Riesz transforms and multipliers for the Bessel-Grushin operator

TL;DR

This paper proves weak type (1,1) boundedness for spectral multipliers of the Bessel-Grushin operator and studies associated Riesz transforms' boundedness on L^p spaces, advancing harmonic analysis for this class of operators.

Contribution

It establishes weak type (1,1) bounds for spectral multipliers and analyzes Riesz transforms for the Bessel-Grushin operator, introducing new weighted estimates and extending harmonic analysis tools.

Findings

Spectral multipliers are of weak type (1,1) under certain Sobolev conditions.

Weighted Plancherel estimates are developed for the operator.

L^p-boundedness of Riesz transforms is shown for the case n=1.

Abstract

We establish that the spectral multiplier $\frak{M}(G_{\alpha})$ associated to the differential operator $$ G_{\alpha}=- \Delta_x +\sum_{j=1}^m{{\alpha_j^2-1/4}\over{x_j^2}}-|x|^2 \Delta_y \; \text{on} (0,\infty)^m \times \R^n,$$ which we denominate Bessel-Grushin operator, is of weak type $(1,1)$ provided that $\frak{M}$ is in a suitable local Sobolev space. In order to do this we prove a suitable weighted Plancherel estimate. Also, we study $L^p$-boundedness properties of Riesz transforms associated to $G_{\alpha}$, in the case $n=1$.