Weighted Plancherel estimates and sharp spectral multipliers for the Grushin operators

Abstract

We study the Grushin operators acting on ${\mathbb{R}}_{x^{\prime }}^{d_1}\times {\mathbb{R}}_{x"}^{d_2}$ 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. Müller and E.M. Stein and by W. Hebisch.