Weighted Plancherel estimates and sharp spectral multipliers for the Grushin operators

We study the Grushin operators acting on ${\mathbb R}^{d_1}_{x'}\times {\mathbb R}^{d_2}_{x''}$ and defined by the formula\[L=-\sum_{j=1}^{d_1}\partial_{x'_j}^2 - \left(\sum_{j=1}^{d_1}|x'_j|^2\right)\sum_{k=1}^{d_2}\partial_{x''_k}^2.\]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 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 Müller and Stein and by Hebisch.

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Published 15 March 2013