We show new local L-p-smoothing estimates for the Schrodinger equation in modulation spaces via decoupling inequalities. Furthermore, we probe necessary conditions by Knapp-type examples for space-time estimates of so...
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We show new local L-p-smoothing estimates for the Schrodinger equation in modulation spaces via decoupling inequalities. Furthermore, we probe necessary conditions by Knapp-type examples for space-time estimates of solutions with initial data in modulation and L-p-spaces. The examples show sharpness of the smoothing estimates up to the endpoint regularity in a certain range. Moreover, the examples rule out global Strichartz estimates for initial data in L-p(R-d) for d >= 1 and p >= 2, which was previously known for d > 2. The estimates are applied to show new local and global well-posedness results for the cubic nonlinear Schrodinger equation on the line. Lastly, we show l(2)-decoupling inequalities for variable coefficient versions of elliptic and non-elliptic Schrodinger phase functions. (C) 2021 Elsevier Inc. All rights reserved.
modulation spaces M-p,q(s) were introduced by Feichtinger [3] in 1983. Later, considering the scaling property of the modulation spaces, Sugimoto and Wang [11] defined the scaling limit of the modulation spaces, which...
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modulation spaces M-p,q(s) were introduced by Feichtinger [3] in 1983. Later, considering the scaling property of the modulation spaces, Sugimoto and Wang [11] defined the scaling limit of the modulation spaces, which contains both the modulation spaces and Benyi and Oh's modulation spaces [1], and these spaces also have some applications in nonlinear Schrodinger equations. So, it is important to consider the relationship between these new spaces and some classical Banach spaces such as L-p spaces, Fourier L-p spaces and Besov-Triebel-Sobolev spaces. We study the embedding properties of the scaling limit of the modulation spaces, including the homogeneous case and nonhomogeneous case. (C) 2022 Elsevier Inc. All rights reserved.
In this paper we derive time-decay and Strichartz estimates for the generalized Benjamin-Bona-Mahony equation on the framework of modulation spaces M-p,q(s). We use these results to analyze the existence of local and ...
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In this paper we derive time-decay and Strichartz estimates for the generalized Benjamin-Bona-Mahony equation on the framework of modulation spaces M-p,q(s). We use these results to analyze the existence of local and global solutions of the corresponding Cauchy problem with rough data in modulation spaces. As consequence of the existence theorems in modulation spaces, some results in Sobolev spaces are derived. (C) 2021 Elsevier Inc. All rights reserved.
Inspired by the recent article Skrettingland (J. Fourier Anal. Appl. 28(2), 1-34 (2022)), this paper is devoted to the study of a suitable class of windows in the framework of bounded linear operators on L-2(R-d). We ...
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Inspired by the recent article Skrettingland (J. Fourier Anal. Appl. 28(2), 1-34 (2022)), this paper is devoted to the study of a suitable class of windows in the framework of bounded linear operators on L-2(R-d). We establish a natural and complete characterization for the window class such that the corresponding STFT leads to equivalent norms on modulation spaces. The positive bounded linear operators are also characterized by its Cohen's class distributions such that the corresponding quantities form equivalent norms on modulation spaces. As a generalization, we introduce a family of operator classes corresponding to the operator-valued modulation spaces. Some applications of our main theorems to the localization operators are also concerned.
Let A(beta)(1) (T) denote the set of all Lebesgue integrable functions F on the torus T such that Sigma(m is an element of Z)vertical bar boolean AND F(m)vertical bar (1 + vertical bar m vertical bar)beta < infinit...
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Let A(beta)(1) (T) denote the set of all Lebesgue integrable functions F on the torus T such that Sigma(m is an element of Z)vertical bar boolean AND F(m)vertical bar (1 + vertical bar m vertical bar)beta < infinity, where {boolean AND F(m)}(m is an element of Z) denote the Fourier coefficients of F. We consider necessary and sufficient conditions for all functions F is an element of A(beta)(1)(T) to operate on all real-valued functions in the modulation spaces M-s(p,q) (R).
We characterize Gelfand-Shilov spaces, their distribution spaces and modulation spaces in terms of estimates of their Zak transforms. We use these results for general investigations of quasi-periodic functions and dis...
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We characterize Gelfand-Shilov spaces, their distribution spaces and modulation spaces in terms of estimates of their Zak transforms. We use these results for general investigations of quasi-periodic functions and distributions. We also establish necessary and sufficient conditions for linear operators in order for these operators should be conjugations by Zak transforms.
We consider nonlinear Schrodinger equations in Fourier-Lebesgue and modulation spaces involving negative regularity. The equations are posed on the whole space, and involve a smooth power nonlinearity. We prove two ty...
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We consider nonlinear Schrodinger equations in Fourier-Lebesgue and modulation spaces involving negative regularity. The equations are posed on the whole space, and involve a smooth power nonlinearity. We prove two types of norm inflation results. We first establish norm inflation results below the expected critical regularities. We then prove norm inflation with infinite loss of regularity under less general assumptions. To do so, we recast the theory of multiphase weakly nonlinear geometric optics for nonlinear Schrodinger equations in a general abstract functional setting.
We prove new well-posedness results for energy-critical nonlinear Schrodinger equations in modulation spaces. This covers initial data with infinite mass and energy. The proof is carried out via bilinear refinements a...
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We prove new well-posedness results for energy-critical nonlinear Schrodinger equations in modulation spaces. This covers initial data with infinite mass and energy. The proof is carried out via bilinear refinements and adapted function spaces. (c) 2022 Elsevier Inc. All rights reserved.
We define and investigate a-modulation spaces M-p,q(s,alpha) (G) associated to a step two stratified Lie group G with rational structure constants. This is an extension of the Euclidean alpha-modulation spaces M-p,q(s...
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We define and investigate a-modulation spaces M-p,q(s,alpha) (G) associated to a step two stratified Lie group G with rational structure constants. This is an extension of the Euclidean alpha-modulation spaces M-p,q(s,alpha) (R-n) that act as intermediate spaces between the modulation spaces (alpha = 0) in time-frequency analysis and the Besov spaces (alpha = 1) in harmonic analysis. We will illustrate that the group structure and dilation structure on G affect the boundary cases alpha = 0, 1 where the spaces M-p,q(s) (G) and B-p,q(s) (G) have non-standard translation and dilation symmetries. Moreover, we show that the spaces M-p,q(s,alpha) (G) are non-trivial and generally distinct from their Euclidean counterparts. Finally, we examine how the metric geometry of the coverings Q(G) underlying the alpha = 0 case M-p,q(s) (G) allows for the existence of geometric embeddings F : M-p,q(s) (R-k) -> M-p,q(s) (G), as long as k (that only depends on G) is small enough. Our approach naturally gives rise to several open problems that is further elaborated at the end of the paper.
In the last twenty years modulation spaces, introduced by H. G. Feichtinger in 1983, have been successfully addressed to the study of signal analysis, PDE's, pseudodifferential operators, quantum mechanics, by hun...
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In the last twenty years modulation spaces, introduced by H. G. Feichtinger in 1983, have been successfully addressed to the study of signal analysis, PDE's, pseudodifferential operators, quantum mechanics, by hundreds of contributions. In 2011 M. de Gosson showed that the time-frequency representation Short-time Fourier Transform (STFT), which is the tool to define modulation spaces, can be replaced by the Wigner distribution. This idea was further generalized to Tau-Wigner representations in [11]. In this paper time-frequency representations are viewed as images of symplectic matrices via metaplectic operators. This new perspective highlights that the protagonists of time -frequency analysis are metaplectic operators and symplectic matrices A is an element of Sp(2d,R). We find conditions on A for which the related symplectic time-frequency representation WA can replace the STFT and give equivalent norms for weighted modulation spaces. In particular, we study the case of covariant matrices A, i.e., their corresponding WA are members of the Cohen class. Finally, we show that symplectic time-frequency representa-tions WA can be efficiently employed in the study of Schrodinger equations. In fact, modulation spaces and WA representations are the frame for a new definition of wave front set, providing a sharp result for propagation of micro -singularities in the case of the quadratic Hamiltonians. This new approach may have further applications in quantum mechanics and PDE's.(c) 2023 Elsevier Inc. All rights reserved.
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