Archivio Istituzionale della RicercaLet L be a sub-Laplacian on a connected Lie group G of polynomial growth. It is well known that, if F:R→C is in the Schwartz class S(R), then the convolution kernel KF(L) of the operator F(L) is in the Schwartz class S(G). Here we prove a sort of converse implication for a class of groups G including all solvable noncompact groups of polynomial growth. We also discuss the problem whether integrability of KF(L) implies continuity of F.
Convolution kernels versus spectral multipliers for sub-Laplacians on groups of polynomial growth / Martini, A.; Ricci, F.; Tolomeo, L.. - In: JOURNAL OF FUNCTIONAL ANALYSIS. - ISSN 0022-1236. - STAMPA. - 277:6(2019), pp. 1603-1638. [10.1016/j.jfa.2019.05.024]
Convolution kernels versus spectral multipliers for sub-Laplacians on groups of polynomial growth
Martini A.;Ricci F.;
2019
Abstract
Let L be a sub-Laplacian on a connected Lie group G of polynomial growth. It is well known that, if F:R→C is in the Schwartz class S(R), then the convolution kernel KF(L) of the operator F(L) is in the Schwartz class S(G). Here we prove a sort of converse implication for a class of groups G including all solvable noncompact groups of polynomial growth. We also discuss the problem whether integrability of KF(L) implies continuity of F.
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https://hdl.handle.net/11583/2949494