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Fractional integral operator on modulation and Wiener amalgam spaces
Journal of Inequalities and Applications volume 2015, Article number: 344 (2015)
Abstract
The purpose of this paper is to investigate the mapping properties of the fractional integral operators on weighted modulation spaces. Based on this result, we also study the boundedness of the bilinear fractional integral operators on product Wiener amalgam spaces. Our results show that, besides modulation spaces, Wiener amalgam spaces are good substitutions for Lebesgue spaces.
1 Introduction
Time-frequency analysis is a modern branch of harmonic analysis. It has many applications in signal analysis and wireless communication (see [1, 2]). Time-frequency analysis is a form of local Fourier analysis that treats time and frequency simultaneously. Inspired by this idea, modulation and Wiener amalgam spaces have been introduced and used to measure the time-frequency concentration of a function or a tempered distribution (see [3–7]). During the last ten years, these two function spaces have not only become useful function spaces for time-frequency analysis, they have also been employed to study boundedness properties of pseudo-differential operators, Fourier multipliers, Fourier integral operators, and well-posedness of solutions to PDEs. For more details of the applications of these two function spaces, the reader is referred to [8–19] and the references therein.
In this paper, we are mainly concerned with the mapping properties of the fractional integral operator on weighted modulation spaces. Using this result, we also prove the boundedness of the bilinear fractional integral operator on product Wiener amalgam spaces. From our results, we will see that, besides modulation spaces, Wiener amalgam spaces are good substitutions for Lebesgue spaces.
The fractional integral operator \(I_{\alpha}\) is defined by
where \(0<\alpha<n\). The well-known Hardy-Littlewood-Sobolev theorem says that \(I_{\alpha}\) is bounded from \(L^{p}(\mathbb{R}^{n})\) to \(L^{q}(\mathbb{R}^{n})\) for \(1< p< q<\infty\) and \(\frac{1}{q}=\frac{1}{p}-\frac{\alpha}{n}\). This theorem plays important roles in partial differential equations.
In recent years, many authors were interested in the mapping properties of the fractional integral operator on modulation spaces. For example, in [20], Tomita regarded the operator \(I_{\alpha}\) as a special case of pseudo-differential operator and proved the following results. Set \(0<\alpha<n\), \(0<\varepsilon<1-\frac{\alpha}{n}\), and \(0<\rho<\frac{\alpha}{n}\). If \(1< p_{1},p_{2},q_{1},q_{2}<\infty\) satisfy
then there exists a constant \(C_{\alpha,\varepsilon,\rho}>0\), such that
for all \(f\in\mathcal{S}(\mathbb{R}^{n})\).
Subsequently, Sugimoto and Tomita improved the results in [20] and obtained necessary and sufficient conditions for the boundedness of the operator \(I_{\alpha}\) (see [21]). Using the definitions of the discrete form for modulation spaces, which will be given by Definition 2.4 in the next section, they proved that \(I_{\alpha}\) is bounded from \(M^{p_{1},q_{1}}({\mathbb{R}}^{n})\) to \(M^{p_{2},q_{2}}({\mathbb{R}}^{n})\) if and only if
where \(0<\alpha<n\) and \(1< p_{1}, p_{2}, q_{1}, q_{2}<\infty\).
Recently, in [22], by using the norm of Hardy spaces \(H^{p}\), Chen and Zhong introduced the modulation Hardy spaces \(M^{p,q}\) for \(0< p\leq1\) and \(0< q\leq\infty\). They proved that if \(\frac {np}{n-\alpha p}\leq r\leq\infty\), then \(I_{\alpha}\) is bounded from \(M^{p,q}_{-\alpha}\) to \(M^{r,q}\) for \(0< p\leq1\) and \(0< q\leq\infty\). Moreover, for \(1< p<\infty\) and \(0< q_{1},q_{2}\leq\infty\), they obtained \(I_{\alpha}\) is bounded from \(M^{p,q_{1}}\) to \(M^{\infty, q_{2}}\) if and only if
In their proofs, Chen and Zhong also used the definition of the discrete form for modulation spaces.
Inspired by Sugimoto and Tomita, using the definition of integral form for modulation spaces, which will be given by Definition 2.3 in Section 2, we prove the following result.
Theorem 1.1
For \(0<\alpha<n\) and \(s_{1}, s_{2}\in\mathbb{R}\), let \(1< p_{1}, p_{2}<\infty\) and \(0< q_{1}, q_{2}\leq\infty\). If
then the fractional integral operator \(I_{\alpha}\) is bounded from \(M^{p_{1},q_{1}}_{s_{1}}({\mathbb{R}}^{n})\) to \(M^{p_{2},q_{2}}_{s_{2}}({\mathbb{R}}^{n})\).
Another purpose of this paper is to study the mapping properties of the bilinear fractional integral operator \(B_{\alpha}\), which is defined by
on modulation spaces. It was showed in [22] that if \(1< p<\frac {n}{n-\alpha}\) and \(0< q\leq\infty\), then
In particular, for \(p\leq q\),
Inspired by Chen and Zhong, in this paper, we investigate the boundedness of the bilinear fractional integral operator \(B_{\alpha}\) on Wiener amalgam spaces. Our result is as follows.
Theorem 1.2
For \(0<\alpha<n\), \(1< p_{0},p_{1}<\infty\), and \(1\leq q_{0},q_{1},p_{2},q_{2}<\infty\), let \(p'_{0}\) and \(q'_{0}\) denote the conjugate index of \(p_{0}\) and \(q_{0}\), respectively. Suppose \(q_{0}'\geq p_{0}'\), \(\frac{1}{p_{0}}\leq\frac{1}{p_{1}}-\frac {n-\alpha}{n}\), \(\frac{1}{q_{0}}<\frac{1}{q_{1}}+\frac{n-\alpha }{n}\), and \(\frac{1}{q_{0}'}+\frac{1}{p_{0}'}=\frac{1}{p_{2}}+\frac {1}{q_{2}}\), then the operator \(B_{\alpha}\) maps \(W(\mathcal{F}L^{p_{1}},L^{q_{1}})\times W(\mathcal {F}L^{p_{2}},L^{p_{0}'})\) to \(W(\mathcal{F}L^{q_{2}'},L^{q_{0}})\).
In what follows, we always denote C to be a positive constant that may be different at each place, but is independent of the essential variables.
This paper is organized as follows. In Section 2, we give the definitions and basic properties of modulation and Wiener amalgam spaces. Section 3 is devoted to the proofs of our main results.
2 Basic definitions and important lemmas
The following notations will be used throughout this paper. Let \(\mathcal{S}(\mathbb{R}^{n})\) be the Schwartz space of all complex-valued rapidly decreasing infinitely differentiable functions on \(\mathbb{R}^{n}\) and \(\mathcal{S}'(\mathbb{R}^{n})\) be the topological dual of \(\mathcal{S}(\mathbb{R}^{n})\). For a function f in \(\mathcal{S}(\mathbb{R}^{n})\), its Fourier transform is defined by \(\hat{f}(\omega)=\int f(t)e^{-2\pi i\omega\cdot t}\,dt\), and its inverse Fourier transform is \(\check{f}(t)=\hat{f}(-t)\). The translation and the modulation operators are defined by
for every \(x,\omega\in\mathbb{R}^{n}\). For \(s\in\mathbb{R}\) and \(x\in\mathbb{R}^{n}\), the weight function \(\langle x\rangle ^{s}=(1+|x|^{2})^{\frac{s}{2}}\).
Definition 2.1
Let g be a non-zero Schwartz function and \(1\leq p,q\leq\infty\) and \(s\in\mathbb{R}\), the weighted modulation space \(M^{p,q}_{s}(\mathbb{R}^{n})\) is defined as the closure of the Schwartz class with respect to the norm
with obvious modifications for p or \(q=\infty\), where \(V_{g}f(x,w)\) is the so-called short time Fourier transform (STFT), which is defined by
i.e. the Fourier transform \(\mathcal{F}\) applied to \(f\overline{T_{x}g}\).
Recently, the above definition has been generalized by Kobayashi in [23] to the case \(0< p,q\leq\infty\). In his definition, the function g is restricted to the space \(\Phi^{\delta}(\Bbb{R}^{n})\), which is defined as follows.
Definition 2.2
For \(\delta>0\), we define \(\Phi^{\delta}(\Bbb{R}^{n})\) to be the space of all \(g\in\mathcal{S}(\mathbb{R}^{n})\) satisfying
We may choose a sufficiently small δ, such that the function space \(\Phi^{\delta}(\Bbb{R}^{n})\) is not empty.
Definition 2.3
Given a \(g\in\Phi^{\delta}(\Bbb{R}^{n})\), and \(0< p,q\leq\infty\), \(s\in\Bbb{R}\), we define the modulation space \(M^{p,q}_{s}(\Bbb{R}^{n})\) to be the space of all tempered distributions \(f\in\mathcal{S'}(\mathbb {R}^{n})\) such that the quasi-norm
is finite, with obvious modifications for p or \(q=\infty\).
For a more general definition, involving different kinds of weight functions, both in the time and the frequency variables we refer the reader to [24]. Definitions 2.1 and 2.3 are the integral form. We also have the definition of discrete form for modulation spaces, which is very useful in studying unimodular Fourier multipliers.
Definition 2.4
Let \(1\leq p,q\leq\infty\), \(s\in\mathbb{R}\), and \(\phi\in\mathcal {S}(\mathbb{R}^{n})\) be such that
for all \(\xi\in\mathbb{R}^{n}\). Denote \(\phi_{k}(\xi)=\phi(\xi-k)\) and let \(\phi_{k}(D)\) be the Fourier multiplier operator given by \(\widehat{\phi_{k}(D)f}(\xi)=\phi_{k}(\xi)\hat{f}(\xi)\). Then the weighted modulation space \(M^{p,q}_{s}(\mathbb{R}^{n})\) consists of all \(f\in\mathcal{S}'(\mathbb {R}^{n})\) such that
with obvious modifications for p or \(q=\infty\).
There is yet another definition of modulation spaces, which is given by Gabor frames and plays a key role in studying simultaneously the local time and global frequency behavior of functions (see [24]).
The following basic properties of weighted modulation spaces, which play important roles in this article, can be found in [23, 25].
Lemma 2.5
Let \(0< p,q\leq\infty\) and \(g\in\Phi^{\delta}(\Bbb{R}^{n})\). Then
-
(1)
Different test functions \(g_{1}, g_{2}\in\Phi^{\delta}(\Bbb{R}^{n})\) define the same spaces and equivalent quasi-norms on \(M^{p,q}_{s}(\Bbb{R}^{n})\).
-
(2)
Let \(0< p_{0}\leq p_{1}\leq\infty\), \(0< q_{0}\leq q_{1}\leq\infty\) and \(s\in\Bbb{R}\), then
$$M^{p_{0},q_{0}}_{s}\bigl(\mathbb{R}^{n}\bigr) \hookrightarrow M^{p_{1},q_{1}}_{s}\bigl(\mathbb{R}^{n} \bigr). $$ -
(3)
If \(0< p,q<\infty\), then \(\mathcal{S}(\mathbb{R}^{n})\) is dense in \(M^{p,q}_{s}(\Bbb{R}^{n})\).
-
(4)
For \(1\leq p,q<\infty\) and \(s\in\Bbb{R}\), \((M^{p,q}_{s})'=M^{p',q'}_{-s}\).
To prove our main results, we also need the definition of the Wiener amalgam space \(W(\mathcal{F}L^{p},L^{q})(\mathbb{R}^{n})\).
For \(1\leq p\leq\infty\), let \(\mathcal{F}L^{p}(\mathbb{R}^{n})\) be the space of tempered distributions with their Fourier transforms in \(L^{p}(\mathbb{R}^{n})\), that is,
with norm \(\|f\|_{\mathcal{F}L^{p}}=\|\hat{f}\|_{L^{p}(\mathbb{R}^{n})}\).
Definition 2.6
For \(1\leq p,q<\infty\), a tempered distribution f is in the Wiener amalgam spaces \(W(\mathcal{F}L^{p},L^{q})(\mathbb{R}^{n})\), if f is locally in \(\mathcal{F}L^{p}(\mathbb{R}^{n})\), that is, for every non-zero \(g\in C_{0}^{\infty}(\mathbb{R}^{n})\), \(\mathcal{F}(fT_{x}g)\in L^{p}(\mathbb{R}^{n})\), and
is finite, with obvious modifications for p or \(q=\infty\). This definition is independent of the choice of the test function \(g\in C_{0}^{\infty}(\mathbb{R}^{n})\).
Both the modulation spaces and the Wiener amalgam spaces are mixed-norm function spaces. The following lemma gives the relationship between them.
Lemma 2.7
Let \(\mathcal{F}\) be the Fourier transform and \(1\leq p,q\leq\infty\), then \(M^{p,q}(\mathbb{R}^{n})= \mathcal{F} W(\mathcal {F}L^{p}, L^{q})(\mathbb{R}^{n})\).
Proof
Choose \(g\in\mathcal{S}(\mathbb{R}^{n})\) such that \(g\neq0\) and \(f\in \mathcal{S}'(\mathbb{R}^{n})\). For every \(x, \xi\in\mathbb{R}^{n}\), the definition of the short time Fourier transform implies that
and
Therefore, Definitions 2.1 and 2.6 yield
The proof of Lemma 2.7 is completed. □
3 Proof of the main results
In this section, we are going to prove our main results. First, we show the proof of Theorem 1.1.
Set \(K_{\alpha}(x)=\frac{1}{\gamma(\alpha)}\frac{1}{|x|^{n-\alpha}}\), where \(\gamma(\alpha)=\frac{\pi^{\frac{n}{2}}\Gamma(\frac{\alpha }{2})2^{\alpha}}{\Gamma(\frac{n-\alpha}{2})}\). The fractional integral operator \(I_{\alpha}\), which is defined by \(I_{\alpha}(f)(x)=(K_{\alpha}\ast f)(x)\), may be realized on the transform side as a Fourier multiplier
where \(m_{\alpha}(\xi)=\widehat{K_{\alpha}}(\xi)\).
Proof
We consider the following three cases to obtain Theorem 1.1.
Case 1: \(\frac{1}{p_{2}}=\frac{1}{p_{1}}-\frac{\alpha}{n}\) and \(q_{1}>q_{2}\).
Let \(g, \chi\in\Phi^{\delta}({\mathbb{R}}^{n})\) satisfy the condition \(g=\chi\ast\chi\). Then \(M_{\xi}g=M_{\xi}\chi\ast M_{\xi}\chi\). Young’s inequalities and the Hardy-Littlewood-Sobolev theorem give
Using the Hölder inequalities for the exponent \(\frac {q_{1}}{q_{2}}\) and \((\frac{q_{1}}{q_{2}} )'\), we get
Denote
Since \(s_{2}< s_{1}+n (\frac{1}{q_{1}}-\frac{1}{q_{2}} )\), an easy computation shows that
Therefore,
Case 2: \(\frac{1}{p_{2}}=\frac{1}{p_{1}}-\frac{\alpha}{n}\) and \(q_{1}\leq q_{2}\). Since \(0<\frac{\alpha}{n}<1\), we can choose \(0<\widetilde{q_{2}}\leq\infty\) such that \(q_{1}>\widetilde{q_{2}}\) and \(\frac{1}{\widetilde{q_{2}}}<\frac{1}{q_{1}}+\frac{\alpha}{n}\). According to the proof of case 1, we can see that \(I_{\alpha}\) is bounded from \(M^{p_{1},q_{1}}_{s_{1}}\) to \(M^{p_{2},\tilde{q_{2}}}_{s_{2}}\). By the condition \(q_{2}\geq q_{1}>\tilde{q_{2}}\) and the fact \(M_{s_{2}}^{p_{2},\widetilde{q_{2}}}\hookrightarrow M_{s_{2}}^{p_{2},{q_{2}}}\), the proof of Case 2 is completed.
Case 3: \(\frac{1}{p_{2}}<\frac{1}{p_{1}}-\frac{\alpha}{n}\). In this case, \(0<\frac{1}{p_{1}}-\frac{\alpha}{n}<1\). Take \(1<\widetilde{p_{2}}<\infty\) such that \(\frac{1}{\widetilde{p_{2}}}=\frac{1}{p_{1}}-\frac{\alpha}{n}\), then \(p_{2}>\widetilde{p_{2}}\). Using the embedding result \(M_{s_{2}}^{\widetilde{p_{2}},{q_{2}}}\hookrightarrow M_{s_{2}}^{p_{2},{q_{2}}}\) and the proof of Case 1 and 2, we finish the proof of Theorem 1.1. □
Now, we turn our attention to the proof of Theorem 1.2. From a Fourier point of view, the bilinear fractional integral operator can be rewritten as
where
For the proof of Theorem 1.2, we need some lemmas.
Lemma 3.1
Let \(H(t,y)=e^{-2\pi it\cdot y}\mathcal{F}^{-1}f(y)\) and \(\widehat {H_{2}}\) be the Fourier transform for the second variable of H, then \(\widehat{H_{2}}(t,\xi)=f(t+\xi)\) and \(F(t,2\xi)=2^{-\alpha}I_{n-\alpha}(H(t,\cdot))(\xi)\).
Proof
We only prove the second equality. The first one is very easy, we omit the details here. Note that if we set \(y'=2y\), \(F(t,2\xi)\) can be rewritten as
We finish the proof of Lemma 3.1. □
Lemma 3.2
Denote \(F^{2}_{2}(t,\xi)=F(t,2\xi)\), then \(\|F^{2}_{2}\| _{M^{p,q}}=C_{n,p,q}\|F\|_{M^{p,q}}\), where \(C_{n,p,q}=2^{-n(1+\frac {1}{p}-\frac{1}{q})}\).
Proof
Taking \(g\in\mathcal{S}({\mathbb{R}}^{2n})\setminus\{0\}\), then we have
Set \(t'=2t\), then
where \(\phi(s,t')=g(s,\frac{t'}{2})\in\mathcal{S}({\mathbb {R}}^{2n})\setminus\{0\}\) is another window function. Definition 2.1 gives
Let \(\xi'=2\xi\), \(\eta'=\frac{\eta}{2}\). It is easy to check that
We complete the proof of Lemma 3.2. □
Lemma 3.3
Suppose \(\phi_{0}\), \(\phi_{1}\), g, h to be the non-zero Schwartz functions and \(W(g,h)(t,s)=\overline{g(t)}h(s)e^{2\pi its}\), then
Proof
The definition of the short time Fourier transform yields
The proof of Lemma 3.3 is completed. □
Lemma 3.4
For \(1< p_{0}<\infty\) and \(1\leq q_{0}, p_{2}, q_{2}<\infty\), let \(p'_{0}\) and \(q'_{0}\) denote the conjugate index of \(p_{0}\) and \(q_{0}\), respectively. If \(q_{0}'\geq p_{0}'\) and \(\frac{1}{q_{0}'}+\frac{1}{p_{0}'}=\frac{1}{p_{2}}+\frac{1}{ q_{2}}\), then \(W(g,h)\) defined in Lemma 3.3 is bounded from \(W(\mathcal {F}L^{p_{2}},L^{p_{0}'})\times M^{q_{2},q_{0}'}\) to \(M^{p_{0}',q_{0}'}\).
Proof
For each \(g\in W(\mathcal{F}L^{p_{2}},L^{p_{0}'})\) and \(h\in M^{q_{2},q_{0}'}\), Lemma 3.3 indicates
Note that \(V_{\phi_{0}}g(u_{0},u-v_{0})=e^{-2\pi iu_{0}(u-v_{0})}V_{\widehat{\phi_{0}}}\hat{g}(u-v_{0},-u_{0})\). If we denote \(P(x,y)=\widetilde{P(-x,y)}\), then
Since \(\frac{1}{q_{0}'}+\frac{1}{p_{0}'}=\frac{1}{p_{2}}+\frac{1}{q_{2}}\), using Minkowski’s integral inequalities, Young’s inequalities, and Lemma 2.7, we obtain
The proof of Lemma 3.4 is completed. □
We give the proof of Theorem 1.2.
Proof
Choose \(h\in\mathcal{S}({\mathbb{R}}^{n})\setminus\{0\}\) and denote by \(\langle \cdot,\cdot\rangle \) the inner product, then
where \(W(g,h)=\overline{g(t)}h(\xi)e^{2\pi it\xi}\). Hölder’s inequalities show
Lemma 3.1, Lemma 3.2, and Theorem 1.1 yield
On the other hand, Lemma 3.4 gives
Therefore,
which implies \(\widehat{B_{\alpha}(f,g)}\in M^{q_{2}',q_{0}}\) and
Lemma 2.7 indicates \(\|B_{\alpha}(f,g)\|_{W(\mathcal {F}L^{q_{2}'},L^{q_{0}})}=\|\widehat{B_{\alpha}(f,g)}\| _{M^{q_{2}',q_{0}}}\), we conclude
The proof of Theorem 1.2 is finished. □
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Acknowledgements
This work is supported by the National Nature Science Foundation of China (No. 11201003) and NNSF (No. KJ2014A087) of Anhui Province in China.
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Cheng, M., Tao, W. Fractional integral operator on modulation and Wiener amalgam spaces. J Inequal Appl 2015, 344 (2015). https://doi.org/10.1186/s13660-015-0872-4
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DOI: https://doi.org/10.1186/s13660-015-0872-4