Weighted Plancherel estimates and sharp spectral multipliers for the Grushin operators

We study the Grushin operators acting on ℝd1x' × ℝd2x″ and defined by the formula L = -Σd1j=1∂2x'j- (Σd1j=1 |x'j|2) Σd2k=1 ∂2x″k. We obtain weighted Plancherel estimates for the considered operators. As a consequence we prove Lp spectral multiplier results and Bochner-Riesz summability for the Grushin operators. These results are sharp if d1 ≥ d2. We discuss also an interesting phenomenon for weighted Plancherel estimates for d1 < d2. 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. © International Press 2012.