We study the Grushin operators acting on ℝd 1x' × ℝd 2 x″ and defined by the formula L = -Σd1 j=1∂2 x'j- (Σd1 j=1 |x'j|2) Σd2 k=1 ∂2 x″ 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.
|Number of pages||14|
|Journal||Mathematical Research Letters|
|Publication status||Published - 2012|