Elena Cordero, Fabio Nicola
We prove sharp estimates for the dilation operator $f(x)\longmapsto f(λx)$, when acting on Wiener amalgam spaces $W(L^p,L^q)$. Scaling arguments are also used to prove the sharpness of the known convolution and pointwise relations for modulation spaces $M^{p,q}$, as well as the optimality of an estimate for the Schrödinger propagator on modulation spaces.
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In this paper, we study the boundedness of the Schrödinger operator $e^{i Δ}$ on Wiener amalgam spaces and determine its optimal condition.
In this paper various properties of global and local changes of variables as well as properties of canonical transforms are investigated on modulation and Wiener amalgam spaces. We establish several relations among localisations of modulation and Wiener amalgam spaces and, as a consequence, we obtain several versions of local and global Beurling-Helson type theorems. We also establish a number of positive results such as local boundedness of canonical transforms on modulation spaces, properties of homogeneous changes of variables, and local continuity of Fourier integral operators on Fourier L…
We give a complete characterization of the continuity of pseudodifferential operators with symbols in modulation spaces $M^{p,q}$, acting on a given Lebesgue space $L^r$. Namely, we find the full range of triples $(p,q,r)$, for which such a boundedness occurs. More generally, we completely characterize the same problem for operators acting on Wiener amalgam space $W(L^r,L^s)$ and even on modulation spaces $M^{r,s}$. Finally the action of pseudodifferential operators with symbols in $W(\Fur L^1,L^\infty)$ is also investigated.
The paper deals with trace operators of Wiener amalgam spaces using frequency-uniform decomposition operators and maximal inequalities, obtaining sharp results. Additionally, we provide the embeddings between standard and anisotropic Wiener amalgam spaces.
In this article, notations are included in Section 1. In Section 2, we define the grand Wiener amalgam space by using the classical Wiener amalgam space [9, 15, 16, 17] and the generalized grand Lebesgue space [18, 13] . Section 3, concerns the inclusions between these spaces and some applications. In last section Section 4, we prove the Holders inequality for grand Wiener amalgam space. We also find the associate space and dual of this space, and we prove that the grand Wiener amalgam space is not reflexive.
We define Wiener amalgam spaces of (quasi)analytic ultradistributions whose local components belong to a general class of translation and modulation invariant Banach spaces of ultradistributions and their global components are either weighted $L^p$ or weighted $\mathcal{C}_0$ spaces. We provide a discrete characterisation via so called uniformly concentrated partitions of unity. Finally, we study the complex interpolation method and we identify the strong duals for most of these Wiener amalgam spaces.
This paper provides sufficient conditions for the boundedness of Weyl operators on modulation spaces. The Weyl symbols belong to Wiener amalgam spaces, or generalized modulation spaces, as recently renamed by their inventor Hans Feichtinger. This is the first result which relates symbols in Wiener amalgam spaces to operators acting on classical modulation spaces.
Dilation estimates for Wiener amalgam spaces of Orlicz type
本文研究Orlicz型Wiener amalgam空间的伸缩性质，将Lebesgue情形推广到Orlicz空间，并证明Zak变换在相应空间上的连续性。
We generalize the theory of Wiener amalgam spaces on locally compact groups to quasi-Banach spaces. As a main result we provide convolution relations for such spaces. Also we weaken the technical assumption that the global component is invariant under right translations, which is even new for the classical Banach space case. To illustrate our theory we discuss in detail an example on the $ax+b$ group.
In this paper we focus on the almost-diagonalization properties of $τ$-pseudodifferential operators using techniques from time-frequency analysis. Our function spaces are modulation spaces and the special class of Wiener amalgam spaces arising by considering the action of the Fourier transform of modulation spaces. A particular example is provided by the Sjöstrand class, for which Gröchenig exhibited the almost diagonalization of Weyl operators. We shall show that such result can be extended to any $τ$-pseudodifferential operator, for $τ\in [0,1]$, also with symbol in weighted Wiener amalgam s…
Wiener-Luxemburg amalgam spaces
本文提出Wiener-Luxemburg amalgam空间，改进经典Wiener amalgam空间在重排不变Banach函数空间中的缺陷，研究其性质、嵌入与伴随空间，并应用该理论解决Hardy-Littlewood-Pólya原理对拟Banach函数空间是否成立的问题。
In this paper we show that functions from the Paley-Wiener amalgam space $(PW,l^1)=\{f\in L^2(\mathbb{R}): \sum\|\hat{f}(ξ+2πm) \|_{L^2([-π,π])} < \infty\}$ enjoy similar recovery properties as the classical Paley-Wiener space. Specifically, if $\{ϕ_α(x): α\in A\}$ is a regular family of interpolators and $\{x_n: n\in \mathbb{Z}\}$ is a complete interpolating sequence for $L^2([-π,π])$, then the family $\{ e^{2πi m x}ϕ_α(x-x_n): m,n\in \mathbb{Z}, α\in A \} $ may be used to recover $f\in(PW,l^1)$.
Wiener--Luxemburg amalgam spaces revisited
本文针对 Wiener–Luxemburg 融合空间仅适用于重排不变空间的问题，构建通用框架将其拓展至任意拟Banach函数空间，构造新型拟范数解决结构难题并完成相关性质刻画。
We use a generalization of Wiener's $1/f$ theorem to prove that for a Gabor frame with the generator in the Wiener amalgam space $W(L^{\infty}, \ell^{1}_ν)(\mathbb{R}^{d})$, the corresponding frame operator is invertible on this space. Therefore, for such a Gabor frame, the generator of the canonical dual belongs also to $W(L^{\infty}, \ell^{1}_ν)(\mathbb{R}^{d}) $
It is well known that the Gabor expansions converge to identity operator in weak* sense on the Wiener amalgam spaces as sampling density tends to infinity. In this paper we prove the convergence of Gabor expansions to identity operator in the operator norm as well as weak* sense on $W(L^p, \ell^q)$ as the sampling density tends to infinity. Also we show the validity of the Janssen's representation and the Wexler-Raz biorthogonality condition for Gabor frame operator on $W(L^p, \ell^q)$.
We study the action of metaplectic operators on Wiener amalgam spaces, giving upper bounds for their norms. As an application, we obtain new fixed-time estimates in these spaces for Schrödinger equations with general quadratic Hamiltonians and Strichartz estimates for the Schrödinger equation with potentials $V(x)=\pm|x|^2$.
We establish the sharp conditions for the embedding between Wiener amalgam spaces $W_{p,q}^s$ and some classical spaces, including Sobolev spaces $L^{s,r}$, local Hardy spaces $h_{r}$, Besov spaces $B_{p,q}^s$, which partially improve and extend the main result obtained by Guo et al. in J. Funct. Anal., 273(1):404-443, 2017. In addition, we give the full characterization of inclusion between Wiener amalgam spaces $W_{p,q}^s$ and $α$-modulation spaces $M_{p,q}^{s,α}$. Especially, in the case of $α=0$ with $M_{p,q}^{s,α} = M_{p,q}^s$, we give the sharp conditions of the most general case of thes…
We study the local-in-time regularity of the Brownian motion with respect to localized variants of modulation spaces M^{p, q}_s and Wiener amalgam spaces W^{p, q}_s. We show that the periodic Brownian motion belongs locally in time to M^{p, q}_s (T) and W^{p, q}_s (T) for (s-1)q < -1, and the condition on the indices is optimal. Moreover, with the Wiener measure μon T, we show that (M^{p, q}_s (T), μ) and (W^{p, q}_s (T), μ) form abstract Wiener spaces for the same range of indices, yielding large deviation estimates. We also establish the endpoint regularity of the periodic Brownian motion wi…
We show that multi-window Gabor frames with windows in the Wiener algebra $W(L^{\infty}, \ell^{1})$ are Banach frames for all Wiener amalgam spaces. As a byproduct of our results we positively answer an open question that was posed by [Krishtal and Okoudjou, Invertibility of the Gabor frame operator on the Wiener amalgam space, J. Approx. Theory, 153(2), 2008] and concerns the continuity of the canonical dual of a Gabor frame with a continuous generator in the Wiener algebra. The proofs are based on a recent version of Wiener's $1/f$ lemma.
We study continuity properties on modulation spaces for $τ$-pseudodifferential operators with symbols $a$ in Wiener amalgam spaces. We obtain boundedness results for $τ\in (0,1)$ whereas, in the end-points $τ=0$ and $τ=1$, the corresponding operators are in general unbounded. Furthermore, for $τ\in (0,1)$, we exhibit a function of $τ$ which is an upper bound for the operator norm. The continuity properties of $τ$-pseudodifferential operators, for any $τ\in [0,1]$, with symbols $a$ in modulation spaces are well known. Here we find an upper bound for the operator norm which does not depend on th…
In this paper we first prove the metric approximation property for weighted mixed-norm $L_w^{(p_1,\dots ,p_n)}$ spaces. Using Gabor frame representation this implies that the same property holds in weighted modulation and Wiener amalgam spaces. As a consequence, Grothendieck's theory becomes applicable, and we give criteria for nuclearity and $r$-nuclearity for operators acting on these space as well as derive the corresponding trace formulae. Finally, we apply the notion of nuclearity to functions of the harmonic oscillator on modulation spaces.
The family of harmonic Hilbert spaces is a natural enlargement of those classical $L^{2}$-Sobolev space on $\R^d$ which consist of continuous functions. In the present paper we demonstrate that the use of basic results from the theory of Wiener amalgam spaces allows to establish fundamental properties of harmonic Hilbert spaces even if they are defined over an arbitrary locally compact abelian group $\G$. Even for $\G = \R^{d}$ this new approach improves previously known results. In this paper we present results on minimal norm interpolators over lattices and show that the infinite minimal nor…
The Generalized Grand Wiener Amalgam Spaces and the boundedness of Hardy-Littlewood maximal operators
本文将 grand Wiener 融合空间推广为基于广义 grand Lebesgue 空间的广义 grand Wiener 融合空间，研究其基本性质与嵌入关系，并讨论 Hardy-Littlewood 极大算子在相应空间之间的有界性与无界性。
Characterization of boundedness on Wiener amalgam spaces of multilinear Rihaczek distributions
本文刻画了多线性 Rihaczek 分布在 Wiener amalgam 空间到调制及 Fourier 调制空间上的有界性，建立自提升性质并给出典型情形的精确指数，同时推广了相应伪微分算子的有界性结果。
In the last decade it has become clear that one of the central themes within Gabor analysis (with respect to general time-frequency lattices) is a duality theory for Gabor frames, including the Wexler-Raz biorthogonality condition, the Ron-Shen's duality principle or Janssen's representation of a Gabor frame operator. All these results are closely connected with the so-called {\it Fundamental Identity of Gabor Analysis}, which we derive from an application of Poisson's summation formula for the symplectic Fourier transform. The new aspect of this presentation is the description of range of the…
The Concept of Wiener Amalgam Spaces
本文回顾 Wiener 融合空间的基本性质，展示其在调和分析中的广泛应用，包括改进 Hausdorff-Young 定理、证明 Sobolev 嵌入定理以及确定 Sobolev 代数的点态乘子。
We characterize the dual spaces of the generalized Hardy spaces defined by replacing Lebesgue quasi-norms by Wiener amalgam ones. In these generalized Hardy spaces, we prove that some classical linear operators such as Calderón-Zygmund, convolution and Riesz potential operators are bounded.
On the sums of rearrangement-invariant quasi-Banach function spaces and their relationship to amalgams
本文研究了重排不变拟Banach函数空间和的重排不变性与Fatou性质，通过弱化Luxemburg型表示证明和的拟范数等价性，并刻画其与Wiener--Luxemburg混合的关系，给出该和构成重排不变空间的充分条件。
Let $\mathbb{H}$ be a separable Hilbert space. In this paper we establish a generalization of Walnut's representation and Janssen's representation of the $\mathbb{H}-$valued Gabor frame operator on $\mathbb{H}-$valued weighted amalgam spaces $W_{\mathbb{H}}(L^p,L^q_v)$, $1 \leq p, q \leq \infty$. Also we show that the frame operator is invertible on $W_{\mathbb{H}}(L^p,L^q_v)$, $1 \leq p, q \leq \infty$, if the window function is in the Wiener amalgam space $W_{\mathbb{H}}(L^{\infty},L^1_w)$. Further, we obtain the Walnut representation and invertibility of the frame operator corresponding to …
This paper is devoted to give several characterizations on a more general level for the boundedness of $τ$-Wigner distributions acting from weighted modulation spaces to weighted modulation and Wiener amalgam spaces. As applications, sharp exponents are obtained for the boundedness of $τ$-Wigner distributions on modulation spaces with power weights. We also recapture the main theorems of Wigner distribution obtained in \cite{CorderoNicola2018IMRNI,Cordero2020a}. As consequences, the characterizations of the boundedness on weighted modulation spaces of several types of pseudodifferential operat…
We study decay and smoothness properties for eigenfunctions of compact localization operators. Operators with symbols a in the wide modulation space M^{p,\infty} (containing the Lebesgue space L^p), p<\infty, and windows \f_1,\f_2 in the Schwartz class are known to be compact. We show that their L^2-eigenfuctions with non-zero eigenvalues are indeed highly compressed onto a few Gabor atoms. Similarly, for symbols a in the weighted modulation spaces M^{\infty}_{v_s\otimes 1} (\rdd), s>0 (subspaces of M^{p,\infty}(\rdd), p>2d/s) the L^2-eigenfunctions of the localization operator are actually Sc…
On the weighted Wiener-Lévy theorem: analogue on Euclidean space, strong converse on LCA group and applications to modulation spaces
本文在加权 Fourier 代数中证明了 Wiener-Lévy 定理的强逆定理及其在欧几里得空间和局部紧阿贝尔群上的类似形式，并应用于调制空间等以揭示色散 PDE 动力学中的非线性特征。
We give sufficient conditions on the Lebesgue exponents for compositions of odd numbers of pseudo-differential operators with symbols in modulation spaces. As a byproduct, we obtain sufficient conditions for twisted convolutions of odd numbers of factors to be bounded on Wiener amalgam spaces.
It is the purpose of this paper to give a characterization of the pre-dual of the spaces introduced by I.~Fofana on the basis of Wiener amalgam spaces. Those spaces have a specific dilation behaviour similar to the spaces $L^α(\mathbb R^d)$. The characterization of the pre-dual will be based on the idea of minimal invariant spaces (with respect to such a group of dilation operators).
A Christ-Kiselev maximal theorem in quasi-Banach function lattices
本文在拟Banach函数格中证明Christ-Kiselev极大定理，并推广至加权Lorentz空间、经典Lorentz空间、Wiener amalgam空间、Köthe对偶算子及插值空间，应用于极大Fourier算子。
Factorizations for quasi-Banach time-frequency spaces and Schatten classes
本文推广 Hewitt 的分解定理，建立具有有界左近似单位元的拟 Banach 代数上模的分解性质，并应用于 Wiener amalgam、modulation 空间及 Schatten 符号类。
We study the continuity on the modulation spaces $M^{p,q}$ of Fourier multipliers with symbols of the type $e^{iμ(ξ)}$, for some real-valued function $μ(ξ)$. A number of results are known, assuming that the derivatives of order $\geq 2$ of the phase $μ(ξ)$ are bounded or, more generally, that its second derivatives belong to the Sjöstrand class $M^{\infty,1}$. Here we extend those results, by assuming that the second derivatives lie in the bigger Wiener amalgam space $W(\mathcal{F} L^1,L^\infty)$; in particular they could have stronger oscillations at infinity such as $\cos |ξ|^2$. Actually ou…
It is known that Fourier integral operators arising when solving Schrödinger-type operators are bounded on the modulation spaces $\cM^{p,q}$, for $1\leq p=q\leq\infty$, provided their symbols belong to the Sjöstrand class $M^{\infty,1}$. However, they generally fail to be bounded on $\cM^{p,q}$ for $p\not=q$. In this paper we study several additional conditions, to be imposed on the phase or on the symbol, which guarantee the boundedness on $\cM^{p,q}$ for $p\not=q$, and between $\cM^{p,q}\to\cM^{q,p}$, $1\leq q< p\leq\infty$. We also study similar problems for operators acting on Wiener amalg…
We give sufficient and necessary conditions on the Lebesgue exponents for the Weyl product to be bounded on modulation spaces. The sufficient conditions are obtained as the restriction to $N=2$ of a result valid for the $N$-fold Weyl product. As a byproduct, we obtain sharp conditions for the twisted convolution to be bounded on Wiener amalgam spaces.
We give a new class of equivalent norms for modulation spaces by replacing the window of the short-time Fourier transform by a Hilbert-Schmidt operator. The main result is applied to Cohen's class of time-frequency distributions, Weyl operators and localization operators. In particular, any positive Cohen's class distribution with Schwartz kernel can be used to give an equivalent norm for modulation spaces. We also obtain a description of modulation spaces as time-frequency Wiener amalgam spaces. The Hilbert-Schmidt operator must satisfy a nuclearity condition for these results to hold, and we…
In this note we present the metric approximation property for weighted mixed-norm $L_w^{(p_1,\dots ,p_n)}$ and variable exponent Lebesgue type spaces. As a consequence, this also implies the same property for modulation and Wiener-Amalgam spaces. We then characterise nuclear operators on such spaces and state the corresponding Grothendieck-Lidskii trace formulae. We apply the obtained results to derive criteria for nuclearity and trace formulae for periodic operators on $\mathbb R^n$ and functions of the harmonic oscillator in terms of the global symbol.
Born-Jordan operators are a class of pseudodifferential operators arising as a generalization of the quantization rule for polynomials on the phase space introduced by Born and Jordan in 1925. The weak definition of such operators involves the Born-Jordan distribution, first introduced by Cohen in 1966 as a member of the Cohen class. We perform a time-frequency analysis of the Cohen kernel of the Born -Jordan distribution, using modulation and Wiener amalgam spaces. We then provide sufficient and necessary conditions for Born-Jordan operators to be bounded on modulation spaces. We use modulati…
Let $\ell^p, 1\le p\le \infty$, be the space of all $p$-summable sequences and $C_a$ be the convolution operator associated with a summable sequence $a$. It is known that the $\ell^p$- stability of the convolution operator $C_a$ for different $1\le p\le \infty$ are equivalent to each other, i.e., if $C_a$ has $\ell^p$-stability for some $1\le p\le \infty$ then $C_a$ has $\ell^q$-stability for all $1\le q\le \infty$. In the study of spline approximation, wavelet analysis, time-frequency analysis, and sampling, there are many localized operators of non-convolution type whose stability is one of …
Modulation spaces: from Feichtinger's original definition to modern Gabor and symplectic characterizations
本文综述调制空间从Feichtinger原始定义到现代辛刻画的演进，强调统一局部-全局原理，并展示亚辛Wigner分布与短时傅里叶变换的结构联系及其在非线性色散方程中的应用。
Norm estimates for a broad class of modulation spaces, and continuity of Fourier type operators
本文针对一类广泛调制空间建立范数等价刻画，证明其与 Wiener 融合空间范数等价，并由此推导出符号在加权 M^{∞,r_0} 空间中的伪微分算子作用于该类调制空间时的连续性。
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