A restriction theorem for Grushin operators

“…Other results extending the restriction theorem of Müller to more general nilpotent groups through spectral analysis have been considered in [20,21] and [35,36]. Finally, let us mention that applications of non commutative Fourier analysis have been also used to study the heat equation associated to sublaplacians on groups, see for instance [1].…”

Section: Remark 13mentioning

confidence: 99%

Strichartz Estimates and Fourier Restriction Theorems on the Heisenberg Group

Bahouri

Barilari

Gallagher

2021

J Fourier Anal Appl

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This paper is dedicated to the proof of Strichartz estimates on the Heisenberg group H d for the linear Schrödinger and wave equations involving the sublaplacian. The Schrödinger equation on H d is an example of a totally non-dispersive evolution equation: for this reason the classical approach that permits to obtain Strichartz estimates from dispersive estimates is not available. Our approach, inspired by the Fourier transform restriction method initiated in [45], is based on Fourier restriction theorems on H d , using the non-commutative Fourier transform on the Heisenberg group. It enables us to obtain also an anisotropic Strichartz estimate for the wave equation, for a larger range of indices than was previously known.

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“…Other results extending the restriction theorem of Müller to more general nilpotent groups through spectral analysis have been considered in [18,19] and [32,33]. Finally, let us mention that applications of non commutative Fourier analysis have been also used to study the heat equation associated to sublaplacians on groups, see for instance [1].…”

Section: Proposition 11 ([10]mentioning

confidence: 99%

Strichartz estimates and Fourier restriction theorems on the Heisenberg group

Bahouri,

Barilari,

Gallagher

2019

Preprint

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This paper is dedicated to the proof of Strichartz estimates on the Heisenberg group H d for the linear Schrödinger and wave equations involving the sublaplacian. The Schrödinger equation on H d is an example of a totally non-dispersive evolution equation: for this reason the classical approach that permits to obtain Strichartz estimates from dispersive estimates is not available. Our approach, inspired by the Fourier transform restriction method initiated in [41], is based on Fourier restriction theorems on H d , using the non-commutative Fourier transform on the Heisenberg group. It enables us to obtain also an anisotropic Strichartz estimate for the wave equation, for a larger range of indices than was previously known.

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A restriction theorem for Grushin operators

Abstract: We study the Grushin operators acting on R d1 x × R d2 t and defined by the formulaWe establish a restriction theorem associated with the considered operators. Our result is an analogue of the restriction theorem on the Heisenberg group obtained by D. Muller.

Cited by 3 publications

References 6 publications

Weighted restriction type estimates for Grushin operators and application to spectral multipliers and Bochner–Riesz summability

Weighted restriction type estimates for Grushin operators and application to spectral multipliers and Bochner–Riesz summability

Strichartz Estimates and Fourier Restriction Theorems on the Heisenberg Group

Strichartz estimates and Fourier restriction theorems on the Heisenberg group

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