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Let \(L\) be the Laplacian on \(\mathbb{R}^n\). The investigation of necessary and sufficient conditions for an operator of the form \(F(L)\) to be bounded on \(L^p\) in terms of ``smoothness properties'' of the spectral multiplier \(F\) is a classical research area of harmonic analysis, with long-standing open problems (e.g., the Bochner-Riesz conjecture) and connections with the regularity theory of PDEs.
In settings other than the Euclidean, particularly in the presence of a sub-Riemannian geometric structure, the natural substitute \(L\) for the Laplacian need not be an elliptic operator, and it may be just hypoelliptic. In this context, even the simplest questions related to the \(L^p\)-boundedness of operators of the form \(F(L)\) are far from being completely understood.
I will present some recent results, obtained in joint works with Detlef Müller (Kiel) and Adam Sikora (Sydney), dealing with the case of sublaplacians on \(2\)-step stratified (Carnot) groups, and with Grushin operators.
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References- A. Martini, A. Sikora,
Weighted Plancherel estimates and sharp spectral multipliers for the Grushin operators ,
Math. Res. Lett. 19, 2012, 1075-1088.
- A. Martini, D. Müller,
A sharp multiplier theorem for Grushin operators in arbitrary dimensions , 2012,
to appear in Rev. Mat. Iberoam., arXiv:1210.3564.
- A. Martini, D. Müller,
\(L^p\) spectral multipliers on the free group \(N_{3,2} \)Studia
Math. 217, 2013, 41-55.
- A. Martini,
Spectral multipliers on Heisenberg-Reiter and related groups ,
Ann. Mat. Pura Appl., 2014 (online first), doi:10.1007/s10231-014-0414-6.
- A. Martini, D. Müller,
Spectral multiplier theorems of Euclidean type on new classes of 2-step stratified groups , 2013,
to appear in Proc. Lond. Math. Soc., arXiv:1306.0387.
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