On the Multilinear Fractional Transforms

Abstract

In this paper we first introduce multilinear fractional wavelet transform on ${\mathbb{R}}^n\times {\mathbb{R}}_{+}^n$ using Schwartz functions, i.e., infinitely differentiable complex-valued functions, rapidly decreasing at infinity. We also give multilinear fractional Fourier transform and prove the Hausdorff–Young inequality and Paley-type inequality. We then study boundedness of the multilinear fractional wavelet transform on Lebesgue spaces and Lorentz spaces.

1. Introduction

Throughout this paper we will work on $\mathbb{R}$ with a Lebesgue measure $dx$. We denote $S\left(\mathbb{R}\right)$ as the Schwartz class of functions on $\mathbb{R}$. For $1\le p<\infty$, $L^p\left(\mathbb{R}\right)$ denotes the usual Lebesgue space. Let $\left(X,{∥.∥}_X\right)$ be Banach algebra, i.e., complete normed space, algebra, and ${∥x.y∥}_X\le {∥x∥}_X{∥y∥}_X$ for all $x,y\in X$. A Banach space $\left(B,{∥.∥}_B\right)$ is called Banach module over X if B is a module over X in the algebraic sense for some multiplication, $\left(u,v\right)\to u.v$, and satisfies ${∥u.v∥}_B\le {∥u∥}_X{∥v∥}_B$ for all $u\in X$, $v\in B$. The distribution function of f is defined by:

$${\lambda }_f\left(y\right)=\mu \left(\left\{x\in \mathbb{R}:\left|f\left(x\right)\right|>y\right\}\right), \left(y\ge 0\right).$$

The rearrangement function of f is defined by:

$$f^{\ast }\left(t\right)=inf\left\{y>0:{\lambda }_f\left(y\right)\le t\right\}=sup\left\{y>0:{\lambda }_f\left(y\right)>t\right\}, \left(t\ge 0\right).$$

The average function of f is also given by

$$f^{\ast \ast }\left(t\right)=\frac{1}{t}\stackrel{t}{\underset{0}{\int }}{f}^{\ast }\left(s\right)ds$$

for $t>0$. The Lorentz space $L^{p,q}\left(\mathbb{R}\right)$ is defined as the set of all (equivalence classes) measurable functions f on $\mathbb{R}$ such that ${∥f∥}_{pq}^{\ast }<\infty ,$ where:

$${∥f∥}_{pq}^{\ast }={\left(\stackrel{\infty }{\underset{0}{\int }}{t}^{\frac{q}{p}-1}{\left(f^{\ast }\left(t\right)\right)}^{q}dt\right)}^{\frac{1}{q}},\mathrm{if}0<p,q<\infty$$

and

$${∥f∥}_{pq}^{\ast }=\underset{t>0}{sup}{f}^{\ast }{\left(t\right)}^{\frac{1}{p}},\mathrm{if}0<p\le q=\infty .$$

The Lorentz space is normed vector space with the norm defined by:

$${∥f∥}_{pq}={\left(\stackrel{\infty }{\underset{0}{\int }}{t}^{\frac{q}{p}-1}{\left(f^{\ast \ast }\left(t\right)\right)}^{q}dt\right)}^{\frac{1}{q}},\mathrm{if}0<p,q<\infty$$

and

$${∥f∥}_{pq}=\underset{t>0}{sup}{f}^{\ast \ast }{\left(t\right)}^{\frac{1}{p}},\mathrm{if}0<p\le q=\infty ,$$

in [1,2]. In addition, the Lorentz space $L^{p,q}\left(\mathbb{R}\right)$ is a Banach module over $L^1\left(\mathbb{R}\right)$ [3].

Let us give the definition of the fractional Fourier transform, which is a generalization of the classical Fourier transform and is often used in the fields of quantum physics, signal, and image processing. The fractional Fourier transform with angle $\theta$ of $f\in L^2\left(\mathbb{R}\right)$ is defined by:

$${\mathcal{F}}^{\theta }f\left(\omega \right)={\widehat{f}}^{\theta }\left(\omega \right)=\underset{\mathbb{R}}{\int }f\left(t\right)K^{\theta }\left(t,\omega \right)dt,$$

where the kernel is:

$$K^{\theta }\left(t,\omega \right)=\left\{\begin{array}{c}{C}^{\theta }{e}^{i\left(t^2+{\omega }^2\right)\frac{cot\theta }{2}-it\omega csc\theta },\theta \ne n\pi \hfill \\ \frac{1}{\sqrt{2\pi }}{e}^{-it\omega },\theta =\frac{\pi }{2}\hfill \\ \delta \left(t-\omega \right),\theta =2n\pi \hfill \\ \delta \left(t+\omega \right),\theta =(2n+1)\pi ,n\in \mathbb{Z}\hfill \end{array}\right.$$

(1)

and $C^{\theta }={\left(2\pi isin\theta \right)}^{\frac{-1}{2}}{e}^{\frac{i\theta }{2}}.$ The inversion formula of the fractional Fourier transform is also given by:

$$f\left(t\right)=\underset{\mathbb{R}}{\int }{\widehat{f}}^{\theta }\left(\omega \right)\overline{K^{\theta }\left(t,\omega \right)}d\omega ,$$

where $\overline{K^{\theta }\left(t,\omega \right)}=K^{-\theta }\left(t,\omega \right)$ and $\overline{C^{\theta }}=C^{-\theta }.$ The fractional Fourier transform with angle $\theta =\frac{\pi }{2}$ corresponds to the classical Fourier transform. Indeed the inversion of fractional Fourier transform with angle $\theta$ is the fractional Fourier transform with angle $-\theta$ [4].

In wavelet theory, a is called the scaling parameter, which measures the degree of compression or scale and b is a translation parameter, which determines the time location of the wavelet function. The mother wavelet function is defined by:

$${\Psi }_{b,a}\left(t\right)=T_b{\Psi }^a\left(t\right)=a^{-\frac{1}{2}}\Psi \left(\frac{t-b}{a}\right),b\in \mathbb{R},a\in {\mathbb{R}}_{+},$$

where $T_b$ denotes the translation operator and ${\Psi }^a$ denotes the dilation of $\Psi$ with a scale [5]. The mother wavelet takes its name from two important properties of wavelet theory. The term wavelet means a small wave. This implies that it is a window function of a finite length and oscillatory. The term mother means that functions with different supports are derived from a main function. The mother wavelet is defined as a prototype for generating the window functions, that is, zero-valued outside of a chosen set centered at the orjin, symmetric functions [6]. In addition the fractional mother wavelet function with angle $\theta$ is given by:

$${\Psi }_{b,a,\theta }\left(t\right)=a^{-\frac{1}{2}}\Psi \left(\frac{t-b}{a}\right)e^{-\frac{i}{2}\left(t^2-b^2\right)cot\theta },$$

where $b\in \mathbb{R}$, $a\in {\mathbb{R}}_{+}$. The fractional mother wavelet is a generalization of the mother wavelet. Given any $0\ne \Psi \in L^2\left(\mathbb{R}\right)$ (called wavelet function), the fractional wavelet transform with angle $\theta$ of $f\in L^2\left(\mathbb{R}\right)$ with respect to $\Psi$ is defined by:

$$W_{\Psi }^{\theta }f\left(b,a\right)=\underset{\mathbb{R}}{\int }f\left(t\right)\overline{{\Psi }_{b,a,\theta }\left(t\right)}dt,$$

for all $\left(b,a\right)\in \mathbb{R}\times {\mathbb{R}}_{+}$. Since the fractional wavelet transform provides the time-frequency information of a signal in the successful way, it is one of the important time-frequency operators. The fractional wavelet transform with angle $\theta =\frac{\pi }{2}$ corresponds to the classical wavelet transform. The signal f is reconstructed from its fractional wavelet transform by:

$$f\left(t\right)=\frac{1}{2\pi sin\theta C_{\Psi ,\theta }}\underset{\mathbb{R}}{\int }\underset{{\mathbb{R}}_{+}}{\int }{W}_{\Psi }^{\theta }f\left(b,a\right){\Psi }_{b,a,\theta }\left(t\right)\frac{dbda}{a^2},$$

where $C_{\Psi ,\theta }=\underset{{\mathbb{R}}_{+}}{\int }{\left|\left({\mathcal{F}}^{\theta }\left(e^{-\frac{i}{2}{(.)}^{2}cot\theta }\Psi \right)\right)\left(\omega \right)\right|}^2\frac{1}{\left|\omega \right|}d\omega <\infty$ (called admissibility condition). The fractional convolution of f, $g\in L^2\left(\mathbb{R}\right)$ is given by:

$$\left(f{\ast }_{\theta }g\right)\left(t\right)=\underset{\mathbb{R}}{\int }f\left(x\right)g\left(t-x\right)e^{-\frac{i}{2}\left(t^2-x^2\right)cot\theta }dx.$$

For $\theta =\frac{\pi }{2},$ the fractional convolution is equal to classical convolution. The fractional wavelet transform can also be written as convolution:

$$W_{\Psi }^{\theta }f\left(b,a\right)=\left(f{\ast }_{\theta }{\Psi }^a\right)\left(b\right)=e^{-\frac{i}{2}{b}^{2}cot\theta }\left(e^{\frac{i}{2}{(.)}^{2}cot\theta }f\ast {\Psi }^a\right)\left(b\right),$$

for all $\left(b,a\right)\in \mathbb{R}\times {\mathbb{R}}_{+}$ [7].

The signal analysis capability of the wavelet transform is limited in the time-frequency plane. The wavelet transform is inefficient to process signals that are not well concentrated in the frequency domain, such as chirp-like signals ubiquitous in nature and man-made systems [8]. Therefore, a series of new signal processing tools such as fractional Fourier transform and fractional wavelet transform have been proposed to analyze such signals. Although the fractional Fourier transform overcomes this limitation and can provide signal representations in the fractional field, it fails to obtain the local structures of the signal. The fractional wavelet transform finds a solution to the limitations encountered in these transforms. It covers the advantages of the wavelet transform and the fractional Fourier transform. It has the option to provide multi-analysis and represent signals in the fractional area. This allows the fractional wavelet transform to be applied in many signal processing areas such as speech, vision, communication, and radar [9]. Thus, the fractional wavelet transform is potentially useful with signaling [10].

In recent years bilinear and multilinear time-frequency operators have been derived. Furthermore, boundedness and convergence of time-frequency operators in spaces such as Lebesgue, Lorentz, weighted variable exponent amalgam spaces, and modulation spaces have been studied [11,12,13,14]. However, the boundedness of fractional wavelet transform has also been investigated in various function spaces in the literature [15,16,17].

In wavelet theory, since the dilation operator changes the scale, the continuous wavelet transform gives us local information of a signal at any neighborhood of x time in a size. The dilation operator preserves the shape of the signal while doing this process. If the scale a is close enough to zero, this transform acts like a microscope [5,18]. For this reason, the continuous wavelet transform is a useful tool in the plane $\mathbb{R}\times {\mathbb{R}}_{+}$. Based on this, in this work we introduce multilinear fractional wavelet transform on ${\mathbb{R}}^n\times {\mathbb{R}}_{+}^n$ using Schwartz functions. We then give multilinear fractional Fourier transform and prove the Hausdorff–Young inequality and Paley-type inequality. In the third section and the fourth section, we consider the boundedness of the multilinear fractional wavelet transform on Lebesque and Lorentz spaces under some conditions.

2. Multilinear Fractional Fourier Transform and Multilinear Fractional Wavelet Transform

In this section we will introduce multilinear fractional wavelet transform and multilinear fractional Fourier transform. We then will consider some of their properties.

Let $\Psi =\left({\Psi }_1,...,{\Psi }_n\right)$ be a multi-wavelet. That means ${\Psi }_j$ is a wavelet function for $j=1,...,n$. Now, we can give definitions of the multilinear mother wavelet and the multilinear fractional mother wavelet.

Definition 1.

Let $a=\left(a_1,...,a_n\right)\in {\mathbb{R}}_{+}^n$ be a multi-scaling parameter and let $b=\left(b_1,...,b_n\right)\in {\mathbb{R}}^n$ be a multi-translation parameter, i.e., $a_j$ scaling parameter and $b_j$ translation parameter for $j=1,...,n$. The multilinear mother wavelet is defined by:

$$\begin{array}{cc}\hfill {\Psi }_{b,a}\left(t\right)=\stackrel{n}{\prod _{j=1}}{T}_{b_j}{\Psi }_j^{a_j}\left(t_j\right)& ={\left|a\right|}_n^{-\frac{1}{2}}\stackrel{n}{\prod _{j=1}}{\Psi }_j\left(\frac{t_j-b_j}{a_j}\right),\hfill \end{array}$$

for all $t=\left(t_1,...t_n\right)\in {\mathbb{R}}^n$. The multilinear mother wavelet is derived by the n-fold product of dilation and translation of a wavelet. Throughout the paper, we will take the parameters $a\in {\mathbb{R}}_{+}^n$ and $b\in {\mathbb{R}}^n$ as the multi-scaling parameter and multi-translation parameter, respectively.

The multilinear fractional mother wavelet with multi-angle $\theta =\left({\theta }_1,...,{\theta }_n\right)$ is defined as:

$${\Psi }_{b,a,\theta }\left(t\right)={\left|a\right|}_n^{-\frac{1}{2}}\stackrel{n}{\prod _{j=1}}{\Psi }_j\left(\frac{t_j-b_j}{a_j}\right)e^{-\frac{i}{2}\left(t_j^2-b_j^2\right)cot{\theta }_j},$$

(2)

where $b=\left(b_1,...,b_n\right)$, $t=\left(t_1,...t_n\right)\in {\mathbb{R}}^n$, and $a=\left(a_1,...,a_n\right)\in {\mathbb{R}}_{+}^n$. We shall also use the notation ${\left|a\right|}_n=a_1{a}_2...a_n.$

In this work f denotes both the vector $\mathbf{f}=\left(f_1,...,f_n\right)$ and the tensor product $\mathbf{f}=f_1\otimes ...\otimes f_n$ such that $\mathbf{f}\left(t\right)=f_1\left(t_1\right)...f_n\left(t_n\right)=\stackrel{n}{\underset{j=1}{\otimes }}{f}_j\left(t_j\right)$, $t=\left(t_1,...,t_n\right)\in {\mathbb{R}}^n$. We write $x.y={\displaystyle \sum _{j=1}^n}{x}_j{y}_j$ for the inner product on ${\mathbb{R}}^n$ and use the notation $x^2=\left(x_1^2,...,x_n^2\right).$ Furthermore, we say that $∥.∥\lesssim ∥.∥$ if there exists $C>0$ such that $∥.∥\le C∥.∥$. We will use similar notations throughout this work.

Definition 2.

Let $\mathbf{f}=\left(f_1,...,f_n\right)$ and $\mathbf{g}=\left(g_1,...,g_n\right)\in S{\left(\mathbb{R}\right)}^n=S\left(\mathbb{R}\right)\times ...\times S\left(\mathbb{R}\right)$. For all $t=\left(t_1,...t_n\right)\in {\mathbb{R}}^n$, the multilinear fractional convolution of $\mathbf{f}$ and $\mathbf{g}$ is defined as:

$$\begin{array}{cc}\hfill \left(\mathbf{f}{\ast }_{\theta }\mathbf{g}\right)\left(t\right)& =\underset{{\mathbb{R}}^n}{\int }\stackrel{n}{\prod _{j=1}}{f}_j\left(x_j\right)g_j\left(t_j-x_j\right)e^{-\frac{i}{2}\left(t_j^2-x_j^2\right)cot{\theta }_j}dx\hfill \\ \hfill & =\underset{\mathbb{R}}{\int }{f}_1\left(x_1\right)g_1\left(t_1-x_1\right)e^{-\frac{i}{2}\left(t_1^2-x_1^2\right)cot{\theta }_1}d{x}_1...\hfill \\ \hfill & ...\underset{\mathbb{R}}{\int }{f}_n\left(x_n\right)g_n\left(t_n-x_n\right)e^{-\frac{i}{2}\left(t_n^2-x_n^2\right)cot{\theta }_n}d{x}_n\hfill \\ \hfill & =\stackrel{n}{\prod _{j=1}}\left(f_j{\ast }_{{\theta }_j}{g}_j\right)\left(t_j\right),\hfill \end{array}$$

where ${\ast }_{{\theta }_j}$, $\left(j=1,...,n\right)$ denotes the fractional convolution.

Definition 3.

Let $\mathbf{f}=\left(f_1,...,f_n\right)$ and $\Psi =\left({\Psi }_1,...,{\Psi }_n\right)\in S{\left(\mathbb{R}\right)}^n$. Assume that ${\Psi }_{b,a,\theta }$ is a multilinear fractional mother wavelet given by (2). The multilinear fractional wavelet transform of $\mathbf{f}$ with a multi-angle θ is defined by:

$$\begin{array}{cc}\hfill W_{\Psi }^{\theta }\mathbf{f}\left(b,a\right)& =\underset{{\mathbb{R}}^n}{\int }\mathbf{f}\left(t\right)\overline{{\Psi }_{b,a,\theta }\left(t\right)}dt=⟨\mathbf{f},{\Psi }_{b,a,\theta }⟩\hfill \\ \hfill & ={\left|a\right|}_n^{-\frac{1}{2}}\underset{{\mathbb{R}}^n}{\int }\mathbf{f}\left(t\right)\stackrel{n}{\prod _{j=1}}\overline{{\Psi }_j\left(\frac{t_j-b_j}{a_j}\right)}{e}^{\frac{i}{2}\left(t_j^2-b_j^2\right)cot{\theta }_j}dt\hfill \end{array}$$

for all $\left(b,a\right)\in {\mathbb{R}}^n\times {\mathbb{R}}_{+}^n$.

Definition 4.

Let $\Psi =\left({\Psi }_1,...,{\Psi }_n\right)$ be a multi-wavelet. If there exists the condition:

$$C_{\Psi ,\theta }=\underset{{\mathbb{R}}^n}{\int }\stackrel{n}{\prod _{j=1}}\frac{{\left|{\mathcal{F}}^{{\theta }_j}\left(e^{-\frac{i}{2}{(.)}^{2}cot{\theta }_j}{\Psi }_j\right)\left({\omega }_j\right)\right|}^2}{\left|{\omega }_j\right|}d\omega <\infty ,$$

then it is said that Ψ has a multi-fractional admissiblity condition.

In this paper, we will assume that multi-wavelet $\Psi$ has multi-fractional admissiblity condition and ${\Psi }_{b,a,\theta }$ is a multilinear fractional mother wavelet given by (2).

Theorem 1.

Let $\Psi =\left({\Psi }_1,...,{\Psi }_n\right)\in S{\left(\mathbb{R}\right)}^n$. Then,

$$W_{\Psi }^{\theta }\mathbf{f}\left(b,a\right)=e^{-\frac{i}{2}{b}^2.cot\theta }\stackrel{n}{\prod _{j=1}}\left(e^{\frac{i}{2}{(.)}^{2}cot{\theta }_j}{f}_j\ast {\Psi }_j^{a_j}\right)\left(b_j\right)$$

and

$$W_{\Psi }^{\theta }\mathbf{f}\left(b,a\right)=\stackrel{n}{\prod _{j=1}}\left(f_j{\ast }_{{\theta }_j}{\Psi }_j^{a_j}\right)\left(b_j\right)=\left(\mathbf{f}{\ast }_{\theta }{\Psi }^a\right)\left(b\right)$$

holds for all $\mathbf{f}=\left(f_1,...,f_n\right)\in L^p{\left(\mathbb{R}\right)}^n=L^p\left(\mathbb{R}\right)\times ...\times L^p\left(\mathbb{R}\right).$

Proof. 

Let $\mathbf{f}=\left(f_1,...,f_n\right)\in S{\left(\mathbb{R}\right)}^n$ be given. For all $\left(b,a\right)\in {\mathbb{R}}^n\times {\mathbb{R}}_{+}^n$, we have:

$$\begin{array}{cc}& W_{\Psi }^{\theta }\mathbf{f}\left(b,a\right)={\left|a\right|}_n^{-\frac{1}{2}}\underset{{\mathbb{R}}^n}{\int }\mathbf{f}\left(t\right)\stackrel{n}{\prod _{j=1}}\overline{{\Psi }_j\left(\frac{t_j-b_j}{a_j}\right)}{e}^{\frac{i}{2}\left(t_j^2-b_j^2\right)cot{\theta }_j}dt\\ \hfill & ={\left|a\right|}_n^{-\frac{1}{2}}\underset{\mathbb{R}}{\int }...\underset{\mathbb{R}}{\int }{f}_1\left(t_1\right)...f_n\left(t_n\right)\overline{{\Psi }_1\left(\frac{t_1-b_1}{a_1}\right)}{e}^{\frac{i}{2}\left(t_1^2-b_1^2\right)cot{\theta }_1}...\hfill \\ \hfill & ...\overline{{\Psi }_n\left(\frac{t_1-b_1}{a_1}\right)}{e}^{\frac{i}{2}\left(t_n^2-b_n^2\right)cot{\theta }_n}d{t}_1...d{t}_n\hfill \\ \hfill & =a_1^{-\frac{1}{2}}\underset{\mathbb{R}}{\int }{f}_1\left(t_1\right)\overline{{\Psi }_1\left(\frac{t_1-b_1}{a_1}\right)}{e}^{\frac{i}{2}\left(t_1^2-b_1^2\right)cot{\theta }_1}d{t}_1...\hfill \\ \hfill & ...a_n^{-\frac{1}{2}}\underset{\mathbb{R}}{\int }\overline{{\Psi }_n\left(\frac{t_1-b_1}{a_1}\right)}{e}^{\frac{i}{2}\left(t_n^2-b_n^2\right)cot{\theta }_n}d{t}_n\hfill \\ \hfill & =W_{{\Psi }_1}^{{\theta }_1}{f}_1\left(b_1,a_1\right)...W_{{\Psi }_n}^{{\theta }_n}{f}_n\left(b_n,a_n\right)=\stackrel{n}{\prod _{j=1}}{W}_{{\Psi }_j}^{{\theta }_j}{f}_j\left(b_j,a_j\right).\hfill \end{array}$$

(3)

In adddition, by [7], it is known that:

$$\begin{array}{cc}\hfill W_{{\Psi }_j}^{{\theta }_j}{f}_j\left(b_j,a_j\right)& =e^{-\frac{i}{2}{b}_j^2.cot{\theta }_j}\left(e^{\frac{i}{2}{(.)}^{2}cot{\theta }_j}{f}_j\ast {\Psi }_j^{a_j}\right)\left(b_j\right),\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill W_{{\Psi }_j}^{{\theta }_j}{f}_j\left(b_j,a_j\right)& =\left(f_j{\ast }_{{\theta }_j}{\Psi }_j^{a_j}\right)\left(b_j\right)\hfill \end{array}$$

(5)

for all $\left(b_j,a_j\right)\in \mathbb{R}\times {\mathbb{R}}_{+}$, $j=1,2,...,n$. Combining (3) and (4), we have:

$$\begin{array}{cc}\hfill & W_{\Psi }^{\theta }\mathbf{f}\left(b,a\right)=\stackrel{n}{\prod _{j=1}}{W}_{{\Psi }_j}^{{\theta }_j}{f}_j\left(b_j,a_j\right)=e^{-\frac{i}{2}{b}_1^2.cot{\theta }_1}\left(e^{\frac{i}{2}{(.)}^{2}cot{\theta }_1}{f}_1\ast {\Psi }_1^{a_1}\right)\left(b_1\right)...\hfill \\ \hfill & ...e^{-\frac{i}{2}{b}_n^2.cot{\theta }_n}\left(e^{\frac{i}{2}{(.)}^{2}cot{\theta }_n}{f}_n\ast {\Psi }_n^{a_n}\right)\left(b_n\right)\hfill \\ \hfill & =e^{-\frac{i}{2}{b}_1^2.cot{\theta }_1}...e^{-\frac{i}{2}{b}_n^2.cot{\theta }_n}\left(e^{\frac{i}{2}{(.)}^{2}cot{\theta }_1}{f}_1\ast {\Psi }_1^{a_1}\right)\left(b_1\right)...\hfill \\ \hfill & ...\left(e^{\frac{i}{2}{(.)}^{2}cot{\theta }_n}{f}_n\ast {\Psi }_n^{a_n}\right)\left(b_n\right)\hfill \\ \hfill & =e^{-\frac{i}{2}{\displaystyle \sum _{j=1}^n}{b}_j^2.cot{\theta }_j}\stackrel{n}{\prod _{j=1}}\left(e^{\frac{i}{2}{(.)}^{2}cot{\theta }_j}{f}_j\ast {\Psi }_j^{a_j}\right)\left(b_j\right)\hfill \\ \hfill & =e^{-\frac{i}{2}{b}^2.cot\theta }\stackrel{n}{\prod _{j=1}}\left(e^{\frac{i}{2}{(.)}^{2}cot{\theta }_j}{f}_j\ast {\Psi }_j^{a_j}\right)\left(b_j\right).\hfill \end{array}$$

(6)

In addition, by (5), we obtain:

$$\begin{array}{cc}\hfill W_{\Psi }^{\theta }\mathbf{f}\left(b,a\right)& =\left(f_1{\ast }_{{\theta }_1}{\Psi }_1^{a_1}\right)\left(b_1\right)...\left(f_n{\ast }_{{\theta }_n}{\Psi }_n^{a_n}\right)\left(b_n\right)\hfill \\ \hfill & =\stackrel{n}{\prod _{j=1}}\left(f_j{\ast }_{{\theta }_j}{\Psi }_j^{a_j}\right)\left(b_j\right)=\left(\mathbf{f}{\ast }_{\theta }{\Psi }^a\right)\left(b\right).\hfill \end{array}$$

(7)

Since $S{\left(\mathbb{R}\right)}^n$ is dense in $L^p{\left(\mathbb{R}\right)}^n$, which means closure of $S{\left(\mathbb{R}\right)}^n$ equals to $L^p{\left(\mathbb{R}\right)}^n$ [19], we achieve the equalities (6) and (7) for all $\mathbf{f}=\left(f_1,...,f_n\right)\in L^p{\left(\mathbb{R}\right)}^n.$ □

Definition 5.

Let $\mathbf{f}=\left(f_1,...,f_n\right)\in S{\left(\mathbb{R}\right)}^n$. The multilinear fractional Fourier transform of $\mathbf{f}$ with a multi-angle θ is defined as:

$$\begin{array}{cc}\hfill {\mathcal{F}}^{\theta }\mathbf{f}\left(\omega \right)& =\underset{{\mathbb{R}}^n}{\int }\mathbf{f}\left(t\right){\mathbf{K}}^{\mathit{\theta }}\left(t,\omega \right)dt\hfill \\ \hfill & =\underset{{\mathbb{R}}^n}{\int }\mathbf{f}\left(t\right)\stackrel{n}{\prod _{j=1}}{K}_j^{{\theta }_j}\left(t_j,{\omega }_j\right)dt,\hfill \end{array}$$

where the $K_j^{{\theta }_j}\left(t_j,{\omega }_j\right)$ is kernel given by (1). Sometimes we will denote the multilinear fractional Fourier transform of $\mathbf{f}$ with a multi-angle θ with the symbol ${\widehat{\mathbf{f}}}^{\theta }$. Since the classical fractional Fourier transform is injective and surjective from $S\left(\mathbb{R}\right)$ to $S\left(\mathbb{R}\right)$ (see in [20]), the multilinear fractional Fourier transform is injective and surjective from $S{\left(\mathbb{R}\right)}^n$ to $S\left({\mathbb{R}}^n\right)$.

Now we can give definition of the multilinear inverse fractional Fourier transform of $\mathbf{f}$ with a multi-angle θ:

$$\begin{array}{cc}\hfill {\mathcal{F}}^{{}^{-1}\theta }\mathbf{f}\left(t\right)& =\underset{{\mathbb{R}}^n}{\int }{\mathcal{F}}^{\theta }\mathbf{f}\left(u\right)\overline{{\mathbf{K}}^{\mathit{\theta }}\left(t,u\right)}du\hfill \\ \hfill & =\underset{{\mathbb{R}}^n}{\int }{\mathcal{F}}^{\theta }\mathbf{f}\left(u\right)\stackrel{n}{\prod _{j=1}}\overline{K_j^{{\theta }_j}\left(t_j,u_j\right)}du\hfill \\ \hfill & =\underset{{\mathbb{R}}^n}{\int }{\mathcal{F}}^{\theta }\mathbf{f}\left(u\right)\stackrel{n}{\prod _{j=1}}{K}_j^{-{\theta }_j}\left(t_j,u_j\right)du.\hfill \end{array}$$

In addition, the multilinear inverse fractional Fourier transform of $\mathbf{f}$ with a multi-angle θ is shown with the symbol ${\check{\mathbf{f}}}^{\theta }$. Furthermore, since ${\mathcal{F}}^{\theta }\mathbf{f}\left(u\right)\in L^1\left({\mathbb{R}}^n\right)$ and by Theorem 3.1 in [20], it is easily shown that ${\mathcal{F}}^{{}^{-1}\theta }\left({\mathcal{F}}^{\theta }\mathbf{f}\right)=\mathbf{f}$ and ${\mathcal{F}}^{\theta }\left({\mathcal{F}}^{{}^{-1}\theta }\mathbf{f}\right)=\mathbf{f}$.

Theorem 2.

(Hausdorff-Young inequality for multilinear fractional Fourier transform)

Assume that $1\le p\le 2$ and $\frac{1}{p}+\frac{1}{p^{\prime }}=1$. If $\mathbf{f}=\left(f_1,...,f_n\right)\in L^p{\left(\mathbb{R}\right)}^n$, then ${\mathcal{F}}^{\theta }\mathbf{f}={\widehat{\mathbf{f}}}^{\theta }\in L^{p^{\prime }}\left({\mathbb{R}}^n\right)$ and:

$${∥{\mathcal{F}}^{\theta }\mathbf{f}∥}_{L^{p^{\prime }}\left({\mathbb{R}}^n\right)}\le \stackrel{n}{\prod _{j=1}}\frac{1}{\sqrt{\left|sin{\theta }_j\right|}}\stackrel{n}{\prod _{j=1}}{∥f_j∥}_{L^p\left(\mathbb{R}\right)}.$$

Proof. 

Take any $\mathbf{f}=\left(f_1,...,f_n\right)\in S{\left(\mathbb{R}\right)}^n$. It is clear that:

$$\underset{\mathbb{R}}{\int }\left|f_j\left(t_j\right)\right|\left|K_j^{{\theta }_j}\left(t_j,{\omega }_j\right)\right|\le \left|C^{{\theta }_j}\right|{∥f_j∥}_{L^1\left(\mathbb{R}\right)}$$

(8)

for $j=1,...,n$. By Parseval’s relation in [7], we also know:

$${∥{\mathcal{F}}_j^{{\theta }_j}{f}_j∥}_{L^2\left(\mathbb{R}\right)}={∥f_j∥}_{L^2\left(\mathbb{R}\right)}$$

(9)

for $j=1,...,n$. Then, using (8), we have:

$$\begin{array}{cc}\hfill & {∥{\mathcal{F}}^{\theta }\mathbf{f}∥}_{L^{\infty }\left({\mathbb{R}}^n\right)}=\underset{\omega \in {\mathbb{R}}^n}{sup}\left|\underset{{\mathbb{R}}^n}{\int }\mathbf{f}\left(t\right)\stackrel{n}{\prod _{j=1}}{K}_j^{{\theta }_j}\left(t_j,{\omega }_j\right)dt\right|\hfill \\ \hfill & \le \underset{\omega \in {\mathbb{R}}^n}{sup}\underset{{\mathbb{R}}^n}{\int }\left|\mathbf{f}\left(t\right)\right|\stackrel{n}{\prod _{j=1}}\left|K_j^{{\theta }_j}\left(t_j,{\omega }_j\right)\right|dt\hfill \\ \hfill & =\underset{\omega \in {\mathbb{R}}^n}{sup}\left\{\underset{\mathbb{R}}{\int }...\underset{\mathbb{R}}{\int }\left|f_1\left(t_1\right)\right|...\left|f_n\left(t_n\right)\right|\stackrel{n}{\prod _{j=1}}\left|K_j^{{\theta }_j}\left(t_j,{\omega }_j\right)\right|d{t}_1...d{t}_n\right\}\hfill \\ \hfill & =\underset{\omega \in {\mathbb{R}}^n}{sup}\left\{\underset{\mathbb{R}}{\int }\left|f_1\left(t_1\right)\right|\left|K_1^{{\theta }_1}\left(t_1,{\omega }_1\right)\right|d{t}_1...\underset{\mathbb{R}}{\int }\left|f_n\left(t_n\right)\right|\left|K_n^{{\theta }_n}\left(t_n,{\omega }_n\right)\right|d{t}_n\right\}\hfill \\ \hfill & =\underset{\omega \in {\mathbb{R}}^n}{sup}\stackrel{n}{\prod _{j=1}}\underset{\mathbb{R}}{\int }\left|f_j\left(t_j\right)\right|\left|K_j^{{\theta }_j}\left(t_j,{\omega }_j\right)\right|d{t}_j\hfill \\ \hfill & \le \underset{\omega \in {\mathbb{R}}^n}{sup}\stackrel{n}{\prod _{j=1}}\underset{\mathbb{R}}{\int }\left|f_j\left(t_j\right)\right|\left|C^{{\theta }_j}\right|d{t}_j\hfill \\ \hfill & =\stackrel{n}{\prod _{j=1}}\left|C^{{\theta }_j}\right|{∥f_j∥}_{L^1\left(\mathbb{R}\right)}=C_{\theta }\stackrel{n}{\prod _{j=1}}{∥f_j∥}_{L^1\left(\mathbb{R}\right)},\hfill \end{array}$$

(10)

where $C_{\theta }={\displaystyle \prod _{j=1}^n}\left|C^{{\theta }_j}\right|$. Now, we will use (9), then:

$$\begin{array}{cc}\hfill & {∥{\mathcal{F}}^{\theta }\mathbf{f}∥}_{L^2\left({\mathbb{R}}^n\right)}={∥\underset{{\mathbb{R}}^n}{\int }\mathbf{f}\left(t\right)\stackrel{n}{\prod _{j=1}}{K}_j^{{\theta }_j}\left(t_j,{\omega }_j\right)dt∥}_{L^2\left({\mathbb{R}}^n\right)}\hfill \\ \hfill & ={∥\left\{\underset{\mathbb{R}}{\int }{f}_1\left(t_1\right)K_1^{{\theta }_1}\left(t_1,{\omega }_1\right)d{t}_1...\underset{\mathbb{R}}{\int }{f}_n\left(t_n\right)K_n^{{\theta }_n}\left(t_n,{\omega }_n\right)d{t}_n\right\}∥}_{L^2\left({\mathbb{R}}^n\right)}\hfill \\ \hfill & ={∥\stackrel{n}{\prod _{j=1}}\underset{\mathbb{R}}{\int }{f}_j\left(t_j\right)K_j^{{\theta }_j}\left(t_j,{\omega }_j\right)d{t}_j∥}_{L^2\left({\mathbb{R}}^n\right)}\hfill \\ \hfill & ={∥\stackrel{n}{\prod _{j=1}}{\mathcal{F}}_j^{{\theta }_j}{f}_j∥}_{L^2\left({\mathbb{R}}^n\right)}={\left\{\underset{{\mathbb{R}}^n}{\int }{\left|\stackrel{n}{\prod _{j=1}}{\mathcal{F}}_j^{{\theta }_j}{f}_j\left({\omega }_j\right)\right|}^{2}d\omega \right\}}^{\frac{1}{2}}\hfill \\ \hfill & ={\left\{\underset{\mathbb{R}}{\int }...\underset{\mathbb{R}}{\int }{\left|{\mathcal{F}}_1^{{\theta }_1}{f}_1\left({\omega }_1\right)\right|}^2...{\left|{\mathcal{F}}_n^{{\theta }_n}{f}_n\left({\omega }_n\right)\right|}^{2}d{\omega }_1...d{\omega }_n\right\}}^{\frac{1}{2}}\hfill \\ \hfill & ={\left\{\underset{\mathbb{R}}{\int }{\left|{\mathcal{F}}_1^{{\theta }_1}{f}_1\left({\omega }_1\right)\right|}^{2}d{\omega }_1...\underset{\mathbb{R}}{\int }{\left|{\mathcal{F}}_n^{{\theta }_n}{f}_n\left({\omega }_n\right)\right|}^{2}d{\omega }_n\right\}}^{\frac{1}{2}}\hfill \\ \hfill & =\stackrel{n}{\prod _{j=1}}{∥{\mathcal{F}}_j^{{\theta }_j}{f}_j∥}_{L^2\left(\mathbb{R}\right)}=\stackrel{n}{\prod _{j=1}}{∥f_j∥}_{L^2\left(\mathbb{R}\right)}.\hfill \end{array}$$

(11)

So the estimates (10) and (11) say that the fractional Fourier transform $\mathbf{f}\to {\mathcal{F}}^{\theta }\mathbf{f}$ is bounded from $L^1{\left(\mathbb{R}\right)}^n$ into $L^{\infty }\left({\mathbb{R}}^n\right)$ and from $L^2{\left(\mathbb{R}\right)}^n$ into $L^2\left({\mathbb{R}}^n\right)$ with operator norms $M_1\le C_{\theta }$, $M_2=1$ respectively. If we take $\frac{2}{p^{\prime }}=\alpha$, then we have $0\le \alpha \le 1$, $\frac{\alpha }{2}+\frac{1-\alpha }{1}=\frac{1}{p}$, $\frac{\alpha }{2}+\frac{1-\alpha }{\infty }=\frac{1}{p^{\prime }}$. So we find $1\le p\le 2$. Now if we use the interpolation theorem for multilinear operators in [21], we obtain that the fractional Fourier transform $\mathbf{f}\to {\mathcal{F}}^{\theta }\mathbf{f}$ is bounded from $L^p{\left(\mathbb{R}\right)}^n$ into $L^{p^{\prime }}\left({\mathbb{R}}^n\right)$ with the norm:

$$M_{\alpha }\le M_1^{\alpha }{M}_2^{1-\alpha }=C_{\theta }^{\alpha }.$$

(12)

On the other hand, we have:

$$\begin{array}{cc}\hfill & C_{\theta }^{\alpha }=\stackrel{n}{\prod _{j=1}}{\left|C^{{\theta }_j}\right|}^{\alpha }=\stackrel{n}{\prod _{j=1}}{\left|\frac{e^{\frac{i{\theta }_j}{2}}}{\sqrt{2\pi isin{\theta }_j}}\right|}^{\alpha }\hfill \\ \hfill & ={\left(\frac{1}{\sqrt{2\pi }}\right)}^{n\alpha }{\left(\frac{1}{\left|\sqrt{i}\right|}\right)}^{n\alpha }\stackrel{n}{\prod _{j=1}}{\left(\frac{1}{\sqrt{\left|sin{\theta }_j\right|}}\right)}^{\alpha }\hfill \\ \hfill & ={\left(\frac{1}{\sqrt{2\pi }}\right)}^{n\alpha }{\left(\frac{1}{\left|e^{i\left(\frac{\pi }{4}+k\pi \right)}\right|}\right)}^{n\alpha }\stackrel{n}{\prod _{j=1}}{\left(\frac{1}{\sqrt{\left|sin{\theta }_j\right|}}\right)}^{\alpha }\hfill \\ \hfill & \le \stackrel{n}{\prod _{j=1}}\frac{1}{\sqrt{\left|sin{\theta }_j\right|}}.\hfill \end{array}$$

(13)

Therefore by (12) and (13), we achieve:

$${∥{\mathcal{F}}^{\theta }\mathbf{f}∥}_{L^{p^{\prime }}\left({\mathbb{R}}^n\right)}\le \stackrel{n}{\prod _{j=1}}\frac{1}{\sqrt{\left|sin{\theta }_j\right|}}{∥f_j∥}_{L^p\left(\mathbb{R}\right)}.$$

□

Theorem 3.

(Paley-type inequality for linear fractional Fourier transform)

Assume that $1\le p\le 2$, $0<q\le \infty$ and $\frac{1}{p}+\frac{1}{p^{\prime }}=1$. If $f\in L^{p,q}\left(\mathbb{R}\right)$, then ${\mathcal{F}}^{\theta }f\in L^{p^{\prime },q}\left(\mathbb{R}\right)$ and:

$${∥{\mathcal{F}}^{\theta }f∥}_{L^{p^{\prime },q}}\lesssim \frac{1}{\sqrt{\left|sin\theta \right|}}{∥f∥}_{L^{p,q}}.$$

Proof. 

It is known that ${\mathcal{F}}^{\theta }:L^1\left(\mathbb{R}\right)\to L^{\infty }\left(\mathbb{R}\right)$ and ${\mathcal{F}}^{\theta }:L^2\left(\mathbb{R}\right)$ into $L^2\left(\mathbb{R}\right)$ are bounded with the operator norms $M_1\le$$C_{\theta }$ and $M_2=$ 1, respectively. Then, by real interpolation [21], we achieve:

$${\mathcal{F}}^{\theta }:L^{p,q}={\left(L^1,L^2\right)}_{\alpha ,q}\to L^{p^{\prime },q}={\left(L^{\infty },L^2\right)}_{\alpha ,q},$$

where $0\le \alpha \le 1$, $\frac{\alpha }{2}+\frac{1-\alpha }{1}=\frac{1}{p}$, $\frac{\alpha }{2}+\frac{1-\alpha }{\infty }=\frac{1}{p^{\prime }}$ with the norm:

$$M_{\alpha }\le C{M}_1^{\alpha }{M}_2^{1-\alpha }=C{C}_{\theta }^{\alpha }.$$

(14)

Hence, we obtain that ${\mathcal{F}}^{\theta }:L^{p,q}\to L^{p^{\prime },q}$ is bounded, where $1\le p\le 2$, $0<q\le \infty$. Moreover combining (13) and (14), we conclude:

$${∥{\mathcal{F}}^{\theta }f∥}_{L^{p^{\prime },q}}\lesssim \frac{1}{\sqrt{\left|sin\theta \right|}}{∥f∥}_{L^{p,q}}.$$

Therefore, the proof is completed. □

Theorem 4.

(Paley-type inequality for multilinear fractional Fourier transform)

Assume that $\frac{1}{q}+(n-1)=\frac{1}{q_1}+...+\frac{1}{q_n}$, $1<p\le 2$, $1\le q,q_j\le \infty$, $\left(j=1,...,n\right)$ and $\frac{1}{p}+\frac{1}{p^{\prime }}=1$. If $\mathbf{f}\in L^{p,q}{\left(\mathbb{R}\right)}^n$, then ${\mathcal{F}}^{\theta }\mathbf{f}\in L^{p^{\prime },q}\left({\mathbb{R}}^n\right)$ and

$${∥{\mathcal{F}}^{\theta }\mathbf{f}∥}_{L^{p^{\prime },q}\left({\mathbb{R}}^n\right)}\lesssim \stackrel{n}{\prod _{j=1}}\frac{1}{\sqrt{\left|sin{\theta }_j\right|}}{∥f_j∥}_{L^{p,q_j}\left(\mathbb{R}\right)}.$$

Proof. 

Take any $\mathbf{f}=\left(f_1,...,f_n\right)\in$$S{\left(\mathbb{R}\right)}^n$. Then, we have:

$$\begin{array}{cc}\hfill & \left({\mathcal{F}}^{{\theta }_1}{f}_1,...,{\mathcal{F}}^{{\theta }_n}{f}_n\right)\to \left(\underset{j=1}{\stackrel{n}{\otimes }}{\mathcal{F}}^{{\theta }_j}{f}_j\right)\left(\omega \right)={\mathcal{F}}^{{\theta }_1}{f}_1\left({\omega }_1\right)...{\mathcal{F}}^{{\theta }_n}{f}_n\left({\omega }_n\right)\hfill \\ \hfill & =\underset{\mathbb{R}}{\int }{f}_1\left(u_1\right)K_1^{{\theta }_1}\left(u_1,{\omega }_1\right)d{u}_1...\underset{\mathbb{R}}{\int }{f}_n\left(u_n\right)K_n^{{\theta }_n}\left(u_n,{\omega }_n\right)d{u}_n\hfill \\ \hfill & =\underset{\mathbb{R}}{\int }....\underset{\mathbb{R}}{\int }{f}_1\left(u_1\right)...f_n\left(u_n\right)\stackrel{n}{\prod _{j=1}}{K}_j^{{\theta }_j}\left(u_j,{\omega }_j\right)d{u}_1...d{u}_n\hfill \\ \hfill & =\underset{{\mathbb{R}}^n}{\int }\mathbf{f}\left(u\right)\stackrel{n}{\prod _{j=1}}{K}_j^{{\theta }_j}\left(u_j,{\omega }_j\right)du={\mathcal{F}}^{\theta }\mathbf{f}\left(\omega \right).\hfill \end{array}$$

So we can say that the multilinear fractional Fourier transform is a tensor product operator. Furthermore, we can write:

$${∥{\mathcal{F}}^{\theta }\mathbf{f}∥}_{L^1\left({\mathbb{R}}^n\right)}=\stackrel{n}{\prod _{j=1}}{∥{\mathcal{F}}^{{\theta }_j}{f}_j∥}_{L^1\left(\mathbb{R}\right)}$$

and

$${∥{\mathcal{F}}^{\theta }\mathbf{f}∥}_{L^{\infty }\left({\mathbb{R}}^n\right)}\le \stackrel{n}{\prod _{j=1}}{∥{\mathcal{F}}^{{\theta }_j}{f}_j∥}_{L^{\infty }\left(\mathbb{R}\right)}$$

for all $\mathbf{f}=\left(f_1,...,f_n\right)\in$$S{\left(\mathbb{R}\right)}^n$. Hence, we find that the multilinear fractional Fourier transform is of restricted weak types (1,...,1;1) and (∞,...∞;∞). If we can generalize Theorem 7.7 in [21] for multilinear fractional Fourier transform, then there exists $C>0$ such that:

$${∥{\mathcal{F}}^{\theta }\mathbf{f}∥}_{L^{p^{\prime },q}\left({\mathbb{R}}^n\right)}\le C\stackrel{n}{\prod _{j=1}}{∥{\mathcal{F}}^{{\theta }_j}{f}_j∥}_{L^{p^{\prime },q_j}\left(\mathbb{R}\right)},$$

(15)

where $\frac{1}{q}+(n-1)=\frac{1}{q_1}+...+\frac{1}{q_n}$, $1<p\le 2,\frac{1}{p}+\frac{1}{p^{\prime }}=1$ and $1\le q,q_j\le \infty$, $\left(j=1,...,n\right)$. Using Theorem 3 and by (15), we also get:

$${∥{\mathcal{F}}^{\theta }\mathbf{f}∥}_{L^{p^{\prime },q}\left({\mathbb{R}}^n\right)}\lesssim \stackrel{n}{\prod _{j=1}}\frac{1}{\sqrt{\left|sin{\theta }_j\right|}}{∥f_j∥}_{L^{p,q_j}\left(\mathbb{R}\right)}.$$

□

Theorem 5.

Let $\mathbf{f}=\left(f_1,...,f_n\right)$, $\mathbf{g}=\left(g_1,...,g_n\right)\in$$L^2{\left(\mathbb{R}\right)}^n$. If $\Psi =\left({\Psi }_1,...,{\Psi }_n\right)$, ${\rm Y}=\left({\phi }_1,...,{\phi }_n\right)$ are two multi-wavelets in $S{\left(\mathbb{R}\right)}^n$, then:

$$\underset{{\mathbb{R}}^n}{\int }\underset{{\mathbb{R}}_{+}^n}{\int }{W}_{\Psi }^{\theta }\mathbf{f}\left(b,a\right)\overline{W_{{\rm Y}}^{\theta }\mathbf{g}\left(b,a\right)}\frac{dbda}{{\left|a\right|}_n^2}={\left(2\pi \right)}^n\stackrel{n}{\prod _{j=1}}sin{\theta }_j{C}_{{\Psi }_j,{\phi }_j}⟨f_j,g_j⟩,$$

where

$$C_{{\Psi }_j,{\phi }_j}=\underset{{\mathbb{R}}_{+}}{\int }\overline{{\mathcal{F}}^{{\theta }_j}\left(e^{\frac{-i}{2}{\left(.\right)}^{2}cot{\theta }_j}{\Psi }_j\right)}\left(a_j\right){\mathcal{F}}^{{\theta }_j}\left(e^{\frac{-i}{2}{\left(.\right)}^{2}cot{\theta }_j}{\phi }_j\right)\left(a_j\right)a_j^{-1}d{a}_j<\infty$$

for $j=1,...,n.$

Proof. 

Take any $\mathbf{f}=\left(f_1,...,f_n\right)$, $\mathbf{g}=\left(g_1,...,g_n\right)\in$$S{\left(\mathbb{R}\right)}^n$. Assume that $\Psi =\left({\Psi }_1,...,{\Psi }_n\right)$, ${\rm Y}=\left({\phi }_1,...,{\phi }_n\right)$ are two multi-wavelets in $S{\left(\mathbb{R}\right)}^n$. It is known that

$$\underset{\mathbb{R}}{\int }\underset{{\mathbb{R}}_{+}}{\int }{W}_{{\Psi }_j}^{{\theta }_j}{f}_j\left(b_j,a_j\right)\overline{W_{{\phi }_j}^{{\theta }_j}{g}_j\left(b_j,a_j\right)}\frac{d{b}_{j}d{a}_j}{a_j^2}=2\pi sin{\theta }_j{C}_{{\Psi }_j,{\phi }_j}⟨f_j,g_j⟩,$$

(16)

where

$$C_{{\Psi }_j,{\phi }_j}=\underset{{\mathbb{R}}_{+}}{\int }\overline{{\mathcal{F}}^{{\theta }_j}\left(e^{\frac{-i}{2}{\left(.\right)}^{2}cot{\theta }_j}{\Psi }_j\right)}\left(a_j\right){\mathcal{F}}^{{\theta }_j}\left(e^{\frac{-i}{2}{\left(.\right)}^{2}cot{\theta }_j}{\phi }_j\right)\left(a_j\right)a_j^{-1}d{a}_j<\infty$$

for $j=1,...,n$ [7]. Then by (16), we get:

$$\begin{array}{cc}\hfill & \underset{{\mathbb{R}}^n}{\int }\underset{{\mathbb{R}}_{+}^n}{\int }\left(\stackrel{n}{\prod _{j=1}}{W}_{{\Psi }_j}^{{\theta }_j}{f}_j\left(b_j,a_j\right)\right)\overline{\left(\stackrel{n}{\prod _{j=1}}{W}_{{\phi }_j}^{{\theta }_j}{g}_j\left(b_j,a_j\right)\right)}\frac{d{b}_{j}d{a}_j}{{\left|a\right|}_n^2}\hfill \\ \hfill & =\underset{\mathbb{R}}{\int }...\underset{\mathbb{R}}{\int }\underset{{\mathbb{R}}_{+}}{\int }...\underset{{\mathbb{R}}_{+}}{\int }{W}_{{\Psi }_1}^{{\theta }_1}{f}_1\left(b_1,a_1\right)\overline{W_{{\phi }_1}^{{\theta }_1}{g}_1\left(b_1,a_1\right)}...\hfill \\ \hfill & ...W_{{\Psi }_n}^{{\theta }_n}{f}_n\left(b_n,a_n\right)\overline{W_{{\phi }_n}^{{\theta }_n}{g}_n\left(b_n,a_n\right)}\frac{d{b}_{1}d{a}_1}{a_1^2}...\frac{d{b}_{n}d{a}_n}{a_n^2}\hfill \\ \hfill & =\underset{\mathbb{R}}{\int }\underset{{\mathbb{R}}_{+}}{\int }{W}_{{\Psi }_1}^{{\theta }_1}{f}_1\left(b_1,a_1\right)\overline{W_{{\phi }_1}^{{\theta }_1}{g}_1\left(b_1,a_1\right)}\frac{d{b}_{1}d{a}_1}{a_1^2}...\hfill \\ \hfill & ...\underset{\mathbb{R}}{\int }\underset{{\mathbb{R}}_{+}}{\int }{W}_{{\Psi }_n}^{{\theta }_n}{f}_n\left(b_n,a_n\right)\overline{W_{{\phi }_n}^{{\theta }_n}{g}_n\left(b_n,a_n\right)}.\frac{d{b}_{n}d{a}_n}{a_n^2}\hfill \\ \hfill & =\stackrel{n}{\prod _{j=1}}\underset{{\mathbb{R}}^n}{\int }\underset{{\mathbb{R}}_{+}^n}{\int }{W}_{{\Psi }_j}^{{\theta }_j}{f}_j\left(b_j,a_j\right)\overline{W_{{\phi }_j}^{{\theta }_j}{g}_j\left(b_j,a_j\right)}\frac{d{b}_{j}d{a}_j}{a_j^2}\hfill \\ \hfill & ={\left(2\pi \right)}^n\stackrel{n}{\prod _{j=1}}sin{\theta }_j{C}_{{\Psi }_j,{\phi }_j}⟨f_j,g_j⟩.\hfill \end{array}$$

□

Theorem 6.

Let $\mathbf{f}=\left(f_1,...,f_n\right)$, $\mathbf{g}=\left(g_1,...,g_n\right)\in$$L^2{\left(\mathbb{R}\right)}^n$. If $\Psi =\left({\Psi }_1,...,{\Psi }_n\right)$ is a multi-wavelet in $S{\left(\mathbb{R}\right)}^n$, then:

$$\underset{{\mathbb{R}}^n}{\int }\underset{{\mathbb{R}}_{+}^n}{\int }{W}_{\Psi }^{\theta }\mathbf{f}\left(b,a\right)\overline{W_{\Psi }^{\theta }\mathbf{g}\left(b,a\right)}\frac{dbda}{{\left|a\right|}_n^2}={\left(2\pi \right)}^n\stackrel{n}{\prod _{j=1}}sin{\theta }_j{C}_{{\Psi }_j,{\theta }_j}⟨f_j,g_j⟩.$$

Proof. 

If we take $\Psi ={\rm Y}$ in Theorem 5, we get the desired result. □

By setting $\mathbf{f}=\mathbf{g}$ in Theorem 6, the following theorem is obtained.

Theorem 7.

Let $\mathbf{f}=\left(f_1,...,f_n\right)\in$$L^2{\left(\mathbb{R}\right)}^n$. If $\Psi =\left({\Psi }_1,...,{\Psi }_n\right)$ is a multi-wavelet in $S{\left(\mathbb{R}\right)}^n$, then:

$$\underset{{\mathbb{R}}^n}{\int }\underset{{\mathbb{R}}_{+}^n}{\int }{\left|W_{\Psi }^{\theta }\mathbf{f}\left(b,a\right)\right|}^2\frac{dbda}{{\left|a\right|}_n^2}={\left(2\pi \right)}^n\stackrel{n}{\prod _{j=1}}sin{\theta }_j{C}_{{\Psi }_j,{\theta }_j}{∥f_j∥}_2^2.$$

Let us derive the inversion formula for the multilinear fractional wavelet transform.

Theorem 8.

Let $\mathbf{f}=\left(f_1,...,f_n\right)\in$$L^2{\left(\mathbb{R}\right)}^n$. If $\Psi =\left({\Psi }_1,...,{\Psi }_n\right)$ is a multi-wavelet in $S{\left(\mathbb{R}\right)}^n$, then $\mathbf{f}$ can be reconstructed by the following equation:

$$\mathbf{f}\left(t\right)={\left(2\pi \right)}^{-n}\stackrel{n}{\prod _{j=1}}\frac{1}{sin{\theta }_j{C}_{{\Psi }_j,{\theta }_j}}\underset{{\mathbb{R}}^n}{\int }\underset{{\mathbb{R}}_{+}^n}{\int }{W}_{\Psi }^{\theta }\mathbf{f}\left(b,a\right){\Psi }_{b,a,\theta }\left(t\right)\frac{dbda}{{\left|a\right|}_n^2}.$$

Proof. 

Take any $\mathbf{f}=\left(f_1,...,f_n\right)\in$$S{\left(\mathbb{R}\right)}^n$. We can write:

$$f_j\left(t_j\right)=\frac{1}{2\pi sin{\theta }_j{C}_{{\Psi }_j,{\theta }_j}}\underset{\mathbb{R}}{\int }\underset{{\mathbb{R}}_{+}}{\int }{W}_{{\Psi }_j}^{{\theta }_j}{f}_j\left(b_j,a_j\right){\Psi }_{b_j,a_j,{\theta }_j}\left(t_j\right)\frac{d{b}_{j}d{a}_j}{a_j^2},$$

(17)

for $j=1,...,n$. Then by (17), we achieve:

$$\begin{array}{cc}\hfill & \mathbf{f}\left(t\right)=f_1\left(t_1\right)....f_n\left(t_n\right)\hfill \\ \hfill & =\frac{1}{2\pi sin{\theta }_1{C}_{{\Psi }_1,{\theta }_1}}\underset{\mathbb{R}}{\int }\underset{{\mathbb{R}}_{+}}{\int }{W}_{{\Psi }_1}^{{\theta }_1}{f}_1\left(b_1,a_1\right){\Psi }_{b_1,a_1,{\theta }_1}\left(t_1\right)\frac{d{b}_{1}d{a}_1}{a_1^2}...\hfill \\ \hfill & ...\frac{1}{2\pi sin{\theta }_n{C}_{{\Psi }_n,{\theta }_n}}\underset{\mathbb{R}}{\int }\underset{{\mathbb{R}}_{+}}{\int }{W}_{{\Psi }_n}^{{\theta }_n}{f}_n\left(b_n,a_n\right){\Psi }_{b_n,a_n,{\theta }_n}\left(t_n\right)\frac{d{b}_{n}d{a}_n}{a_n^2}\hfill \\ \hfill & ={\left(2\pi \right)}^{-n}\stackrel{n}{\prod _{j=1}}\frac{1}{sin{\theta }_j{C}_{{\Psi }_j,{\theta }_j}}\underset{\mathbb{R}}{\int }...\underset{\mathbb{R}}{\int }\underset{{\mathbb{R}}_{+}}{\int }...\underset{{\mathbb{R}}_{+}}{\int }{W}_{{\Psi }_1}^{{\theta }_1}{f}_1\left(b_1,a_1\right)...\hfill \\ \hfill & ...W_{{\Psi }_n}^{{\theta }_n}{f}_n\left(b_n,a_n\right){\Psi }_{b_1,a_1,{\theta }_1}\left(t_1\right)...{\Psi }_{b_n,a_n,{\theta }_n}\left(t_n\right)\frac{d{b}_{1}d{a}_1}{a_1^2}...\frac{d{b}_{n}d{a}_n}{a_n^2}\hfill \\ \hfill & ={\left(2\pi \right)}^{-n}\stackrel{n}{\prod _{j=1}}\frac{1}{sin{\theta }_j{C}_{{\Psi }_j,{\theta }_j}}\underset{{\mathbb{R}}^n}{\int }\underset{{\mathbb{R}}_{+}^n}{\int }\stackrel{n}{\prod _{j=1}}{W}_{{\Psi }_j}^{{\theta }_j}{f}_j\left(b_j,a_j\right)\stackrel{n}{\prod _{j=1}}{\Psi }_{b_j,a_j,{\theta }_j}\left(t_j\right)\frac{dbda}{{\left|a\right|}_n^2}\hfill \\ \hfill & ={\left(2\pi \right)}^{-n}\stackrel{n}{\prod _{j=1}}\frac{1}{sin{\theta }_j{C}_{{\Psi }_j,{\theta }_j}}\underset{{\mathbb{R}}^n}{\int }\underset{{\mathbb{R}}_{+}^n}{\int }{W}_{\Psi }^{\theta }\mathbf{f}\left(b,a\right){\Psi }_{b,a,\theta }\left(t\right)\frac{dbda}{{\left|a\right|}_n^2}.\hfill \end{array}$$

□

3. Boundedness of the Multilinear Fractional Wavelet Transform on Lebesgue Spaces

Theorem 9.

Let $q\ge 2$, $q^{\prime }\le r_j\le q$, $(j=1,...,n)$ and $\frac{1}{q}+\frac{1}{q^{\prime }}=1$. Assume that $\Psi =\left({\Psi }_1,...,{\Psi }_n\right)$ is any multi-wavelet in $S{\left(\mathbb{R}\right)}^n$. Then the multilinear fractional wavelet transform $W_{\Psi }^{\theta }\mathbf{f}:$$L^{r_1}\left(\mathbb{R}\right)\times ...\times L^{r_n}\left(\mathbb{R}\right)\to L^q\left({\mathbb{R}}^n\right)$ defined by $\mathbf{f}\to W_{\Psi }^{\theta }\mathbf{f}\left(.,a\right)$ is bounded for any $a\in {\mathbb{R}}_{+}^n$. Furthermore,

$${∥W_{\Psi }^{\theta }\mathbf{f}\left(.,a\right)∥}_q\lesssim \stackrel{n}{\prod _{j=1}}{∥f_j∥}_{r_j}$$

holds for all $\mathbf{f}=\left(f_1,...,f_n\right)\in L^{r_1}\left(\mathbb{R}\right)\times ...\times L^{r_n}\left(\mathbb{R}\right)$.

Proof. 

Let $a\in {\mathbb{R}}_{+}^n$. We take $q=2$. Then,

$$\begin{array}{cc}\hfill & {∥W_{\Psi }^{\theta }\mathbf{f}\left(.,a\right)∥}_{L^2\left({\mathbb{R}}^n\right)}^2={∥e^{-\frac{i}{2}{b}^2.cot\theta }\stackrel{n}{\prod _{j=1}}\left(e^{\frac{i}{2}{(.)}^{2}cot{\theta }_j}{f}_j\ast {\Psi }_j^{a_j}\right)∥}_{L^2\left({\mathbb{R}}^n\right)}^2\hfill \\ \hfill & ={∥\stackrel{n}{\prod _{j=1}}\left(e^{\frac{i}{2}{(.)}^{2}cot{\theta }_j}{f}_j\ast {\Psi }_j^{a_j}\right)∥}_{L^2\left({\mathbb{R}}^n\right)}^2=\underset{{\mathbb{R}}^n}{\int }{\left|\stackrel{n}{\prod _{j=1}}\left(e^{\frac{i}{2}{(.)}^{2}cot{\theta }_j}{f}_j\ast {\Psi }_j^{a_j}\right)\left(b_j\right)\right|}^{2}db\hfill \\ \hfill & =\underset{\mathbb{R}}{\int }...\underset{\mathbb{R}}{\int }{\left|\stackrel{n}{\prod _{j=1}}\left(e^{\frac{i}{2}{(.)}^{2}cot{\theta }_j}{f}_j\ast {\Psi }_j^{a_j}\right)\left(b_j\right)\right|}^{2}d{b}_1...d{b}_n\hfill \\ \hfill & =\underset{\mathbb{R}}{\int }...\underset{\mathbb{R}}{\int }{\left|\left(e^{\frac{i}{2}{(.)}^{2}cot{\theta }_1}{f}_1\ast {\Psi }_1^{a_1}\right)\left(b_1\right)\right|}^2...{\left|\left(e^{\frac{i}{2}{(.)}^{2}cot{\theta }_n}{f}_n\ast {\Psi }_n^{a_n}\right)\left(b_n\right)\right|}^{2}d{b}_1...d{b}_n\hfill \\ \hfill & =\stackrel{n}{\prod _{j=1}}{∥\left(e^{\frac{i}{2}{(.)}^{2}cot{\theta }_j}{f}_j\ast {\Psi }_j^{a_j}\right)∥}_{L^2\left(\mathbb{R}\right)}^2.\hfill \end{array}$$

(18)

Since $L^2\left(\mathbb{R}\right)$ is a Banach module over $L^1\left(\mathbb{R}\right)$, we have:

$${∥\left(e^{\frac{i}{2}{(.)}^{2}cot{\theta }_j}{f}_j\ast {\Psi }_j^{a_j}\right)∥}_{L^2\left(\mathbb{R}\right)}\le {∥e^{\frac{i}{2}{(.)}^{2}cot{\theta }_j}{f}_j∥}_{L^2\left(\mathbb{R}\right)}{∥{\Psi }_j^{a_j}∥}_{L^1\left(\mathbb{R}\right)}$$

$$={∥f_j∥}_{L^2\left(\mathbb{R}\right)}{∥{\Psi }_j^{a_j}∥}_{L^1\left(\mathbb{R}\right)}=a_j^{\frac{1}{2}}{∥f_j∥}_{L^2\left(\mathbb{R}\right)}{∥{\Psi }_j∥}_{L^1\left(\mathbb{R}\right)}$$

(19)

for $j=1,...,n.$ Then by (18) and (19), we find:

$$\begin{array}{cc}\hfill {∥W_{\Psi }^{\theta }\mathbf{f}\left(.,a\right)∥}_{L^2\left({\mathbb{R}}^n\right)}& \le {\left|a\right|}_n^{\frac{1}{2}}\stackrel{n}{\prod _{j=1}}{∥{\Psi }_j∥}_{L^1\left(\mathbb{R}\right)}\stackrel{n}{\prod _{j=1}}{∥f_j∥}_{L^2\left(\mathbb{R}\right)}\hfill \\ \hfill & =C_1\stackrel{n}{\prod _{j=1}}{∥f_j∥}_{L^2\left(\mathbb{R}\right)}<\infty ,\hfill \end{array}$$

(20)

where $C_1={\left|a\right|}_n^{\frac{1}{2}}{\displaystyle \prod _{j=1}^n}{∥{\Psi }_j∥}_{L^1\left(\mathbb{R}\right)}.$ That means the multilinear fractional wavelet transform $W_{\Psi }^{\theta }\mathbf{f}\left(.,a\right)$ is bounded from $L^2{\left(\mathbb{R}\right)}^n$ to $L^2\left({\mathbb{R}}^n\right)$. Now, we take $q=\infty$, then:

$$\begin{array}{cc}\hfill {∥W_{\Psi }^{\theta }\mathbf{f}\left(.,a\right)∥}_{L^{\infty }\left({\mathbb{R}}^n\right)}& =\underset{b\in {\mathbb{R}}^n}{sup}\left|W_{\Psi }^{\theta }\mathbf{f}\left(.,a\right)\right|\hfill \\ \hfill & =\underset{b\in {\mathbb{R}}^n}{sup}\left|\underset{{\mathbb{R}}^n}{\int }\mathbf{f}\left(t\right)\overline{{\Psi }_{b,a,\theta }\left(t\right)}dt\right|\hfill \\ \hfill & \le \underset{b\in {\mathbb{R}}^n}{sup}\underset{{\mathbb{R}}^n}{\int }\left|\mathbf{f}\left(t\right)\right|\left|{\Psi }_{b,a,\theta }\left(t\right)\right|dt\hfill \end{array}$$

(21)

holds for any $a\in {\mathbb{R}}_{+}^n.$ Let $\frac{1}{p_j}+\frac{1}{p_j^{\prime }}=1$, $j=1,...,n$. Using Hölder inequality, we have

$$\begin{array}{cc}\hfill & \underset{{\mathbb{R}}^n}{\int }\left|\mathbf{f}\left(t\right)\right|\left|{\Psi }_{b,a,\theta }\left(t\right)\right|dt=\underset{{\mathbb{R}}^n}{\int }\left|\mathbf{f}\left(t\right)\right|\stackrel{n}{\prod _{j=1}}\left|{\Psi }_{b_j,a_j,{\theta }_j}\left(t_j\right)\right|dt\hfill \\ \hfill & =\underset{\mathbb{R}}{\int }...\underset{\mathbb{R}}{\int }\left|f_1\left(t_1\right)\right|...\left|f_n\left(t_n\right)\right|\left|{\Psi }_{b_1,a_1,{\theta }_1}\left(t_1\right)\right|...\left|{\Psi }_{b_n,a_n,{\theta }_n}\left(t_n\right)\right|d{t}_1...d{t}_n\hfill \\ \hfill & =\underset{\mathbb{R}}{\int }\left|f_1\left(t_1\right)\right|\left|{\Psi }_{b_1,a_1,{\theta }_1}\left(t_1\right)\right|d{t}_1...\underset{\mathbb{R}}{\int }\left|f_n\left(t_n\right)\right|\left|{\Psi }_{b_n,a_n,{\theta }_n}\left(t_n\right)\right|d{t}_n\hfill \\ \hfill & \le a_1^{\frac{1}{p_1^{\prime }}-\frac{1}{2}}{∥f_1∥}_{L^{p_1}\left(\mathbb{R}\right)}{∥{\Psi }_1∥}_{L^{p_1^{\prime }}\left(\mathbb{R}\right)}...a_n^{\frac{1}{p_n^{\prime }}-\frac{1}{2}}{∥f_n∥}_{L^{p_n}\left(\mathbb{R}\right)}{∥{\Psi }_n∥}_{L^{p_n^{\prime }}\left(\mathbb{R}\right)}\hfill \\ \hfill & =\stackrel{n}{\prod _{j=1}}{a}_j^{\frac{1}{p_j^{\prime }}-\frac{1}{2}}{∥{\Psi }_j∥}_{L^{p_j^{\prime }}\left(\mathbb{R}\right)}\stackrel{n}{\prod _{j=1}}{∥f_j∥}_{L^{p_j}\left(\mathbb{R}\right)}.\hfill \end{array}$$

(22)

So by (21) and (22), we find:

$${∥W_{\Psi }^{\theta }\mathbf{f}\left(.,a\right)∥}_{L^{\infty }\left({\mathbb{R}}^n\right)}\le \stackrel{n}{\prod _{j=1}}{a}_j^{\frac{1}{p_j^{\prime }}-\frac{1}{2}}{∥{\Psi }_j∥}_{L^{p_j^{\prime }}\left(\mathbb{R}\right)}\stackrel{n}{\prod _{j=1}}{∥f_j∥}_{L^{p_j}\left(\mathbb{R}\right)}$$

$$=C_2\stackrel{n}{\prod _{j=1}}{∥f_j∥}_{L^{p_j}\left(\mathbb{R}\right)},$$

(23)

where $C_2={\displaystyle \prod _{j=1}^n}{a}_j^{\frac{1}{p_j^{\prime }}-\frac{1}{2}}{∥{\Psi }_1∥}_{L^{p_1^{\prime }}\left(\mathbb{R}\right)}$ for any $a\in {\mathbb{R}}_{+}^n.$ Hence, we say that the multilinear fractional wavelet transform $W_{\Psi }^{\theta }\mathbf{f}\left(.,a\right)$ is bounded from $L^{p_1}\left(\mathbb{R}\right)\times ...\times L^{p_n}\left(\mathbb{R}\right)$ to $L^{\infty }\left({\mathbb{R}}^n\right)$. We get $q,r_j$, $j=1,...,n$ such that $2\le q\le \infty$ and $2\le r_j\le p_j$, $j=1,...,n$. If we use multilinear interpolation theorem in [21], then by (20) and (23), we obtain that $W_{\Psi }^{\theta }\mathbf{f}\left(.,a\right)$ is bounded from $L^{r_1}\left(\mathbb{R}\right)\times ...\times L^{r_n}\left(\mathbb{R}\right)$ to $L^q\left({\mathbb{R}}^n\right)$ such that:

$$\frac{\alpha }{2}+\frac{1-\alpha }{\infty }=\frac{1}{q}$$

(24)

$$\frac{\alpha }{2}+\frac{1-\alpha }{p_j}=\frac{1}{r_j},$$

(25)

for $0\le \alpha \le 1$, $j=1,...n$. Moreover, using (24) and (25), we have:

$$\frac{1}{q}\le \frac{1}{q}+\frac{1-\alpha }{p_j}\le \frac{1}{q}+1-\alpha =\frac{1}{q}+1-\frac{2}{q}=\frac{1}{q^{\prime }}.$$

So we achieve $q^{\prime }\le r_j\le q$. □

Theorem 10.

Let $\Psi =\left({\Psi }_1,...,{\Psi }_n\right)\in S{\left(\mathbb{R}\right)}^n$ be multi-wavelet and let be any $a\in {\mathbb{R}}_{+}^n$. Then,

$${∥W_{\Psi }^{\theta }\mathbf{f}\left(.,a\right)∥}_{L^p\left({\mathbb{R}}^n\right)}\le {\left|a\right|}_n^{\frac{1}{2}}\stackrel{n}{\prod _{j=1}}{∥f_j∥}_{L^p\left(\mathbb{R}\right)}\stackrel{n}{\prod _{j=1}}{∥{\Psi }_j∥}_{L^1\left(\mathbb{R}\right)}$$

holds for all $\mathbf{f}=\left(f_1,...,f_n\right)\in L^p{\left(\mathbb{R}\right)}^n$.

Proof. 

Take any $a\in {\mathbb{R}}_{+}^n$. By Theorem 1, we have:

$$\begin{array}{cc}\hfill {∥W_{\Psi }^{\theta }\mathbf{f}\left(.,a\right)∥}_{L^p\left({\mathbb{R}}^n\right)}& ={∥e^{-\frac{i}{2}{b}^2.cot\theta }\stackrel{n}{\prod _{j=1}}\left(e^{\frac{i}{2}{(.)}^{2}cot{\theta }_j}{f}_j\ast {\Psi }_j^{a_j}\right)∥}_{L^p\left({\mathbb{R}}^n\right)}\hfill \\ \hfill & ={∥\stackrel{n}{\prod _{j=1}}\left(e^{\frac{i}{2}{(.)}^{2}cot{\theta }_j}{f}_j\ast {\Psi }_j^{a_j}\right)∥}_{L^p\left({\mathbb{R}}^n\right)}\hfill \\ \hfill & =\stackrel{n}{\prod _{j=1}}{∥\left(e^{\frac{i}{2}{(.)}^{2}cot{\theta }_j}{f}_j\ast {\Psi }_j^{a_j}\right)∥}_{L^p\left(\mathbb{R}\right)}.\hfill \end{array}$$

Since $L^p\left(\mathbb{R}\right)$ is Banach module over $L^1\left(\mathbb{R}\right)$, we obtain:

$$\begin{array}{cc}\hfill {∥W_{\Psi }^{\theta }\mathbf{f}\left(.,a\right)∥}_{L^p\left({\mathbb{R}}^n\right)}& \le \stackrel{n}{\prod _{j=1}}{∥\left(e^{\frac{i}{2}{(.)}^{2}cot{\theta }_j}{f}_j\right)∥}_{L^p\left(\mathbb{R}\right)}{∥{\Psi }_j^{a_j}∥}_{L^1\left(\mathbb{R}\right)}\hfill \\ \hfill & =\stackrel{n}{\prod _{j=1}}{∥f_j∥}_{L^p\left(\mathbb{R}\right)}\stackrel{n}{\prod _{j=1}}{a}_j^{\frac{1}{2}}{∥{\Psi }_j∥}_{L^1\left(\mathbb{R}\right)}\hfill \\ \hfill & ={\left|a\right|}_n^{\frac{1}{2}}\stackrel{n}{\prod _{j=1}}{∥f_j∥}_{L^p\left(\mathbb{R}\right)}\stackrel{n}{\prod _{j=1}}{∥{\Psi }_j∥}_{L^1\left(\mathbb{R}\right)}.\hfill \end{array}$$

□

Corollary 1.

Let $\Psi =\left({\Psi }_1,...,{\Psi }_n\right)\in S{\left(\mathbb{R}\right)}^n$ be multi-wavelet and let be any $a\in {\mathbb{R}}_{+}^n$. Then,

$${∥W_{\Psi }^{\theta }\mathbf{f}\left(.,a\right)∥}_{L^p\left({\mathbb{R}}^n\right)}=O\left({\left|a\right|}_n^{\frac{1}{2}}\right)$$

holds for all $\mathbf{f}=\left(f_1,...,f_n\right)\in L^p{\left(\mathbb{R}\right)}^n$.

Proof. 

The proof is immediately obtained by Theorem 10. □

Theorem 11.

Let $\Psi =\left({\Psi }_1,...,{\Psi }_n\right)$, ${\rm Y}=\left({\phi }_1,...,{\phi }_n\right)\in S{\left(\mathbb{R}\right)}^n$ be multi-wavelets and let be any $a\in {\mathbb{R}}_{+}^n$. Then,

$${∥W_{\Psi }^{\theta }\mathbf{f}\left(.,a\right)-W_{{\rm Y}}^{\theta }\mathbf{f}\left(.,a\right)∥}_{L^p\left({\mathbb{R}}^n\right)}\le {\left|a\right|}_n^{\frac{1}{2}}\stackrel{n}{\prod _{j=1}}{∥f_j∥}_{L^p\left(\mathbb{R}\right)}\stackrel{n}{\prod _{j=1}}{∥{\Psi }_j-{\phi }_j∥}_{L^1\left(\mathbb{R}\right)}$$

holds for all $\mathbf{f}=\left(f_1,...,f_n\right)\in L^p{\left(\mathbb{R}\right)}^n$.

Proof. 

Assume that $\Psi =\left({\Psi }_1,...,{\Psi }_n\right)$, ${\rm Y}=\left({\phi }_1,...,{\phi }_n\right)\in S{\left(\mathbb{R}\right)}^n$ are multi-wavelets. For any $a\in {\mathbb{R}}_{+}^n$, we find:

$$\begin{array}{cc}\hfill {∥W_{\Psi }^{\theta }\mathbf{f}\left(.,a\right)-W_{{\rm Y}}^{\theta }\mathbf{f}\left(.,a\right)∥}_{L^p\left({\mathbb{R}}^n\right)}& ={∥⟨\mathbf{f},{\Psi }_{b,a,\theta }⟩-⟨\mathbf{f},{{\rm Y}}_{b,a,\theta }⟩∥}_{L^p\left({\mathbb{R}}^n\right)}\hfill \\ \hfill & ={∥⟨\mathbf{f},{\Psi }_{b,a,\theta }-{{\rm Y}}_{b,a,\theta }⟩∥}_{L^p\left({\mathbb{R}}^n\right)}.\hfill \end{array}$$

(26)

We also have:

$$\begin{array}{cc}\hfill & \left({\Psi }_{b,a,\theta }-{{\rm Y}}_{b,a,\theta }\right)\left(t\right)={\left|a\right|}_n^{-\frac{1}{2}}\stackrel{n}{\prod _{j=1}}{\Psi }_j\left(\frac{t_j-b_j}{a_j}\right)e^{-\frac{i}{2}\left(t_j^2-b_j^2\right)cot{\theta }_j}-\hfill \\ \hfill & -{\left|a\right|}_n^{-\frac{1}{2}}\stackrel{n}{\prod _{j=1}}{\phi }_j\left(\frac{t_j-b_j}{a_j}\right)e^{-\frac{i}{2}\left(t_j^2-b_j^2\right)cot{\theta }_j}\hfill \\ \hfill & ={\left|a\right|}_n^{-\frac{1}{2}}\stackrel{n}{\prod _{j=1}}\left({\Psi }_j-{\phi }_j\right)\left(\frac{t_j-b_j}{a_j}\right)e^{-\frac{i}{2}\left(t_j^2-b_j^2\right)cot{\theta }_j}={\left(\Psi -{\rm Y}\right)}_{b,a,\theta }\left(t\right)\hfill \end{array}$$

(27)

for $t=\left(t_1,...t_n\right)\in {\mathbb{R}}^n$. Combining the equalities (26), (27) and using Theorem 1, we achieve:

$$\begin{array}{cc}\hfill & {∥W_{\Psi }^{\theta }\mathbf{f}\left(.,a\right)-W_{{\rm Y}}^{\theta }\mathbf{f}\left(.,a\right)∥}_{L^p\left({\mathbb{R}}^n\right)}={∥⟨\mathbf{f},{\left(\Psi -{\rm Y}\right)}_{b,a,\theta }⟩∥}_{L^p\left({\mathbb{R}}^n\right)}={∥W_{\Psi -{\rm Y}}^{\theta }\mathbf{f}\left(.,a\right)∥}_{L^p\left({\mathbb{R}}^n\right)}\hfill \\ \hfill & ={∥e^{-\frac{i}{2}{b}^2.cot\theta }\stackrel{n}{\prod _{j=1}}\left(e^{\frac{i}{2}{(.)}^{2}cot{\theta }_j}{f}_j\ast {\left({\Psi }_j-{\phi }_j\right)}^{a_j}\right)∥}_{L^p\left({\mathbb{R}}^n\right)}\hfill \\ \hfill & \le \stackrel{n}{\prod _{j=1}}{∥\left(e^{\frac{i}{2}{(.)}^{2}cot{\theta }_j}{f}_j\right)∥}_{L^p\left(\mathbb{R}\right)}{∥{\left({\Psi }_j-{\phi }_j\right)}^{a_j}∥}_{L^1\left(\mathbb{R}\right)}\hfill \\ \hfill & ={\left|a\right|}_n^{\frac{1}{2}}\stackrel{n}{\prod _{j=1}}{∥f_j∥}_{L^p\left(\mathbb{R}\right)}\stackrel{n}{\prod _{j=1}}{∥{\Psi }_j-{\phi }_j∥}_{L^1\left(\mathbb{R}\right)}.\hfill \end{array}$$

□

Theorem 12.

Let $\Psi =\left({\Psi }_1,...,{\Psi }_n\right)\in S{\left(\mathbb{R}\right)}^n$ be multi-wavelet and let be any $a\in {\mathbb{R}}_{+}^n$. Then,

$${∥W_{\Psi }^{\theta }\mathbf{f}\left(.,a\right)-W_{\Psi }^{\theta }\mathbf{g}\left(.,a\right)∥}_{L^p\left({\mathbb{R}}^n\right)}\le {\left|a\right|}_n^{\frac{1}{2}}\stackrel{n}{\prod _{j=1}}{∥f_j-g_j∥}_{L^p\left(\mathbb{R}\right)}\stackrel{n}{\prod _{j=1}}{∥{\Psi }_j∥}_{L^1\left(\mathbb{R}\right)}$$

holds for all $\mathbf{f}=\left(f_1,...,f_n\right)$, $\mathbf{g}=\left(g_1,...g_n\right)\in L^p{\left(\mathbb{R}\right)}^n$.

Proof. 

Take any $\mathbf{f}=\left(f_1,...,f_n\right)$, $\mathbf{g}=\left(g_1,...g_n\right)\in L^p{\left(\mathbb{R}\right)}^n$. Then by Theorem 1, we obtain:

$$\begin{array}{cc}\hfill & {∥W_{\Psi }^{\theta }\mathbf{f}\left(.,a\right)-W_{\Psi }^{\theta }\mathbf{g}\left(.,a\right)∥}_{L^p\left({\mathbb{R}}^n\right)}={∥⟨\mathbf{f},{\Psi }_{b,a,\theta }⟩-⟨g,{\Psi }_{b,a,\theta }⟩∥}_{L^p\left({\mathbb{R}}^n\right)}\hfill \\ \hfill & ={∥⟨\mathbf{f}-\mathbf{g},{\Psi }_{b,a,\theta }⟩∥}_{L^p\left({\mathbb{R}}^n\right)}\hfill \\ \hfill & ={∥e^{-\frac{i}{2}{b}^2.cot\theta }\stackrel{n}{\prod _{j=1}}\left(e^{\frac{i}{2}{(.)}^{2}cot{\theta }_j}\left(f_j-g_j\right)\ast {\Psi }_j^{a_j}\right)∥}_{L^p\left({\mathbb{R}}^n\right)}\hfill \\ \hfill & \le \stackrel{n}{\prod _{j=1}}{∥\left(e^{\frac{i}{2}{(.)}^{2}cot{\theta }_j}\left(f_j-g_j\right)\right)∥}_{L^p\left(\mathbb{R}\right)}{∥{\Psi }_j^{a_j}∥}_{L^1\left(\mathbb{R}\right)}\hfill \\ \hfill & ={\left|a\right|}_n^{\frac{1}{2}}\stackrel{n}{\prod _{j=1}}{∥f_j-g_j∥}_{L^p\left(\mathbb{R}\right)}\stackrel{n}{\prod _{j=1}}{∥{\Psi }_j∥}_{L^1\left(\mathbb{R}\right)}\hfill \end{array}$$

for any $a\in {\mathbb{R}}_{+}^n.$ □

Corollary 2.

Let $\Psi =\left({\Psi }_1,...,{\Psi }_n\right)$, ${\rm Y}=\left({\phi }_1,...,{\phi }_n\right)\in S{\left(\mathbb{R}\right)}^n$ be multi-wavelets and let be any $a\in {\mathbb{R}}_{+}^n$. Then,

$${∥W_{\Psi }^{\theta }\mathbf{f}\left(.,a\right)-W_{{\rm Y}}^{\theta }\mathbf{g}\left(.,a\right)∥}_{L^p\left({\mathbb{R}}^n\right)}\le {\left|a\right|}_n^{\frac{1}{2}}\left\{\begin{array}{c}{\displaystyle \prod _{j=1}^n}{∥f_j-g_j∥}_{L^p\left(\mathbb{R}\right)}{\displaystyle \prod _{j=1}^n}{∥{\Psi }_j∥}_{L^1\left(\mathbb{R}\right)}+\\ +{\displaystyle \prod _{j=1}^n}{∥g_j∥}_{L^p\left(\mathbb{R}\right)}{\displaystyle \prod _{j=1}^n}{∥{\Psi }_j-{\phi }_j∥}_{L^1\left(\mathbb{R}\right)}\end{array}\right\}$$

holds for all $\mathbf{f}=\left(f_1,...,f_n\right)$, $\mathbf{g}=\left(g_1,...g_n\right)\in L^p{\left(\mathbb{R}\right)}^n$.

Proof. 

Assume that $\Psi =\left({\Psi }_1,...,{\Psi }_n\right)$, ${\rm Y}=\left({\phi }_1,...,{\phi }_n\right)\in S{\left(\mathbb{R}\right)}^n$ are multi-wavelets. Take any $\mathbf{f}=\left(f_1,...,f_n\right)$, $\mathbf{g}=\left(g_1,...g_n\right)\in L^p{\left(\mathbb{R}\right)}^n$. Then by Theorem 11 and Theorem 12, we have:

$$\begin{array}{cc}\hfill & {∥W_{\Psi }^{\theta }\mathbf{f}\left(.,a\right)-W_{{\rm Y}}^{\theta }\mathbf{g}\left(.,a\right)∥}_{L^p\left({\mathbb{R}}^n\right)}\hfill \\ \hfill & \le {∥W_{\Psi }^{\theta }\mathbf{f}\left(.,a\right)-W_{\Psi }^{\theta }\mathbf{g}\left(.,a\right)∥}_{L^p\left({\mathbb{R}}^n\right)}+{∥W_{\Psi }^{\theta }\mathbf{g}\left(.,a\right)-W_{{\rm Y}}^{\theta }\mathbf{g}\left(.,a\right)∥}_{L^p\left({\mathbb{R}}^n\right)}\hfill \\ \hfill & \le {\left|a\right|}_n^{\frac{1}{2}}\left\{\stackrel{n}{\prod _{j=1}}{∥f_j-g_j∥}_{L^p\left(\mathbb{R}\right)}\stackrel{n}{\prod _{j=1}}{∥{\Psi }_j∥}_{L^1\left(\mathbb{R}\right)}+\stackrel{n}{\prod _{j=1}}{∥g_j∥}_{L^p\left(\mathbb{R}\right)}\stackrel{n}{\prod _{j=1}}{∥{\Psi }_j-{\phi }_j∥}_{L^1\left(\mathbb{R}\right)}\right\}\hfill \end{array}$$

for any $a\in {\mathbb{R}}_{+}^n$. □

4. Boundedness of the Multilinear Fractional Wavelet Transform on Lorentz Spaces

The multilinear fractional wavelet transform is a tensor product operator such that:

$$\left(W_{{\Psi }_1}^{{\theta }_1}{f}_j,...,W_{{\Psi }_n}^{{\theta }_n}{f}_n\right)\to \left(\underset{j=1}{\stackrel{n}{\otimes }}{W}_{{\Psi }_j}^{{\theta }_j}{f}_j\right)\left(b,a\right)=W_{{\Psi }_1}^{{\theta }_1}{f}_j\left(b_1,a_1\right)...W_{{\Psi }_n}^{{\theta }_n}{f}_n\left(b_n,a_n\right)$$

for all $a=\left(a_1,...,a_n\right)\in {\mathbb{R}}_{+}^n$ and $b=\left(b_1,...,b_n\right)\in {\mathbb{R}}^n$. Take $\Psi =\left({\Psi }_1,...,{\Psi }_n\right)\in S{\left(\mathbb{R}\right)}^n$. It satisfies:

$${∥W_{\Psi }^{\theta }\mathbf{f}\left(.,a\right)∥}_{L^1\left({\mathbb{R}}^n\right)}=\stackrel{n}{\prod _{j=1}}{∥W_{{\Psi }_j}^{{\theta }_j}{f}_j\left(.,a_j\right)∥}_{L^1\left(\mathbb{R}\right)}$$

and

$${∥W_{\Psi }^{\theta }\mathbf{f}\left(.,a\right)∥}_{L^{\infty }\left({\mathbb{R}}^n\right)}\le \stackrel{n}{\prod _{j=1}}{∥W_{{\Psi }_j}^{{\theta }_j}{f}_j\left(.,a_j\right)∥}_{L^{\infty }\left(\mathbb{R}\right)}$$

for all $\mathbf{f}=\left(f_1,...,f_n\right)\in$ $S{\left(\mathbb{R}\right)}^n$. Then multilinear fractional wavelet transform is of restricted weak types (1,...,1;1) and (∞,...,∞;∞). So we can generalize Theorem 7.7 in [21] for multilinear fractional wavelet transform. Hence, there exists $C>0$ such that:

$${∥W_{\Psi }^{\theta }\mathbf{f}\left(.,a\right)∥}_{L^{p,q}\left({\mathbb{R}}^n\right)}\le C\stackrel{n}{\prod _{j=1}}{∥W_{{\Psi }_j}^{{\theta }_j}{f}_j\left(.,a_j\right)∥}_{L^{p,q_j}\left(\mathbb{R}\right)},$$

(28)

where $\frac{1}{q}+(n-1)=\frac{1}{q_1}+...+\frac{1}{q_n}$, $1<p<\infty$ and $1\le q,q_j\le \infty$, $\left(j=1,...,n\right)$.

Theorem 13.

Let $\frac{1}{q}+(n-1)=\frac{1}{q_1}+...+\frac{1}{q_n}$, $1<p<\infty$, $1\le q,q_j\le \infty$ $\left(j=1,...,n\right)$ and $\Psi =\left({\Psi }_1,...,{\Psi }_n\right)\in S{\left(\mathbb{R}\right)}^n$, $a=\left(a_1,...,a_n\right)\in {\mathbb{R}}_{+}^n$. Then,

(i) 

The multilinear fractional wavelet transform $W_{\Psi }^{\theta }\mathbf{f}$$\left(.,a\right)$ is bounded from $L^{p,q_1}\left(\mathbb{R}\right)\times ...\times L^{p,q_n}\left(\mathbb{R}\right)$ into $L^{p,q}\left({\mathbb{R}}^n\right)$;

(ii) 

In addition if $\frac{1}{p}+1=\frac{1}{r}+\frac{1}{s}$, $\frac{1}{q_j}=\frac{1}{c_j}+\frac{1}{d_j}$ $\left(j=1,...,n\right)$, then the multilinear fractional wavelet transform $W_{\Psi }^{\theta }\mathbf{f}$$\left(.,a\right)$ is bounded from $L^{r,c_1}\left(\mathbb{R}\right)\times ...\times L^{r,c_n}\left(\mathbb{R}\right)$ into $L^{p,q}\left({\mathbb{R}}^n\right)$.

Proof. 

(i) Take any $\mathbf{f}=\left(f_1,...,f_n\right)\in$$L^{p,q_1}\left(\mathbb{R}\right)\times ...\times L^{p,q_n}\left(\mathbb{R}\right)$. Then by (28), we have:

$$\begin{array}{cc}\hfill {∥W_{\Psi }^{\theta }\mathbf{f}\left(.,a\right)∥}_{L^{p,q}\left({\mathbb{R}}^n\right)}& \le C\stackrel{n}{\prod _{j=1}}{∥W_{{\Psi }_j}^{{\theta }_j}{f}_j\left(.,a_j\right)∥}_{L^{p,q_j}\left(\mathbb{R}\right)}\hfill \\ \hfill & =C\stackrel{n}{\prod _{j=1}}{∥e^{-\frac{i}{2}{b}_j^{2}cot{\theta }_j}\left(e^{\frac{i}{2}{(.)}^{2}cot{\theta }_j}{f}_j\ast {\Psi }_j^{a_j}\right)∥}_{L^{p,q_j}\left(\mathbb{R}\right)}\hfill \\ \hfill & =C\stackrel{n}{\prod _{j=1}}{∥\left(e^{\frac{i}{2}{(.)}^{2}cot{\theta }_j}{f}_j\ast {\Psi }_j^{a_j}\right)∥}_{L^{p,q_j}\left(\mathbb{R}\right)}.\hfill \end{array}$$

(29)

On the other hand since $L^{p,q_j}\left(\mathbb{R}\right)$ is a Banach module over $L^1\left(\mathbb{R}\right)$ [3] and by (29), we get:

$$\begin{array}{cc}\hfill {∥W_{\Psi }^{\theta }\mathbf{f}\left(.,a\right)∥}_{L^{p,q}\left({\mathbb{R}}^n\right)}& \le C\stackrel{n}{\prod _{j=1}}{∥e^{\frac{i}{2}{(.)}^{2}cot{\theta }_j}{f}_j∥}_{L^{p,q_j}\left(\mathbb{R}\right)}{∥{\Psi }_j^{a_j}∥}_{L^1\left(\mathbb{R}\right)}\hfill \\ \hfill & =C{\left|a\right|}_n^{\frac{1}{2}}\stackrel{n}{\prod _{j=1}}{∥f_j∥}_{L^{p,q_j}\left(\mathbb{R}\right)}\stackrel{n}{\prod _{j=1}}{∥{\Psi }_j∥}_{L^1\left(\mathbb{R}\right)}.\hfill \end{array}$$

(30)

This is the desired result.

(ii) Let $\Psi =\left({\Psi }_1,...,{\Psi }_n\right)\in S{\left(\mathbb{R}\right)}^n$ be given. We then have:

$$\begin{array}{cc}\hfill & {\lambda }_{{\Psi }_j^{a_j}}\left(y\right)=\mu \left(\left\{x\in \mathbb{R}:\left|{\Psi }_j^{a_j}\left(x\right)\right|>y\right\}\right)=\mu \left(\left\{x\in \mathbb{R}:a_j^{\frac{-1}{2}}\left|{\Psi }_j\left(\frac{x}{a_j}\right)\right|>y\right\}\right)\hfill \\ \hfill & =\mu \left(\left\{a_{j}u\in \mathbb{R}:\left|{\Psi }_j\left(u\right)\right|>a_j^{\frac{1}{2}}y\right\}\right)=a_j\mu \left(\left\{u\in \mathbb{R}:\left|{\Psi }_j\left(u\right)\right|>a_j^{\frac{1}{2}}y\right\}\right)\hfill \\ \hfill & =a_j{\lambda }_{{\Psi }_j}\left(a_j^{\frac{1}{2}}y\right),j=1,...,n\hfill \end{array}$$

for $y>0$. So the rearrangement of ${\Psi }_j^{a_j}$ is:

$$\begin{array}{cc}\hfill & {\left({\Psi }_j^{a_j}\right)}^{\ast }\left(t\right)=inf\left\{y>0:{\lambda }_{{\Psi }_j^{a_j}}\left(y\right)\le t\right\}=inf\left\{y>0:a_j{\lambda }_{{\Psi }_j}\left(a_j^{\frac{1}{2}}y\right)\le t\right\}\hfill \\ \hfill & =inf\left\{a_j^{\frac{-1}{2}}z>0:{\lambda }_{{\Psi }_j}\left(z\right)\le \frac{t}{a_j}\right\}=a_j^{\frac{-1}{2}}{\left({\Psi }_j\right)}^{\ast }\left(\frac{t}{a_j}\right),j=1,...,n\hfill \end{array}$$

for $t>0$. Thus, we have ${∥{\Psi }_j^{a_j}∥}_{L^{s,d_j}\left(\mathbb{R}\right)}=a_j^{\frac{1}{s}-\frac{1}{2}}{∥{\Psi }_j∥}_{L^{s,d_j}\left(\mathbb{R}\right)}$, $j=1,...,n$. Therefore, using (30) and by Theorem 7.6 in [21], we have:

$$\begin{array}{cc}\hfill {∥W_{\Psi }^{\theta }\mathbf{f}\left(.,a\right)∥}_{L^{p,q}\left({\mathbb{R}}^n\right)}& \le C\stackrel{n}{\prod _{j=1}}{∥\left(e^{\frac{i}{2}{(.)}^{2}cot{\theta }_j}{f}_j\ast {\Psi }_j^{a_j}\right)∥}_{L^{p,q_j}\left(\mathbb{R}\right)}\hfill \\ \hfill & \le C{C}_1\stackrel{n}{\prod _{j=1}}{∥e^{\frac{i}{2}{(.)}^{2}cot{\theta }_j}{f}_j∥}_{L^{r,c_j}\left(\mathbb{R}\right)}{∥{\Psi }_j^{a_j}∥}_{L^{s,d_j}\left(\mathbb{R}\right)}\hfill \\ \hfill & =C{C}_1{\left|a\right|}_n^{\frac{1}{s}-\frac{1}{2}}\stackrel{n}{\prod _{j=1}}{∥f_j∥}_{L^{r,c_j}\left(\mathbb{R}\right)}\stackrel{n}{\prod _{j=1}}{∥{\Psi }_j∥}_{L^{s,d_j}\left(\mathbb{R}\right)},\hfill \end{array}$$

(31)

where $\frac{1}{p}+1=\frac{1}{r}+\frac{1}{s}$, $\frac{1}{q_j}=\frac{1}{c_j}+\frac{1}{d_j}$ $\left(j=1,...,n\right)$. This completes the proof. □

Now, using Theorem 13, we can give the following corollary.

Corollary 3.

Let $\frac{1}{q}+(n-1)=\frac{1}{q_1}+...+\frac{1}{q_n}$, $1<p<\infty$, $1\le q,q_j\le \infty$ $\left(j=1,...,n\right)$ and $\Psi =\left({\Psi }_1,...,{\Psi }_n\right)\in S{\left(\mathbb{R}\right)}^n$, $a=\left(a_1,...,a_n\right)\in {\mathbb{R}}_{+}^n$. Then,

(i) 

The multilinear fractional wavelet transform $W_{\Psi }^{\theta }\mathbf{f}$$\left(.,a\right)$ is bounded from

$\left(L^{p,q_1}\left(\mathbb{R}\right)\times ...\times L^{p,q_n}\left(\mathbb{R}\right)\right)\times \left(L^1\left(\mathbb{R}\right)\times ...\times L^1\left(\mathbb{R}\right)\right)$ into $L^{p,q}\left({\mathbb{R}}^n\right)$;

(ii) 

In addition if $\frac{1}{p}+1=\frac{1}{r}+\frac{1}{s}$, $\frac{1}{q_j}=\frac{1}{c_j}+\frac{1}{d_j}$ $\left(j=1,...,n\right)$, then, the multilinear fractional wavelet transform $W_{\Psi }^{\theta }\mathbf{f}$$\left(.,a\right)$ is bounded from $\left(L^{r,c_1}\left(\mathbb{R}\right)\times ...\times L^{r,c_n}\left(\mathbb{R}\right)\right)\times \left(L^{s,d_1}\left(\mathbb{R}\right)\times ...\times L^{s,d_n}\left(\mathbb{R}\right)\right)$ into $L^{p,q}\left({\mathbb{R}}^n\right)$.

Theorem 14.

Let $\frac{1}{q}+(n-1)=\frac{1}{q_1}+...+\frac{1}{q_n}$, $\frac{1}{p}+1=\frac{1}{r}+\frac{1}{s}$, $\frac{1}{q_j}=\frac{1}{c_j}+\frac{1}{d_j}$ $\left(j=1,...,n\right)$, $1\le s^{\prime }\le 2$ and $\frac{1}{s}+\frac{1}{s^{\prime }}=1$. Assume that $\Psi =\left({\Psi }_1,...,{\Psi }_n\right)\in S{\left(\mathbb{R}\right)}^n$. Then, the multilinear fractional wavelet transform $W_{{\mathcal{F}}^{\theta }\Psi }^{\theta }\mathbf{f}\left(.,a\right)$ is bounded from $\left(L^{r,c_1}\left(\mathbb{R}\right)\times ...\times L^{r,c_n}\left(\mathbb{R}\right)\right)\times \left(L^{s^{\prime },d_1}\left(\mathbb{R}\right)\times ...\times L^{s^{\prime },d_n}\left(\mathbb{R}\right)\right)$ into $L^{p,q}\left({\mathbb{R}}^n\right)$ for any $a\in {\mathbb{R}}_{+}^n$. Moreover,

$${∥W_{{\mathcal{F}}^{\theta }\Psi }^{\theta }\mathbf{f}\left(.,a\right)∥}_{L^{p,q^{\prime }}\left({\mathbb{R}}^n\right)}\lesssim {\left|a\right|}_n^{\frac{1}{s}-\frac{1}{2}}\stackrel{n}{\prod _{j=1}}\frac{1}{\sqrt{\left|sin{\theta }_j\right|}}\stackrel{n}{\prod _{j=1}}{∥f_j∥}_{L^{r,c_j}\left(\mathbb{R}\right)}\stackrel{n}{\prod _{j=1}}{∥{\Psi }_j∥}_{L^{s^{\prime },d_j}\left(\mathbb{R}\right)}$$

holds for all $f\in \left(L^{r,c_1}\left(\mathbb{R}\right)\times ...\times L^{r,c_n}\left(\mathbb{R}\right)\right).$

Proof. 

Take any $\mathbf{f}=\left(f_1,...,f_n\right)\in$$L^{r,c_1}\left(\mathbb{R}\right)\times ...\times L^{r,c_n}\left(\mathbb{R}\right)$. Since ${\Psi }_j\in S\left(\mathbb{R}\right)$, we have ${\mathcal{F}}^{{\theta }_j}{\Psi }_j\in S\left(\mathbb{R}\right)\subset L^{s,d_j}\left(\mathbb{R}\right)$, $j=1,...,n$. Then, by Theorem 3 and the inequality (31), we conclude:

$$\begin{array}{cc}\hfill {∥W_{{\mathcal{F}}^{\theta }\Psi }^{\theta }\mathbf{f}\left(.,a\right)∥}_{L^{p,q}\left({\mathbb{R}}^n\right)}& \le C{\left|a\right|}_n^{\frac{1}{s}-\frac{1}{2}}\stackrel{n}{\prod _{j=1}}{∥f_j∥}_{L^{r,c_j}\left(\mathbb{R}\right)}\stackrel{n}{\prod _{j=1}}{∥{\mathcal{F}}^{{\theta }_j}{\Psi }_j∥}_{L^{s,d_j}\left(\mathbb{R}\right)}\hfill \\ \hfill & \le C{\left|a\right|}_n^{\frac{1}{s}-\frac{1}{2}}\stackrel{n}{\prod _{j=1}}\frac{1}{\sqrt{\left|sin{\theta }_j\right|}}\stackrel{n}{\prod _{j=1}}{∥f_j∥}_{L^{r,c_j}\left(\mathbb{R}\right)}\stackrel{n}{\prod _{j=1}}{∥{\Psi }_j∥}_{L^{s^{\prime },d_j}\left(\mathbb{R}\right)}\hfill \end{array}$$

where ${\mathcal{F}}^{\theta }\Psi =\left({\mathcal{F}}^{{\theta }_1}{\Psi }_1,...,{\mathcal{F}}^{{\theta }_n}{\Psi }_n\right)$, $a=\left(a_1,...,a_n\right)\in {\mathbb{R}}_{+}^n$. □

Using the inequalities (30) and (31), the following theorems and corollary are proven similar to the proof of Theorems 11 and 12, and Corollary 2.

Theorem 15.

Let $\frac{1}{q}+(n-1)=\frac{1}{q_1}+...+\frac{1}{q_n}$, $1<p<\infty$, $1\le q,q_j\le \infty$ $\left(j=1,...,n\right)$, $a\in {\mathbb{R}}_{+}^n$ and let $\Psi =\left({\Psi }_1,...,{\Psi }_n\right)$, ${\rm Y}=\left({\phi }_1,...,{\phi }_n\right)\in S{\left(\mathbb{R}\right)}^n$ be multi-wavelets.

(i)Then,

$${∥W_{\Psi }^{\theta }\mathbf{f}\left(.,a\right)-W_{{\rm Y}}^{\theta }\mathbf{f}\left(.,a\right)∥}_{L^{p,q}\left({\mathbb{R}}^n\right)}\lesssim {\left|a\right|}_n^{\frac{1}{2}}\stackrel{n}{\prod _{j=1}}{∥f_j∥}_{L^{p,q_j}\left(\mathbb{R}\right)}\stackrel{n}{\prod _{j=1}}{∥{\Psi }_j-{\phi }_j∥}_{L^1\left(\mathbb{R}\right)}$$

holds for all $\mathbf{f}=\left(f_1,...,f_n\right)\in$$L^{p,q_1}\left(\mathbb{R}\right)\times ...\times L^{p,q_n}\left(\mathbb{R}\right)$.

(ii)In addition if $\frac{1}{p}+1=\frac{1}{r}+\frac{1}{s}$, $\frac{1}{q_j}=\frac{1}{c_j}+\frac{1}{d_j}$ $\left(j=1,...,n\right)$, then,

$${∥W_{\Psi }^{\theta }\mathbf{f}\left(.,a\right)-W_{{\rm Y}}^{\theta }\mathbf{f}\left(.,a\right)∥}_{L^{p,q}\left({\mathbb{R}}^n\right)}\lesssim {\left|a\right|}_n^{\frac{1}{s}-\frac{1}{2}}\stackrel{n}{\prod _{j=1}}{∥f_j∥}_{L^{r,c_j}\left(\mathbb{R}\right)}\stackrel{n}{\prod _{j=1}}{∥{\Psi }_j-{\phi }_j∥}_{L^{s,d_j}\left(\mathbb{R}\right)}$$

holds for all $\mathbf{f}=\left(f_1,...,f_n\right)\in$ $L^{r,c_1}\left(\mathbb{R}\right)\times ...\times L^{r,c_n}\left(\mathbb{R}\right).$

Theorem 16.

Let $\frac{1}{q}+(n-1)=\frac{1}{q_1}+...+\frac{1}{q_n}$, $1<p<\infty$, $1\le q,q_j\le \infty$ $\left(j=1,...,n\right)$, $a\in {\mathbb{R}}_{+}^n$ and let $\Psi =\left({\Psi }_1,...,{\Psi }_n\right)\in S{\left(\mathbb{R}\right)}^n$ be multi-wavelets.

(i)Then,

$${∥W_{\Psi }^{\theta }\mathbf{f}\left(.,a\right)-W_{\Psi }^{\theta }\mathbf{g}\left(.,a\right)∥}_{L^{p,q}\left({\mathbb{R}}^n\right)}\lesssim {\left|a\right|}_n^{\frac{1}{2}}\stackrel{n}{\prod _{j=1}}{∥f_j-g_j∥}_{L^{p,q_j}\left(\mathbb{R}\right)}\stackrel{n}{\prod _{j=1}}{∥{\Psi }_j∥}_{L^1\left(\mathbb{R}\right)}$$

holds for all $\mathbf{f}=\left(f_1,...,f_n\right)$, $\mathbf{g}=\left(g_1,...g_n\right)\in L^{p,q_1}\left(\mathbb{R}\right)\times ...\times L^{p,q_n}\left(\mathbb{R}\right)$.

(ii)In addition if $\frac{1}{p}+1=\frac{1}{r}+\frac{1}{s}$, $\frac{1}{q_j}=\frac{1}{c_j}+\frac{1}{d_j}$ $\left(j=1,...,n\right)$, then,

$${∥W_{\Psi }^{\theta }\mathbf{f}\left(.,a\right)-W_{\Psi }^{\theta }\mathbf{g}\left(.,a\right)∥}_{L^{p,q}\left({\mathbb{R}}^n\right)}\lesssim {\left|a\right|}_n^{\frac{1}{s}-\frac{1}{2}}\stackrel{n}{\prod _{j=1}}{∥f_j-g_j∥}_{L^{r,c_j}\left(\mathbb{R}\right)}\stackrel{n}{\prod _{j=1}}{∥{\Psi }_j∥}_{L^{s,d_j}\left(\mathbb{R}\right)}$$

holds for all $\mathbf{f}=\left(f_1,...,f_n\right)$, $\mathbf{g}=\left(g_1,...g_n\right)\in$ $L^{r,c_1}\left(\mathbb{R}\right)\times ...\times L^{r,c_n}\left(\mathbb{R}\right)$.

Corollary 4.

Let $\frac{1}{q}+(n-1)=\frac{1}{q_1}+...+\frac{1}{q_n}$, $1<p<\infty$, $1\le q,q_j\le \infty$ $\left(j=1,...,n\right)$, $a\in {\mathbb{R}}_{+}^n$ and let $\Psi =\left({\Psi }_1,...,{\Psi }_n\right)$, ${\rm Y}=\left({\phi }_1,...,{\phi }_n\right)\in S{\left(\mathbb{R}\right)}^n$ be multi-wavelets.

(i)Then,

$${∥W_{\Psi }^{\theta }\mathbf{f}\left(.,a\right)-W_{{\rm Y}}^{\theta }\mathbf{g}\left(.,a\right)∥}_{L^{p,q}\left({\mathbb{R}}^n\right)}\lesssim {\left|a\right|}_n^{\frac{1}{2}}\left\{\begin{array}{c}{\displaystyle \prod _{j=1}^n}{∥f_j-g_j∥}_{L^{p,q_j}\left(\mathbb{R}\right)}{\displaystyle \prod _{j=1}^n}{∥{\Psi }_j∥}_{L^1\left(\mathbb{R}\right)}+\\ +{\displaystyle \prod _{j=1}^n}{∥g_j∥}_{L^{p,q_j}\left(\mathbb{R}\right)}{\displaystyle \prod _{j=1}^n}{∥{\Psi }_j-{\phi }_j∥}_{L^1\left(\mathbb{R}\right)}\end{array}\right\}$$

holds for all $\mathbf{f}=\left(f_1,...,f_n\right)$, $\mathbf{g}=\left(g_1,...g_n\right)\in L^{p,q_1}\left(\mathbb{R}\right)\times ...\times L^{p,q_n}\left(\mathbb{R}\right).$

(ii)In addition if $\frac{1}{p}+1=\frac{1}{r}+\frac{1}{s}$, $\frac{1}{q_j}=\frac{1}{c_j}+\frac{1}{d_j}$ $\left(j=1,...,n\right)$, then,

$${∥W_{\Psi }^{\theta }\mathbf{f}\left(.,a\right)-W_{{\rm Y}}^{\theta }\mathbf{g}\left(.,a\right)∥}_{L^{p,q}\left({\mathbb{R}}^n\right)}\lesssim {\left|a\right|}_n^{\frac{1}{s}-\frac{1}{2}}\left\{\begin{array}{c}{\displaystyle \prod _{j=1}^n}{∥f_j-g_j∥}_{L^{r,c_j}\left(\mathbb{R}\right)}{\displaystyle \prod _{j=1}^n}{∥{\Psi }_j∥}_{L^{s,d_j}\left(\mathbb{R}\right)}+\\ +{\displaystyle \prod _{j=1}^n}{∥g_j∥}_{L^{r,c_j}\left(\mathbb{R}\right)}{\displaystyle \prod _{j=1}^n}{∥{\Psi }_j-{\phi }_j∥}_{L^{s,d_j}\left(\mathbb{R}\right)}\end{array}\right\}$$

holds for all $\mathbf{f}=\left(f_1,...,f_n\right)$, $\mathbf{g}=\left(g_1,...g_n\right)\in$ $L^{r,c_1}\left(\mathbb{R}\right)\times ...\times L^{r,c_n}\left(\mathbb{R}\right)$.

5. Conclusions

This paper was motivated to define the multilinear fractional wavelet transform on cartesian product spaces by taking the multi-convolution of Schwartz functions and their dilations. At the same time, based on the tensor product of the fractional Fourier transforms, the multilinear fractional Fourier transform was derived on cartesian product spaces. Thus many properties of the fractional wavelet transform and the fractional Fourier transform were transferred to a multilinear form. Thanks to these new multilinear fractional transforms, it is possible to examine the signal representations and local structures of signals in the n-dimensional fractional space. These features will give you the chance to make these transforms valuable when working with signals in areas such as speech, vision, communication, signal processing, and radar.

In this paper, using multilinear interpolation techniques, we proved that the fractional wavelet transform defined on the cartesian product of Lebesgue spaces is bounded under some conditions for any multi-scaling parameter. We then investigated the boundedness of this transform on cartesian products of Lorentz spaces by using the property of being a tensor product operator. On the other hand, it is known that the Hausdorff–Young and Paley inequalities are fundamental results about the mapping properties of the Fourier transform on Lebesgue and Lorentz spaces. For this reason, we also proved the Hausdorff–Young inequality and Paley-type inequality for multilinear fractional Fourier transform, again using the multilinear interpolation technique. In our paper, the Hausdorff–Young inequality says us that the multilinear fractional Fourier transform maps $L^p{\left(\mathbb{R}\right)}^n$ continuously into $L^{p^{\prime }}\left({\mathbb{R}}^n\right)$, $1\le p\le 2$. The Paley type inequality also tells us that the multilinear fractional Fourier transform maps $L^{p,q}{\left(\mathbb{R}\right)}^n$ continuously into $L^{p^{\prime },q}\left({\mathbb{R}}^n\right)$, $\frac{1}{q}+(n-1)=\frac{1}{q_1}+...+\frac{1}{q_n}$, $1<p\le 2$, $1\le q,q_j\le \infty$, $\left(j=1,...,n\right)$. That means the multilinear fractional Fourier transform is extended to Lebesgue and Lorentz spaces. Thus, these new inequalities become valuable and fundamental results for Fourier analysis.