Riesz transforms and multipliers for the Grushin operator

TL;DR

This paper proves boundedness of Riesz transforms for the Grushin operator, establishes a multiplier theorem, and analyzes Bochner-Riesz means using advanced harmonic analysis tools.

Contribution

It introduces new boundedness results and multiplier theorems for the Grushin operator, expanding harmonic analysis techniques in this context.

Findings

Riesz transforms are bounded on L^p for the Grushin operator

Established an analogue of the H"ormander-Mihlin multiplier theorem

Studied Bochner-Riesz means related to the Grushin operator

Abstract

We show that Riesz transforms associated to the Grushin operator G = -\Delta - |x|^2\partial_t^2 are bounded on L^p(R^n+1). We also establish an analogue of H\"ormander-Mihlin multiplier theorem and study Bochner-Riesz means associated to the Grushin operator. The main tools used are Littlewood-Paley theory and an operator valued Fourier multiplier theorem due to L. Weis.