Mitsuru Sugimoto, Naohito Tomita
We prove that pseudo-differential operators with symbols in the class $S_{1,δ}^0$ ($0<δ<1$) are not always bounded on the modulation space $M^{p,q}$ ($q\neq2$).
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In this paper, we consider the $L^2$-boundedness of pseudo-differential operators with symbols in $α$-modulation spaces.
Let $\mathcal D$ be a Schauder decomposition on some Banach space $X$. We prove that if $\mathcal D$ is not $R$-Schauder, then there exists a Ritt operator $T\in B(X)$ which is a multiplier with respect to $\mathcal D$, such that the set $\{T^n\, :\, n\geq 0\}$ is not $R$-bounded. Likewise we prove that there exists a bounded sectorial operator $A$ of type $0$ on $X$ which is a multiplier with respect to $\mathcal D$, such that the set $\{e^{-tA}\, : \, t\geq 0\}$ is not $R$-bounded.
In this paper, we study the boundedness of pseudo-differential operators with symbols in $S_{ρ,δ}^m$ on the modulation spaces $M^{p,q}$. We discuss the order $m$ for the boundedness $\mathrm{Op}(S_{ρ,δ}^m) \subset \calL(M^{p,q}(\R^n))$ to be true. We also prove the existence of a Calderón-Zygmund operator which is not bounded on the modulation space $M^{p,q}$ with $q \neq 2$. This unboundedness is still true even if we assume a generalized T(1) condition. These results are induced by the unboundedness of pseudo-differential operators on $M^{p,q}$ whose symbols are of the class $S_{1,δ}^0$ with…
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…
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.
Boundedness results for multilinear pseudodifferential operators on products of modulation spaces are derived based on ordered integrability conditions on the short-time Fourier transform of the operators' symbols. The flexibility and strength of the introduced methods is demonstrated by their application to the bilinear and trilinear Hilbert transform.
We introduce new classes of modulation spaces over phase space. By means of the Kohn-Nirenberg correspondence, these spaces induce norms on pseudo-differential operators that bound their operator norms on $L^p$-spaces, Sobolev spaces, and modulation spaces.
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…
In this paper, we give the necessary and sufficient conditions for the boundedness of fractional integral operators on the modulation spaces.
Boundedness and compactness kernel theorem for $α$-modulation spaces
本文建立了α-调制空间的核定理，用分布核在混合α-调制空间中的隶属关系刻画线性算子的有界性，并用闭子空间中的核刻画紧性。
We prove the boundedness of a general class of multipliers and Fourier multipliers, in particular of the Hilbert transform, on quasi-Banach modulation spaces. We also deduce boundedness for multiplications and convolutions for elements in such spaces.
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。
We extended the known result that symbols from modulation spaces $M^{\infty,1}(\mathbb{R}^{2n})$, also known as the Sjöstrand's class, produce bounded operators in $L^2(\mathbb{R}^n)$, to general $L^p$ boundedness at the cost of lost of derivatives. Indeed, we showed that pseudo-differential operators acting from $L^p$-Sobolev spaces $L^p_s(\mathbb{R}^n)$ to $L^p(\mathbb{R}^n)$ spaces with symbols from the modulation space $M^{\infty,1}(\mathbb{R}^{2n})$ are bounded, whenever $s\geq n|1/p-1/2|.$ This estimate is sharp for all $1\leq p\leq\infty$.
We investigate the boundedness of unimodular Fourier multipliers on modulation spaces. Surprisingly, the multipliers with general symbol $e^{i|ξ|^α}$, where $α\in[0, 2]$, are bounded on all modulation spaces, but, in general, fail to be bounded on the usual $L^p$-spaces. As a consequence, the phase-space concentration of the solutions to the free Schrödinger and wave equations are preserved. As a byproduct, we also obtain boundedness results on modulation spaces for singular multipliers $|ξ|^{-δ} \sin (|ξ|^α)$ for $ 0 \leq δ\leq α$.
We study the boundedness of Toeplitz-type operators defined in the context of the Calderón reproducing formula considering the specific wavelets whose Fourier transforms are related to Laguerre polynomials. Some sufficient conditions for simultaneous boundedness of these Calderón-Toeplitz operators on each wavelet subspace for unbounded symbols are given, where investigating the behavior of certain sequence of iterated integrals of symbols is helpful. A number of examples and counterexamples is given.
We consider a class of Fourier integral operators, globally defined on $\mathbb{R}^{d}$, with symbols and phases satisfying product type estimates (the so-called $SG$ or scattering classes). We prove a sharp continuity result for such operators when acting on the modulation spaces $M^p$. The minimal loss of derivatives is shown to be $d|1/2-1/p|$. This global perspective produces a loss of decay as well, given by the same order. Strictly related, striking examples of unboundedness on $L^p$ spaces are presented.
Boundedness of composition operators from Lorentz spaces to Orlicz spaces
本研究针对函数空间间复合算子有界性研究成果较少的问题，给出了从Lorentz空间到Orlicz空间的复合算子有界性的充要条件，还提供了相关反例及更多满足主结果条件的复合映射实例。
We study the notion of molecules in coorbit spaces. The main result states that if an operator, originally defined on an appropriate space of test functions, maps atoms to molecules, then it can be extended to a bounded operator on coorbit spaces. For time-frequency molecules we recover some boundedness results on modulation spaces, for time-scale molecules we obtain the boundedness on homogenous Besov spaces.
In this paper, we study the boundedness of pseudodifferential operators with symbols in the Hörmander class $S^0_{ρ,ρ}$ on $α$-modulation spaces $M_{p,q}^{s,α}$, and consider the relation between $α$ and $ρ$. In particular, we show that pseudodifferential operators with symbols in $S^0_{α,α}$ are bounded on all $α$-modulation spaces $M^{s,α}_{p,q}$, for arbitrary $s\in\mathbb{R}$ and for the whole range of exponents $0 < p,q \leq \infty$.
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.
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…
In this paper, we establish the asymptotic estimates for the norms of the matrix dilation operators on modulation spaces. As an application, we study the boundedness on modulation spaces of Hausdorff operators. The definition of Hausdorff operators are also revisited for fitting our study.
In this article, we study properties of multilinear Fourier integral operators on weighted modulation spaces. In particular, using the theory of Gabor frames, we study boundedness of multilinear Fourier integral operators on products of weighted modulation spaces. Further, we investigate the periodic multilinear Fourier integral operator. Finally, we study continuity of bilinear pseudo-differential operators on modulation spaces for certain symbol classes, namely $\textbf{SG}$-class.
We study the existence of the product of two weighted modulation spaces. For this purpose we discuss two different strategies. The more simple one allows transparent proofs in various situations. However, our second method allows a closer look onto associated norm inequalities under restrictions in the Fourier image. This will give us the opportunity to treat the boundedness of composition operators.
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…
We introduce new quasi-Banach modulation spaces on locally compact abelian (LCA) groups which coincide with the classical ones in the Banach setting and prove their main properties. Then we study Gabor frames on quasi-lattices, significantly extending the original theory introduced by Gröchenig and Strohmer. These issues are the key tools in showing boundedness results for Kohn-Nirenberg and localization operators on modulation spaces and studying their eigenfunctions' properties. In particular, the results in the Euclidean space are recaptured.
We prove that if $ω_1$ and $ω_2$ are moderate weights and $\mascB$ is a suitable (quasi-)Banach function space, then a necessary and sufficient condition for the embedding $i\, :\, M (ω_1,\mascB )\to M (ω_2,\mascB )$ between two modulation spaces to be compact is that the quotient $ω_2/ω_1$ vanishes at infinity. Moreover we show, that the boundedness of $ω_2/ω_1$ a necessary and sufficient condition for the previous embedding to be continuous.
Weighted symbol spaces, magnetic translations, and pseudodifferential operators
该研究聚焦J. Sjöstrand提出的微局部有限正则性赋范符号空间相关伪微分算子，证明其在调制空间的有界性与Weyl合成积的连续性。
We give a self-contained introduction to (quasi-)Banach modulation spaces of ultradistributions, and review results on boundedness for multiplications and convolutions for elements in such spaces. Furthermore, we use these results to study the Gabor product. As an example, we show how it appears in a phase-space formulation of the nonlinear cubic Schrödinger equation.
Area operators on Hardy spaces of Dirichlet series II: counterexamples and compactness criteria
本文否证了面积算子的Carleson型猜想，并完整刻画了其在Dirichlet级数Hardy空间上的有界性与紧性等价性，同时给出一般情形下紧性的充分条件及应用。
Maximal operators on BMO and slices
本文研究 BMO 及其变体空间上 Hardy-Littlewood 极大算子、强极大算子和方向极大算子的连续性与有界性，证明前者不连续但保持 VMO，并给出后两者的有界性反例，同时讨论函数切片性质。
We consider a class of linear Schrödinger equations in R^d with rough Hamiltonian, namely with certain derivatives in the Sjöstrand class $M^{\infty,1}$. We prove that the corresponding propagator is bounded on modulation spaces. The present results improve several contributions recently appeared in the literature and can be regarded as the evolution counterpart of the fundamental result of Sjöstrand about the boundedness of pseudodifferential operators with symbols in that class. Finally we consider nonlinear perturbations of real-analytic type and we prove local wellposedness of the correspo…
A counterexample to the Berger--Coburn conjecture
该研究通过构造一个特定的符号，证明了伯格-科布恩猜想在复维数n≥1时的失效，揭示了热变换在边界时间的无界性与偏移算子有界性的矛盾。
Endpoint boundedness of Orlicz-BMO commutators on Orlicz-Hardy type spaces
本文研究次线性算子与Orlicz-BMO函数生成的交换子在Orlicz-Hardy型空间上的端点有界性，证明了两类有界性结果，指出仅Bochner-Riesz平均算子对应的交换子满足特定端点有界性并给出反例。
Phase space analysis of spectral multipliers for the twisted Laplacian
本文利用元辛等价与转移原理，在调制空间和Wiener amalgam空间中研究扭曲拉普拉斯算子谱乘子的有界性，并完整刻画了其热流及分数阶热流的有界性。
We study a class of almost diagonal matrices compatible with the mixed-norm $α$-modulation spaces $M_{\vec{p},q}^{s,α}(\mathbb{R}^n)$, $α\in [0,1]$, introduced recently by Cleanthous and Georgiadis [Trans.\ Amer.\ Math.\ Soc.\ 373 (2020), no. 5, 3323-3356]. The class of almost diagonal matrices is shown to be closed under matrix multiplication and we connect the theory to the continuous case by identifying a suitable notion of molecules for the mixed-norm $α$-modulation spaces. We show that the "change of frame" matrix for a pair of time-frequency frames for the mixed-norm $α$-modulation consi…
On the boundedness of infinite products of relaxed projections: perturbations resilience and dynamic string-averaging
本研究在鲍施克和董的基础上，通过允许扰动、考虑闭半空间与超平面混合及使用动态加权和等方式扩展其关于松弛投影无穷乘积有界性的结果，还讨论推广局限并建立赋范空间压缩映射无穷乘积相关定理。
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…
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…
Let ${\mathcal X}$ be a doubling metric measure space. If ${\mathcal X}$ has the $δ$-annular decay property for some $δ\in (0,\,1]$, the authors then establish the boundedness of the Lusin-area function, which is defined via kernels modeled on the semigroup generated by the Schrödinger operator, from localized spaces ${\rm BMO}_ρ({\mathcal X})$ to ${\rm BLO}_ρ({\mathcal X})$ without invoking any regularity of considered kernels. The same is true for the $g^\ast_\labda$ function and unlike the Lusin-area function, in this case, ${\mathcal X}$ is not necessary to have the $δ$-annular decay prope…
We study continuity properties in Lebesgue spaces for a class of Fourier integral operators arising in the study of the Boltzmann equation. The phase has a Hölder-type singularity at the origin. We prove boundedness in $L^1$ with a precise loss of decay depending on the Hölder exponent, and we show by counterexamples that a loss occurs even in the case of smooth phases. The results can be seen as a quantitative version of the Beurling-Helson theorem for changes of variables with a Hölder singularity at the origin. The continuity in $L^2$ is studied as well by providing sufficient conditions an…
Localization operators associated with the windowed Opdam-Cherednik transform on modulation spaces
本文以加窗 Opdam-Cherednik 变换为基础，在相应调制空间框架下研究依赖符号与双窗函数的时频局部化算子，建立其有界性、紧性及 Schatten-von Neumann 类归属。
Pseudo-differential operators associated with the gyrator transform on modulation spaces with Shubin-type symbols
本文研究了 gyrator 变换在调制空间上的伪微分算子理论，基于 Shubin 符号类，扩展了之前在 Schwartz 和 Sobolev 空间上的研究成果。
Subdyadic time-frequency analysis: Gabor frames, modulation spaces, and Miyachi multipliers
本文提出基于子二分几何的时间-频率分析框架，构建稳定Gabor框架和调制空间，并发展Miyachi乘法理论，为微局分析和高频数值方案提供统一工具。
A Compact Counterexample to Suárez's Berezin Approximation Question
该文构造紧算子反例，证明高阶 Berezin 变换在算子范数下不收敛于原算子，利用移动矩阵边缘和 Pascal–Hankel 定理揭示其本质障碍。
Kernel theorems for modulation spaces
本文研究调制空间上线性算子的连续性，通过核在混合调制空间中的成员资格完全刻画，揭示了调制空间形式在信号处理中的优势。
Localization Operators On Discrete Modulation Spaces
本文在离散格点上定义短时 Fourier 变换与调制空间，系统研究时频局部化算子的有界性、紧性、Schatten–von Neumann 性质及其与 Landau–Pollak–Slepian 型算子的关系，并证明其在符号条件下可成为仿交换子、仿积和 Fourier 乘子。
The cross-Wigner distribution $W(f,g)$ of two functions or temperate distributions $f,g$ is a fundamental tool in quantum mechanics and in signal analysis. Usually, in applications in time-frequency analysis $f$ and $g$ belong to some modulation space and it is important to know which modulation spaces $W(f,g)$ belongs to. Although several particular sufficient conditions have been appeared in this connection, the general problem remains open. In the present paper we solve completely this issue by providing the full range of modulation spaces in which the continuity of the cross-Wigner distrib…
Boundedness of metaplectic operators within $L^p$ spaces, applications to pseudodifferential calculus, and time-frequency representations
本文推广了 Housdorff-Young 不等式，刻画了所有 metaplectic 算子在 $L^p-L^q$ 空间的有界性条件，并基于此推导了伪微分算子的有界性结果及 metaplectic Wigner 分布在调制空间中的性质。
On $L^p$ estimates for positivity-preserving Riesz transforms related to Schrödinger operators
本文研究了与Schrödinger算子相关的保正性Riesz变换的$L^p$有界性，证明了在特定参数和势能条件下在$L^p$、$L^\infty$和$L^1$空间的有界性，并给出了反例。
Real-Variable Characterizations and Their Applications of Anisotropic Besov Spaces with Matrix $\mathcal A_\infty$ Weights
本文建立带$\mathcal A_{p,\infty}$矩阵权的各向异性Besov空间理论，给出其$\varphi$变换与分子刻画，构造反例说明权条件的最优性，引入临界重标指数并得到相关算子的精确有界性。
Phase-Space Analysis of generalised Fractional Anharmonic and Ornstein-Uhlenbeck Semigroups on Weighted Modulation Spaces
本文研究了广义分数阶非谐振子及其热半群在加权调和空间中的相空间分析，推导了分数阶算子的符号估计，并证明了热半群的有界性和光滑性，同时建立了非线性热方程的全局解的存在性。
In this paper we investigate some questions related to the continuity of maximal operators in $W^{1,1}$ and $BV$ spaces, complementing some well-known boundedness results. Letting $\widetilde M$ be the one-dimensional uncentered Hardy-Littlewood maximal operator, we prove that the map $f \mapsto \big(\widetilde Mf\big)'$ is continuous from $W^{1,1}(\mathbb{R})$ to $L^1(\mathbb{R})$. In the discrete setting, we prove that $\widetilde M: BV(\mathbb{Z}) \to BV(\mathbb{Z})$ is also continuous. For the one-dimensional fractional Hardy-Littlewood maximal operator, we prove by means of counterexample…
The metaplectic action on modulation spaces
该论文研究了 metaplectic 算子在模空间上的映射性质，完整刻画了其良定义性与有界性等价的配对，并给出多项式有界权函数下的转移判据。
The Lifting Property for Frame Multipliers and Toeplitz Operators
本文证明正符号框架乘子是coorbit空间间的Banach同构，推广提升定理至Gabor框架和Fock空间，得到调制空间与加权Fock空间的新同构结果。
Euclidean structures and operator theory in Banach spaces
该研究提出基于巴拿赫空间上欧几里得结构的α-有界性的通用方法，将希尔伯特空间算子结论推广到巴拿赫空间，统一了经典因子分解与延拓定理，构建插值方法，证明扇形算子转移原理并推广Littlewood-Paley理论，还构造了相关反例。
For $m\in\mathbb{R}$ we introduce the symbol classes $S^m$, $m\in\mathbb{R}$, consisting of smooth functions $σ$ on $\mathbb{R}^{2d}$ such that $|\partial^ασ(z)|\leq C_α(1+|z|^2)^{m/2}$, $z\in\mathbb{R}^{2d}$, and we show that can be characterized by an intersection of different types of modulation spaces. In the case $m=0$ we recapture the Hörmander class $S^0_{0,0}$ that can be obtained by intersection of suitable Besov spaces as well. Such spaces contain the Shubin classes $Γ^m_ρ$, $0<ρ\leq1$, and can be viewed as their limit case $ρ=0$. We exhibit almost diagonalization properties for the …
Analytic automorphism group and similar representation of analytic functions
本文受 Gromov 多项式增长定理启发，研究多项式增长的加权 Hardy 空间，证明解析自同构诱导的复合算子有界且 M_z 与 M_φ 相似，并肯定回答 Douglas 问题的广义版本，同时给出中间增长情形下的无界反例及 Jordan 表示与相似分类。
Real-variable theory of function spaces with operator-valued $A_p$ weights in Banach spaces
本文研究了Banach空间中具有算子值$A_p$权重的Besov和Triebel-Lizorkin空间的实变数理论，证明了其可建立，同时反驳了部分'无果定理'，并展示了在非Hilbert空间中的应用。
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