Riesz transforms and multipliers for the Bessel-Grushin operator

Abstract

We establish that the spectral multiplier $\mathfrak{M}(G_{\alpha })$ associated to the differential operator $$G_{\alpha }=-{\Delta }_x+\sum _{j=1}^m\frac{{\alpha }_j^2-1/4}{x_j^2}-|x{|}^2{\Delta }_y\text{on}(0,\infty {)}^m\times {\mathbb{R}}^n,$$ which we denominate Bessel-Grushin operator, is of weak type $(1,1)$ provided that $\mathfrak{M}$ 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 $n=1$.

33 pages

References in corpus (1)

• A sharp multiplier theorem for Grushin operators in arbitrary dimensions