Strichartz estimates for the metaplectic representation

Cauli, Alessandra; Nicola, Fabio; Tabacco, Anita

doi:10.48550/arxiv.1706.03615

Cited by 2 publications

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“…Step. We will show that (6) easily follows from (5). Indeed, the Fourier transform J = F is a metaplectic operator and we recall that the Fourier transform is a topological isomorphism F : [16].…”

Section: Thirdmentioning

confidence: 99%

“…Comparing (4) and ( 5) we observe that in (5) the modulation operator M ξ is replaced by the metaplectic operator S and the integral on the phase space R 2n has become an integral on the cartesian product R n × U(2n, R). The integration parameters (x, ξ) of (4) live in R 2n , with dim R 2n = 2n, whereas the parameters (x, S) of ( 5) live in R n × U(2n, R).…”

Section: Introductionmentioning

confidence: 99%

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A characterization of modulation spaces by symplectic rotations

Cordero

Gosson

Nicola

2020

Journal of Functional Analysis

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This note contains a new characterization of modulation spaces M p (R n ), 1 ≤ p ≤ ∞, 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 p (R n )-norm, the integral in the timefrequency plane with an integral on R n × 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. In this new framework, the Gaussian invariance under symplectic rotations yields to choose Gaussians as suitable window functions. We also provide a similar characterization with the group U (2n, R) being reduced to the n-dimensional torus T n . the symplectic group Sp(n, R) is the subgroup of 2n × 2n invertible matrices GL(2n, R), defined bywhere J is the orthogonal matrix J = 0 n I n −I n 0 n , (I n , 0 n are the n × n identity matrix and null matrix, respectively). Here we consider the subgroup U(2n, R) := Sp(n, R) ∩ O(2n, R) ≃ U(n)2010 Mathematics Subject Classification. 42B35,22C05.

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Section: Thirdmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A characterization of modulation spaces by symplectic rotations

Cordero

Gosson

Nicola

2020

Journal of Functional Analysis

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“…As an application outside representation theory, we present some multidimensional dispersive estimates that go beyond the current fascination with dispersive estimates for the Schrödinger equation (see Tao [20] and also [6] for a representation theoretic perspective).…”

Section: Introductionmentioning

confidence: 99%

Estimates for matrix coefficients of representations

Bruno¹,

Cowling²,

Nicola³

et al. 2019

Preprint

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Estimates for matrix coefficients of unitary representations of semisimple Lie groups have been studied for a long time, starting with the seminal work by Bargmann, by Ehrenpreis and Mautner, and by Kunze and Stein. Two types of estimates have been established: on the one hand, L p estimates, which are a dual formulation of the Kunze-Stein phenomenon, and which hold for all matrix coefficients, and on the other pointwise estimates related to asymptotic expansions at infinity, which are more precise but only hold for a restricted class of matrix coefficients. In this paper we prove a new type of estimate for the irreducibile unitary representations of SL(2, R) and for the so-called metaplectic representation, which we believe has the best features of, and implies, both forms of estimate described above. As an application outside representation theory, we prove a new L 2 estimate of dispersive type for the free Schrödinger equation in R n .

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Strichartz estimates for the metaplectic representation

Cited by 2 publications

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A characterization of modulation spaces by symplectic rotations

A characterization of modulation spaces by symplectic rotations

Estimates for matrix coefficients of representations

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