Spectral multipliers for sub-Laplacians with drift on Lie groups |
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Authors: | Waldemar?Hebisch Email author" target="_blank">Giancarlo?MauceriEmail author Stefano?Meda |
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Institution: | (1) Institut Matematyczny, Uniwersytet Wrocławki, pl, Grunwaldzki 2/4, 50-384 Wrocław, Poland;(2) Dipartimento di Matematica, Università di Genova, via Dodecaneso 35, 16146 Genova, Italy;(3) Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca, via Roberto Cozzi 53, 20125 Milano, Italy |
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Abstract: | We study spectral multipliers of right invariant sub-Laplacians with drift on a connected Lie group G. The operators we consider are self-adjoint with respect to a positive measure , whose density with respect to the left Haar measure λG is a nontrivial positive character of G. We show that if p≠2 and G is amenable, then every spectral multiplier of extends to a bounded holomorphic function on a parabolic region in the complex plane, which depends on p and on the drift. When G is of polynomial growth we show that this necessary condition is nearly sufficient, by proving that bounded holomorphic functions
on the appropriate parabolic region which satisfy mild regularity conditions on its boundary are spectral multipliers of .
Work partially supported by the EC HARP Network “Harmonic Analysis and Related Problems”, the Progetto Cofinanziato MURST
“Analisi Armonica” and the Gruppo Nazionale INdAM per l'Analisi Matematica, la Probabilità e le loro Applicazioni. Part of
this work was done while the second and the third author were visiting the “Centro De Giorgi” at the Scuola Normale Superiore
di Pisa, during a special trimester in Harmonic Analysis. They would like to express their gratitude to the Centro for the
hospitality. |
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Keywords: | 47A60 42B15 47D03 60G15 |
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