Weighted Spectral Cluster Bounds and a Sharp Multiplier Theorem for Ultraspherical Grushin Operators

Author:

Casarino Valentina1,Ciatti Paolo2,Martini Alessio3

Affiliation:

1. Università degli Studi di Padova, Stradella san Nicola 3, I-36100 Vicenza, Italy

2. Università degli Studi di Padova, Via Marzolo 9, I-35100 Padova, Italy

3. School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK

Abstract

Abstract We study degenerate elliptic operators of Grushin type on the $d$-dimensional sphere, which are singular on a $k$-dimensional sphere for some $k < d$. For these operators we prove a spectral multiplier theorem of Mihlin–Hörmander type, which is optimal whenever $2k \leq d$, and a corresponding Bochner–Riesz summability result. The proof hinges on suitable weighted spectral cluster bounds, which in turn depend on precise estimates for ultraspherical polynomials.

Funder

Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni of the Istituto Nazionale di Alta Matematica

Publisher

Oxford University Press (OUP)

Subject

General Mathematics

Cited by 4 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. On some operator-valued Fourier pseudo-multipliers associated to Grushin operators;Journal of Mathematical Analysis and Applications;2024-11

2. A sharp multiplier theorem for solvable extensions of Heisenberg and related groups;Annali di Matematica Pura ed Applicata (1923 -);2023-12-26

3. Sparse Bounds for Pseudo-multipliers Associated to Grushin Operators, II;The Journal of Geometric Analysis;2023-12-07

4. An optimal multiplier theorem for Grushin operators in the plane, I;Revista Matemática Iberoamericana;2022-09-05

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