In this paper we have studied Fourier multipliers and Littlewood-Paley square functions in the context of modulation spaces. We have also proved that any bounded linear operator from modulation space M-p,M- q (R-n), 1...
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In this paper we have studied Fourier multipliers and Littlewood-Paley square functions in the context of modulation spaces. We have also proved that any bounded linear operator from modulation space M-p,M- q (R-n), 1 <= p, q <= infinity, into itself possesses an l(2)-valued extension. This is an analogue of a well known result due to Marcinkiewicz and Zygmund on classical L-p-spaces. (C) 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
The alpha-modulation spaces are a family of spaces that contain the Besov and modulation spaces as special cases. In this paper we prove that brushlet bases can be constructed to form unconditional and even greedy bas...
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The alpha-modulation spaces are a family of spaces that contain the Besov and modulation spaces as special cases. In this paper we prove that brushlet bases can be constructed to form unconditional and even greedy bases for the alpha-modulation spaces. We study m-term nonlinear approximation with brushlet bases, and give complete characterizations of the associated approximation spaces in terms of alpha-modulation spaces. (c) 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.
This note contains a new characterization of modulation spaces M-m(p )(R-n), 1 <= p <= infinity, by symplectic rotations. Precisely, instead to measure the time-frequency content of a function by using translati...
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This note contains a new characterization of modulation spaces M-m(p )(R-n), 1 <= p <= infinity, by symplectic rotations. Precisely, instead to measure the time-frequency content of a function by using translations and modulations of a fixed window as building blocks, we use translations and metaplectic operators corresponding to symplectic rotations. Technically, this amounts to replace, in the computation of the M-m(p)(R-n)-norm, the integral in the time-frequency plane with an integral on R-n x U(2n,R) with respect to a suitable measure, U(2n, R) being the group of symplectic rotations. More conceptually, we are considering a sort of polar coordinates in the time-frequency plane. To have invariance under symplectic rotations we choose a Gaussian as suitable window function. We also provide a similar (and easier) characterization with the group U(2n, R) being reduced to the n-dimensional torus T-n. (C) 2020 Elsevier Inc. All rights reserved.
In this note we consider the nonlinear heat equation associated to the fractional Hermite operator H beta=(-Delta+|x|2)beta, 0<= 1. We show the local solvability of the related Cauchy problem in the framework of mo...
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In this note we consider the nonlinear heat equation associated to the fractional Hermite operator H beta=(-Delta+|x|2)beta, 0<<= 1. We show the local solvability of the related Cauchy problem in the framework of modulation spaces. The result is obtained by combining tools from microlocal and time-frequency analysis. As a byproduct, we compute the Gabor matrix of pseudodifferential operators with symbols in the Hormander class S0,0m, m is an element of R.
In this paper we focus on the almost-diagonalization properties of tau-pseudodifferential operators using techniques from time-frequency analysis. Our function spaces are modulation spaces and the special class of Wie...
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In this paper we focus on the almost-diagonalization properties of tau-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. Such spaces are nowadays called modulation spaces as well. A particular example is provided by the Sjostrand class, for which Grochenig (Rev Mat Iberoam 22(2):703-724, 2006) exhibits the almost diagonalization of Weyl operators. We shall show that such result can be extended to any tau-pseudodifferential operator, for tau is an element of[0,1]. This is not surprising, since the mapping that goes from a Weyl symbol to a tau-symbol is bounded in the Sjostrand class. What is new and quite striking is the almost diagonalization for tau-operators with symbols in weighted Wiener amalgam spaces. In this case the diagonalization depends on the parameter tau. In particular, we have an almost diagonalization for tau is an element of(0,1) whereas the cases tau=0 or tau=1 yield only to weaker results. As a consequence, we infer boundedness, algebra and Wiener properties for tau-pseudodifferential operators on Wiener amalgam and modulation spaces.
In the present paper we obtain estimates in the modulation spaces for the solutions to the Dirac equation with quadratic and sub-quadratic potentials. We derive a representation for the Dirac operator that permits to ...
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In the present paper we obtain estimates in the modulation spaces for the solutions to the Dirac equation with quadratic and sub-quadratic potentials. We derive a representation for the Dirac operator that permits to solve approximately the perturbed Dirac equation and to obtain the desired estimates for the solution.
We discuss continuity for weighted modulation spaces, and prove that many such spaces can be obtained in a canonical way from the corresponding standard modulation spaces. We also discuss the trace operator a-->a(0...
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We discuss continuity for weighted modulation spaces, and prove that many such spaces can be obtained in a canonical way from the corresponding standard modulation spaces. We also discuss the trace operator a-->a(0, .) acting on modulation spaces. The results are used to get inclusions between modulation spaces and Besov spaces, and proving continuity for pseudo-differential operators and Toeplitz operators.
We prove the boundedness of a general class of Fourier multipliers, in particular of the Hilbert transform, on modulation spaces. In general, however, the Fourier multipliers in this class fail to be bounded on LP spa...
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We prove the boundedness of a general class of Fourier multipliers, in particular of the Hilbert transform, on modulation spaces. In general, however, the Fourier multipliers in this class fail to be bounded on LP spaces. The main tools are Gabor frames and methods from time-frequency analysis. (c) 2005 Elsevier Inc. All rights reserved.
This work deals with Schrodinger equations with quadratic and subquadratic Hamiltonians perturbed by a potential. In particular we shall focus on bounded, but not necessarily smooth perturbations, following the footst...
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This work deals with Schrodinger equations with quadratic and subquadratic Hamiltonians perturbed by a potential. In particular we shall focus on bounded, but not necessarily smooth perturbations, following the footsteps of the preceding works (Cordero et al., Generalized metaplectic operators and the Schrodinger equation with a potential in the Sjostrand class, 2013;Cordero et al., Propagation of the Gabor wave front set for Schrodinger equations with non-smooth potentials, 2013). To the best of our knowledge these are the pioneering papers which contain the most general results about the time-frequency concentration of the Schrodinger evolution. We shall give a representation of such evolution as the composition of a metaplectic operator and a pseudodifferential operator having symbol in certain classes of modulation spaces. About propagation of singularities, we use a new notion of wave front set, which allows the expression of optimal results of propagation in our context. To support this claim, many comparisons with the existing literature are performed in this work.
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...
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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.
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