Weighted Plancherel estimates and sharp spectral multipliers for the Grushin operators. (English) Zbl 1288.47044
The Grushin operator on \({\mathbb R}^{d_1}\times{\mathbb R}^{d_2}\) is defined by
\[
L=-\Delta_{x'}-|x'|^2\Delta_{x''},
\]
where \((x',x'')\in{\mathbb R}^{d_1}\times{\mathbb R}^{d_2}\) and \(\Delta_{x'}\) and \(\Delta_{x''}\) are the Laplacians in the variables \(x'\) and \(x''\).
The authors of the article under review find conditions on a function \(F\), under which the operator \(F(L)\) has weak type \((1,1)\) or is bounded on \(L^p\) with \(1<p<\infty\).
where \((x',x'')\in{\mathbb R}^{d_1}\times{\mathbb R}^{d_2}\) and \(\Delta_{x'}\) and \(\Delta_{x''}\) are the Laplacians in the variables \(x'\) and \(x''\).
The authors of the article under review find conditions on a function \(F\), under which the operator \(F(L)\) has weak type \((1,1)\) or is bounded on \(L^p\) with \(1<p<\infty\).
Reviewer: Vladimir V. Peller (East Lansing)
MSC:
47F05 | General theory of partial differential operators |
35R03 | PDEs on Heisenberg groups, Lie groups, Carnot groups, etc. |
47A60 | Functional calculus for linear operators |
42A45 | Multipliers in one variable harmonic analysis |