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Phase space analysis of the Hermite semigroup and applications to nonlinear global well-posedness

ADVANCES IN MATHEMATICS 2021年 392卷 107995-107995页

作者： Bhimani, Divyang G. Manna, Ramesh Nicola, Fabio Thangavelu, Sundaram Trapasso, S. Ivan Indian Inst Sci Educ & Res Dept Math Dr Homi Bhabha Rd Pune 411008 Maharashtra India HBNI Natl Inst Sci Educ & Res Bhubaneswar Sch Math Sci Jatni 752050 Khurda India Politecn Torino Dipartimento Sci Matemat GL Lagrange Corso Duca Abruzzi 24 I-10129 Turin Italy Indian Inst Sci Dept Math Bangalore 560012 Karnataka India

We study the Hermite operator H = -Delta + vertical bar x vertical bar(2) in Rd and its fractional powers H-beta, beta > 0 in phase space. Namely, we represent functions f via the so-called short-time Fourier, alias Fourier-Wigner or Bargmann transform V(g)f (g being a fixed window function), and we measure their regularity and decay by means of mixed Lebesgue norms in phase space of V(g)f, that is in terms of membership to modulation spaces M-P,M-q, 0 < p, q <= infinity. We prove the complete range of fixed-time estimates for the semigroup e-tH(beta) when acting on M-p,M-q, for every 0 < p, q <= infinity, exhibiting the optimal global-in-time decay as well as phase-space smoothing. As an application, we establish global well-posedness for the nonlinear heat equation for H-beta with power-type nonlinearity (focusing or defocusing), with small initial data in modulation spaces or in Wiener amalgam spaces. We show that such a global solution exhibits the same optimal decay e(-ct) as the solution of the corresponding linear equation, where c = d(beta) is the bottom of the spectrum of H. Global existence is in sharp contrast to what happens for the nonlinear focusing heat equation without potential, where blow-up in finite time always occurs for (even small) constant initial data (constant functions belong to M-infinity,M-1 ). (C) 2021 Elsevier Inc. All rights reserved.

关键词： Hermite operator Heat semigroup modulation spaces Pseudodifferential operators Nonlinear heat equation

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Estimate for generalized unimodular multipliers on modulation spaces

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Nonlinear Analysis 2013年 85卷 78-92页

作者： Qingquan Deng Yong Ding Lijing Sun Department of Mathematics Huazhong Normal University Wuhan 430079 China School of Mathematical Sciences Beijing Normal University Beijing 100875 China School of Mathematical Sciences Laboratory of Mathematics and Complex Systems (BNU) Ministry of Education Beijing Normal University Beijing 100875 China Department of Mathematical Science University of Wisconsin-Milwaukee Milwaukee WI 53201 USA

In this paper, we consider the operator e i t ϕ ( h ( D ) ) where h defined on R n is a C ∞ ( R n ∖ { 0 } ) positive homogeneous function with degree λ > 0 and ϕ : R → R is a smooth function satisfying the following. (A 1 ) There exists a constant m 1 > 0 such that for all μ ∈ N 0 : = N ∪ { 0 } | ϕ ( μ ) ( r ) | ≲ r m 1 − μ , r ≥ 1 . (A 2 ) There exists a constant m 2 > 0 such that for all μ ∈ N 0 | ϕ ( μ ) ( r ) | ≲ r m 2 − μ , 0 < r < 1 . We prove the boundedness of the operator e i t ϕ ( h ( D ) ) on the modulation spaces and obtain its asymptotic estimate as t goes to infinity. As applications, we give the grow-up rate of the solutions for the Cauchy problems for the generalized wave, Klein–Gordon and Schrödinger equations.

关键词： 42B30 42B25 42B35 35J15 Generalized unimodular multipliers modulation spaces Generalized wave equation Generalized Klein–Gordon equation Generalized Schrödinger equation

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INCLUSION RELATIONS BETWEEN modulation AND TRIEBEL-LIZORKIN spaces

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PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY 2017年第11期145卷 4807-4820页

作者： Guo, Weichao Wu, Huoxiong Zhao, Guoping Guangzhou Univ Sch Math & Informat Sci Guangzhou 510006 Guangdong Peoples R China Xiamen Univ Sch Math Sci Xiamen 361005 Peoples R China Xiamen Univ Technol Sch Appl Math Xiamen 361024 Peoples R China

In this paper, we obtain the sharp conditions of the inclusion relations between modulation spaces M-p,q(s) and Triebel-Lizorkin spaces F-p,F-r for p <= 1, which greatly improve and extend the results for the embedding relations between local Hardy spaces and modulation spaces obtained by Kobayashi, Miyachi and Tomita in [Studia Math. 192 (2009), 79- 96].

关键词： Triebel-Lizorkin spaces modulation spaces inclusion relation

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NLS in the modulation Space M2,q(R)

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JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS 2019年第4期25卷 1447-1486页

作者： Pattakos, N. Karlsruhe Inst Technol Inst Anal Dept Math D-76128 Karlsruhe Germany

We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic nonlinear Schrodinger equation in the modulation space Ms2, q ( R), 1 = q = 2 and s = 0. In addition, for either s = 0 and 1 = q = 3 2 or 3 2 < q = 2 and s > 2 3 - 1 q we show that the Cauchy problem is unconditionally wellposed in Ms2, q ( R). It is done with the use of the differentiation by parts technique which had been previously used in the periodic setting.

关键词： Normal form method modulation spaces Nonlinear Schrodinger Local wellposedness Unconditional uniqueness

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Strichartz estimates for the metaplectic representation

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REVISTA MATEMATICA IBEROAMERICANA 2019年第7期35卷 2079-2092页

作者： Cauli, Alessandra Nicola, Fabio Tabacco, Anita Politecn Torino Dipartimento Sci Matemat Corso Duca Abruzzi 24 I-10129 Turin Italy

We provide new estimates for the matrix coefficients of the metaplectic representation, inspired by a formal analogy with the Strichartz estimates which hold for several classes of evolution propagators U(t). The one parameter group of unitary operators U(t) is replaced by a unitary representation of a non-compact Lie group, the group element playing the role of time;the case of the metaplectic or oscillatory representation is of special interest in this connection, because the Schrodinger group is a subgroup of the metaplectic group. We prove uniform weak-type sharp estimates for matrix coefficients and Strichartz-type estimates for that representation. The crucial point is the choice of function spaces able to detect such a decay, which in general will depend on the given group action. The relevant function spaces here turn out to be the so-called modulation spaces from time-frequency analysis in Euclidean space, and Lebesgue spaces with respect to Haar measure on the metaplectic group. The proofs make use in an essential way of the covariance of the Wigner distribution with respect to the metaplectic representation.

关键词： Strichartz estimates metaplectic representation matrix coefficients weak-type estimates modulation spaces Wigner distribution

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A remark on the Schrodinger propagator on Wiener amalgam spaces

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MATHEMATISCHE NACHRICHTEN 2019年第2期292卷 350-357页

作者： Kato, Tomoya Tomita, Naohito Osaka Univ Dept Math Grad Sch Sci Toyonaka Osaka 5600043 Japan

In this paper, we study the boundedness of the Schrodinger propagator ei Delta on Wiener amalgam spaces. In particular, we determine the necessary and sufficient conditions for the propagator ei Delta to be bounded fr... 详细信息

关键词： modulation spaces Schrodinger propagator Wiener amalgam spaces

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Well-posedness for the incompressible magneto-hydrodynamic system on modulation spaces

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2012年第2期389卷 741-753页

作者： Liu, Qiao Cui, Shangbin Sun Yat Sen Univ Dept Math Guangzhou 510275 Guangdong Peoples R China

We study the Cauchy problem for an incompressible magneto-hydrodynamics (MHD) system in the modulation space M-q,sigma(s) (R-n) (n >= 2) with initial data (u(0), b(0)) is an element of PMq,sigma s (R-n) x PMq,sigma s (R-n). We prove that this problem is locally well-posed in such a space when 1 <= q <= infinity, 1 <= sigma < infinity and n(sigma-1)/sigma - 1 <= s, and globally well-posed when 1 <= q <= n, n/n-1 <= sigma < infinity and max{n(sigma-1)/sigma -1, n(sigma-2)/sigma} < s < n(sigma-1)/sigma. (C) 2011 Elsevier Inc. All rights reserved.

关键词： Magneto-hydrodynamic system Well-posedness modulation spaces

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ON THE SYMPLECTIC COVARIANCE AND INTERFERENCES OF TIME-FREQUENCY DISTRIBUTIONS

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SIAM JOURNAL ON MATHEMATICAL ANALYSIS 2018年第2期50卷 2178-2193页

作者： Cordero, Elena De Gosson, Maurice Doerfler, Monika Nicola, Fabio Univ Turin Dipartimento Matemat I-10123 Turin Italy Univ Vienna Fac Math A-1090 Vienna Austria Politecn Torino Dipartimento Sci Matemat I-10129 Turin Italy

We study the covariance property of quadratic time-frequency distributions with respect to the action of the extended symplectic group. We show how covariance is related to and in fact in competition with the possibility of damping the interferences which arise due to the quadratic nature of the distributions. We also show that the well-known fully covariance property of the Wigner distribution in fact characterizes it (up to a constant factor) among the quadratic distributions L-2(R-n) -> C-0(R-2n). A similar characterization for the closely related Weyl transform is given as well. The results are illustrated by several numerical experiments for the Wigner and Born-Jordan distributions of the sum of four Gaussian functions in the so-called diamond con figuration.

关键词： time-frequency analysis covariance property symplectic group interferences Wigner distribution Gabor frames modulation spaces

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On the reduction of the interferences in the Born-Jordan distribution

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APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS 2018年第2期44卷 230-245页

作者： Cordero, Elena de Gosson, Maurice Nicola, Fabio Univ Turin Dipartimento Matemat Via Carlo Alberto 10 I-10123 Turin Italy Univ Vienna Fac Math Oskar Morgenstern Pl 1 A-1090 Vienna Austria Politecn Torino Dipartimento Sci Matemat Corso Duca Abruzzi 24 I-10129 Turin Italy

One of the most popular time-frequency representations is certainly the Wigner distribution. To reduce the interferences coming from its quadratic nature, several related distributions have been proposed, among which is the so-called Born-Jordan distribution. It is well known that in the Born-Jordan distribution the ghost frequencies are in fact damped quite well, and the noise is in general reduced. However, the horizontal and vertical directions escape from this general smoothing effect, so that the interferences arranged along these directions are in general kept. Whereas these features are graphically evident on examples and heuristically well understood in the engineering community, there is no at present mathematical explanation of these phenomena, valid for general signals in L-2 and, more in general, in the space S' of temperate distributions. In the present note we provide such a rigorous study using the notion of wave-front set of a distribution. We use techniques from Time-=frequency Analysis, such as the modulation and Wiener amalgam spaces, and also results of microlocal regularity of linear partial differential operators. (C) 2016 Elsevier Inc. All rights reserved.

关键词： Time-frequency analysis Wigner distribution Born-Jordan distribution Interferences Wave-front set modulation spaces Fourier Lebesgue spaces

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Convolutions for localization operators

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JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES 2018年 118卷 288-316页

作者： Luef, Franz Skrettingland, Eirik NTNU Norwegian Univ Sci & Technol Dept Math NO-7491 Trondheim Norway

Quantum harmonic analysis on phase space is shown to be linked with localization operators. The convolution between operators and the convolution between a function and an operator provide a conceptual framework for the theory of localization operators which is complemented by an appropriate Fourier transform, the Fourier-Wigner transform. We use Lieb's uncertainty principle to establish a sharp Hausdorff-Young inequality for the Fourier-Wigner transform. Noncommutative Tauberian theorems due to Werner allow us to extend results of Bayer and Geochenig on localization operators. Furthermore we show that the Arveson spectrum and the theory of Banach modules provide the abstract setting of quantum harmonic analysis. (C) 2017 Elsevier Masson SAS. All rights reserved.

关键词： Localization operators Berezin transform Banach modules Arveson spectrum Tauberian theorems modulation spaces

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