Oscillating multipliers on symmetric and locally symmetric spaces

Effie Papageorgiou

doi:10.4153/s0008414x21000250

Abstract

AbstractWe prove $L^{p}$ -boundedness of oscillating multipliers on symmetric spaces of noncompact type of arbitrary rank, as well as on a wide class of locally symmetric spaces.

Highlights

Introduction and statement of the resultsThe main objective in this article is to study the Lp boundedness of oscillating multipliers on symmetric spaces of arbitrary rank and locally symmetric spaces
To prove the Lp boundedness of the local part Tα0,β on X, we follow the spectral multiplier approach of [2] and use the spherical Fourier transform in order to adapt these ideas to the symmetric space setting
Throughout this article, the different constants will always be denoted by the same letter c

Summary

Introduction and statement of the results

The main objective in this article is to study the Lp boundedness of oscillating multipliers on symmetric spaces of arbitrary rank and locally symmetric spaces. We deal with oscillating multipliers in the setting of noncompact symmetric spaces of arbitrary rank These are Riemannian, nonpositively curved manifolds, with a structure that induces a Fourier-like analysis. For p ≠ 2, there is a certain necessary condition (see [7, Theorem 1] or [3, p.604]), first observed by Clerc and Stein, which has no Euclidean analogue: every multiplier that yields an Lp(X) bounded operator, for some p ∈ (1, ∞), p ≠ 2, extends to an invariant by the Weyl group, bounded, holomorphic function inside the tube T p = a∗ + i∣2/p − 1∣Cρ. To prove the Lp boundedness of the local part Tα0,β on X, we follow the spectral multiplier approach of [2] (see [12, 22]) and use the spherical Fourier transform in order to adapt these ideas to the symmetric space setting. Throughout this article, the different constants will always be denoted by the same letter c

Preliminaries

Lp boundedness of the local part

Proof of Proposition 2

Proof of Lemma 3

The part at infinity