We study the functional calculus associated with a hypoelliptic left-invariant differential operator L on a connected and simply connected nilpotent Lie group G with the aid of the corresponding Rockland operator L-0 on the `local' contraction G(0) of G, as well as of the corresponding Rockland operator L-infinity on the `global' contraction G(infinity) of G. We provide asymptotic estimates of the Riesz potentials associated with L at 0 and at infinity, as well as of the kernels associated with functions of L satisfying Mihlin conditions of every order. We also prove some Mihlin-Hormandermultiplier theorems for L which generalize analogous results to the non-homogeneous case. Finally, we extend the asymptotic study of the density of the `Plancherelmeasure' associated with L from the case of a quasi-homogeneous sub-Laplacian to the case of a quasi-homogeneous sum of even powers.

Functional calculus on non-homogeneous operators on nilpotent groups / M. Calzi, F. Ricci. - In: ANNALI DI MATEMATICA PURA ED APPLICATA. - ISSN 0373-3114. - 200:4(2021), pp. 1517-1571. [10.1007/s10231-020-01047-5]

Functional calculus on non-homogeneous operators on nilpotent groups

M. Calzi
Primo
;
2021

Abstract

We study the functional calculus associated with a hypoelliptic left-invariant differential operator L on a connected and simply connected nilpotent Lie group G with the aid of the corresponding Rockland operator L-0 on the `local' contraction G(0) of G, as well as of the corresponding Rockland operator L-infinity on the `global' contraction G(infinity) of G. We provide asymptotic estimates of the Riesz potentials associated with L at 0 and at infinity, as well as of the kernels associated with functions of L satisfying Mihlin conditions of every order. We also prove some Mihlin-Hormandermultiplier theorems for L which generalize analogous results to the non-homogeneous case. Finally, we extend the asymptotic study of the density of the `Plancherelmeasure' associated with L from the case of a quasi-homogeneous sub-Laplacian to the case of a quasi-homogeneous sum of even powers.
Nilpotent Lie groups; Hypoelliptic differential operators; Multiplier theorem; Heat kernel; Riesz potentials
Settore MAT/05 - Analisi Matematica
2021
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/1013477
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