Weighted Plancherel estimates and sharp spectral multipliers for the Grushin operators

TL;DR

This paper establishes sharp spectral multiplier theorems for Grushin operators on ^{d_1} imes ^{d_2} spaces, using weighted Plancherel estimates, extending results known for sublaplacians on the Heisenberg group.

Contribution

It introduces weighted Plancherel estimates for Grushin operators and proves sharp spectral multiplier and Bochner-Riesz results, generalizing known sublaplacian theorems.

Findings

Spectral multipliers are sharp when d_1 a0a0 d_2.

Weighted Plancherel estimates reveal phenomena for d_1 < d_2.

Results extend sublaplacian spectral multiplier theorems to Grushin operators.

Abstract

We study the Grushin operators acting on $\R^{d_1}_{x'}\times \R^{d_2}_{x"}$ and defined by the formula \[ L=-\sum_{\jone=1}^{d_1}\partial_{x'_\jone}^2 - (\sum_{\jone=1}^{d_1}|x'_\jone|^2) \sum_{\jtwo=1}^{d_2}\partial_{x"_\jtwo}^2. \] We obtain weighted Plancherel estimates for the considered operators. As a consequence we prove $L^p$ spectral multiplier results and Bochner-Riesz summability for the Grushin operators. These multiplier results are sharp if $d_1 \ge d_2$. We discuss also an interesting phenomenon for weighted Plancherel estimates for $d_1 <d_2$. The described spectral multiplier theorem is the analogue of the result for the sublaplacian on the Heisenberg group obtained by D. M\"uller and E.M. Stein and by W. Hebisch.