Paper detail

Weighted Plancherel estimates and sharp spectral multipliers for the Grushin operators

We study the Grushin operators acting on $\R^{d_1}_{x&#39;}\times \R^{d_2}_{x&#34;}$ and defined by the formula \[ L=-\sum_{\jone=1}^{d_1}\partial_{x&#39;_\jone}^2 - (\sum_{\jone=1}^{d_1}|x&#39;_\jone|^2) \sum_{\jtwo=1}^{d_2}\partial_{x&#34;_\jtwo}^2. \] 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.