\biggl( \frac{1}{b}-\delta\biggr)x\biggr]+\Omega_{0}\biggl[\biggl( \frac{1}{a}+\rho \biggr)y\biggr]\biggr]\biggr\vert \,dx \\& \quad \leq C_{\delta,\rho,\alpha} \exp\biggl[\Omega_{0}\biggl[\biggl( \frac{1}{a}+\rho \biggr)y\biggr]-\langle w, y\rangle\csc\alpha\biggr] \\& \qquad {}\times\int _{\mathbb{R}^{n}} \exp\biggl[-M_{0}\biggl[\biggl( \frac {1}{b}-\delta\biggr)x\biggr]+\langle\tau, x\rangle\csc\alpha\biggr] \,dx. \end{aligned}$$
$$\begin{aligned} \bigl\vert \phi(\sigma+ i\tau)\bigr\vert \leq& C'_{\delta,\rho,\alpha} \exp \biggl[-M \biggl[\frac{w\csc\alpha}{\frac{1}{a}+ \rho} \biggr] + \Omega \biggl[ \frac{\tau \csc\alpha}{(\frac{1}{b}+ 2\delta)} \biggr] \biggr] \\ \leq&C'_{\delta,\rho,\alpha} \exp \biggl[ -M \biggl[\biggl( \frac{1}{a}-\delta _{0}\biggr)w\csc\alpha \biggr] +\Omega \biggl[\biggl(\frac{1}{b}+ \rho_{0}\biggr)\tau\csc \alpha \biggr] \biggr], \end{aligned}$$
$$ F_{\alpha}^{-1}\bigl(W^{\Omega_{0},b,}_{M_{0},a}\bigr) \subset W^{\Omega,b}_{M,a}. $$
(3.29)
$$ W^{\Omega,b,p}_{M,a}\subset W^{\Omega,b}_{M,a}. $$
(3.30)
$$ W^{\Omega,b}_{M,a}\subset W^{\Omega,b,p}_{M,a}. $$
(3.31)
$$ W^{\Omega,b,p}_{M,a}=W^{\Omega,b}_{M,a}. $$
(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:
where
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.
$$\begin{aligned}& \frac{\partial u(x,t)}{\partial t} = P(i\triangle_{x})u(x, t),\quad \forall(x, t)\in \mathbb{R}^{n} \times[0, T], \end{aligned}$$
(4.1)
$$\begin{aligned}& u(x, 0)= u_{0}(x), \end{aligned}$$
(4.2)
$$\begin{aligned} \triangle_{x}^{k} =& \triangle_{x_{1}}^{k_{1}} \cdots\triangle _{x_{n}}^{k_{n}} \end{aligned}$$
(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)
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,
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 \(\triangle_{x}\).
$$\begin{aligned}& \frac{\partial}{\partial t}\phi(x, t)= \tilde{P}\bigl(i\triangle_{x}^{*} \bigr)\phi (x, t), \end{aligned}$$
(4.5)
$$\begin{aligned}& \phi(x, t_{0})= \phi_{0}(x)\in\Phi, \end{aligned}$$
(4.6)
Applying the fractional Fourier transform to (4.5) and (4.6), we get
where \(\varPsi_{\alpha}(\sigma, t)= (F_{\alpha}\phi)(x, t) \). A formal solution of (4.7) and (4.8) is given by
Let us write
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
and the arguments of [2, p.53] in (4.10) we obtain
under the assumptions \(t_{0}\leq t\leq t_{0}+T\) and \(2^{p_{0}+1}cT< (p_{0})^{-1}\theta ^{p_{0}}\).
$$\begin{aligned}& \frac{d}{ dt}\varPsi_{\alpha}(\sigma, t)= \tilde{P}(\sigma\csc\alpha )\varPsi(\sigma, t), \end{aligned}$$
(4.7)
$$\begin{aligned}& \varPsi_{\alpha}(\sigma, t_{0})= \varPsi_{\alpha,0}( \sigma), \end{aligned}$$
(4.8)
$$ \varPsi_{\alpha}(\sigma, t)= \exp\bigl[(t-t_{0})\tilde{P}( \sigma\csc\alpha )\bigr]\varPsi_{\alpha,0}(\sigma). $$
(4.9)
$$ Q(\sigma\csc\alpha,t_{0},t)= \exp\bigl[(t-t_{0}) \tilde{P}(\sigma\csc\alpha)\bigr], $$
(4.10)
$$\vert \sigma\csc\alpha \vert ^{p_{0}} \leq2^{p_{0}}\bigl( \vert w\csc\alpha \vert ^{p_{0}} + |\tau \csc\alpha|^{p_{0}} \bigr) $$
$$\bigl\vert Q(\sigma\csc\alpha,t_{0},t)\bigr\vert \leq C \exp \bigl[(p_{0})^{-1}\theta ^{p_{0}}\bigl(\vert w\csc \alpha \vert ^{p_{0}} + |\tau\csc\alpha|^{p_{0}} \bigr)\bigr] $$
If we set
then
Now, let us assume that
Then
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
and the Cauchy problem (4.5) and (4.6) has a unique solution in \(W^{\Omega_{0},\frac{1}{a-\theta}}_{M_{0},\frac{1}{b+\theta }}\).
$$M(w\csc\alpha)= {|w\csc\alpha|}^{p_{0}}/{p_{0}},\qquad \Omega(\tau \csc\alpha)= {|\tau\csc\alpha|}^{p_{0}}/{p_{0}}, $$
$$\bigl\vert Q(\sigma\csc\alpha,t_{0},t)\bigr\vert \leq C \exp \bigl[M(\theta\cdot w\csc\alpha) + \Omega(\theta\cdot\tau\csc\alpha)\bigr]. $$
$$\phi_{0}(x)\in\Phi= W^{\Omega_{0},\frac{1}{a}}_{M_{0},\frac{1}{b}}. $$
$$\psi_{\alpha,0}(\sigma)= (F_{\alpha}\phi_{0}) (x)\in W^{\Omega,b}_{M,a}. $$
$$F^{-1}_{\alpha}\bigl[W^{\Omega,b+\theta}_{M,a-\theta}\bigr]= \Phi_{1}= W^{\Omega _{0},\frac{1}{a-\theta}}_{M_{0},\frac{1}{b+\theta}}, $$
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
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