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.
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.
We discuss when the nonlinear operation f -> F(f) maps the modulation space M-s(pg) (R-n) (1 n = n/q, hence it is true for this space if F is entire. We claim that it is still true for non-analytic F when q >= ...
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We discuss when the nonlinear operation f -> F(f) maps the modulation space M-s(pg) (R-n) (1 <= p,q <= oo) to the same space again. It is known that M-s(pg) (R-n) p= (R-n) is a multiplication algebra when s > n = n/q, hence it is true for this space if F is entire. We claim that it is still true for non-analytic F when q >= 4/3. (C) 2019 Elsevier Inc. All rights reserved.
In this paper we study the Cauchy problem for the generalized Boussinesq equation with initial data in modulation spaces M-p',q(s) (R-n), n >= 1. After a decomposition of the Boussinesq equation in a 2 x 2-nonl...
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In this paper we study the Cauchy problem for the generalized Boussinesq equation with initial data in modulation spaces M-p',q(s) (R-n), n >= 1. After a decomposition of the Boussinesq equation in a 2 x 2-nonlinear system, we obtain the existence of global and local solutions in several classes of functions with values in M-p,q(s) x D(-1)JM(p,q)(s) spaces for suitable p, q and s, including the special case p = 2, q = 1 and s = 0. Finally, we prove some results of scattering and asymptotic stability in the framework of modulation spaces.
We establish both a local and a global well-posedness theories for the nonlinear Hartree-Fock equations and its reduced analog in the setting of the modulation spaces on In addition, we prove similar results when a ha...
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We establish both a local and a global well-posedness theories for the nonlinear Hartree-Fock equations and its reduced analog in the setting of the modulation spaces on In addition, we prove similar results when a harmonic potential is added to the equations. In the process, we prove the boundedeness of certain multilinear operators on products of the modulation spaces which may be of independent interest.
We consider the Cauchy problem of the modified Dispersion-Generalized Benjamin Ono equations on the real line. We prove some optimal local well-posedness results in the modulation spaces. The main new ingredient is an...
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We consider the Cauchy problem of the modified Dispersion-Generalized Benjamin Ono equations on the real line. We prove some optimal local well-posedness results in the modulation spaces. The main new ingredient is an improved version of bilinear Strichartz estimates in modulation space. (c) 2021 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.
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.
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.
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