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A Hörmander-Mikhlin theorem for high rank simple Lie groups

José M. Conde-Alonso, Adrián M. González-Pérez, Javier Parcet, Eduardo Tablate

Abstract

We establish regularity conditions for $L_p$-boundedness of Fourier multipliers on the group von Neumann algebras of higher rank simple Lie groups. This provides a natural Hörmander-Mikhlin criterion in terms of Lie derivatives of the symbol and a metric given by the adjoint representation. In line with Lafforgue/de la Salle's rigidity theorem, our condition imposes certain decay of the symbol at infinity. It refines and vastly generalizes a recent result by Parcet, Ricard and de la Salle for $\SL$. Our approach is partly based on a sharp local Hörmander-Mikhlin theorem for arbitrary Lie groups, which follows in turn from recent estimates by the authors on singular nonToeplitz Schur multipliers. We generalize the latter to arbitrary locally compact groups and refine the cocycle-based approach to Fourier multipliers in group algebras by Junge, Mei and Parcet. A few related open problems are also discussed.

A Hörmander-Mikhlin theorem for high rank simple Lie groups

Abstract

We establish regularity conditions for -boundedness of Fourier multipliers on the group von Neumann algebras of higher rank simple Lie groups. This provides a natural Hörmander-Mikhlin criterion in terms of Lie derivatives of the symbol and a metric given by the adjoint representation. In line with Lafforgue/de la Salle's rigidity theorem, our condition imposes certain decay of the symbol at infinity. It refines and vastly generalizes a recent result by Parcet, Ricard and de la Salle for . Our approach is partly based on a sharp local Hörmander-Mikhlin theorem for arbitrary Lie groups, which follows in turn from recent estimates by the authors on singular nonToeplitz Schur multipliers. We generalize the latter to arbitrary locally compact groups and refine the cocycle-based approach to Fourier multipliers in group algebras by Junge, Mei and Parcet. A few related open problems are also discussed.
Paper Structure (8 sections, 6 theorems, 92 equations)

This paper contains 8 sections, 6 theorems, 92 equations.

Table of Contents

  1. Local transference and Schur multipliers
  2. Local transference
  3. Singular Schur multipliers
  4. The local theorem
  5. Simple Lie groups
  6. HMS multipliers in groups
  7. Singular multipliers in groups
  8. Herz-Schur multipliers

Key Result

Theorem 1.1

Let $\mathrm{G}$ be a locally compact unimodular group and consider a relatively compact neighborhood of the identity $\Omega$ and any open set $\Sigma$ in $\mathrm{G}$ containing the closure of $\Omega$. Let $m : \mathrm{G} \to \mathbf{C}$ be a bounded symbol supported in $\Omega$. Then, the follo Moreover, we may put $C_{\Omega,\Sigma,p} \le 2$ for $\Omega$ small enough depending on the value o

Theorems & Definitions (17)

  • Theorem 1.1: Local transference
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4: HMS multipliers
  • Lemma 3.1
  • Remark 3.2
  • Remark 3.3
  • Theorem 4.1: General HMS multipliers
  • Remark 4.2
  • Remark 4.3
  • ...and 7 more