Weighted Plancherel estimates and sharp spectral multipliers for the Grushin operators

Details Weighted Plancherel estimates and sharp spectral multipliers for the Grushin operators Weighted Plancherel estimates and sharp spectral multipliers for the Grushin operators

2012-04-05

arxiv.org/abs/1204.1159v1

Mathematics

Analysis of PDEs

Scientific paper

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

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