Let G be the semidirect product of N and A, where N is a stratified group and A = R acts on N via automorphic dilations. Homogeneous sub-Laplacians on N and A can be lifted to left-invariant operators on G and their sum is a sub-Laplacian Delta on G. Here we prove weak type (1, 1), L-p-boundedness for p is an element of (1, 2] and H-1 -> L-1 boundedness of the Riesz transforms Y Delta^{-1/2} and Y Delta^{-1} Z, where Y and Z are any horizontal left-invariant vector fields on G, as well as the corresponding dual boundedness results. At the crux of the argument are large-time bounds for spatial derivatives of the heat kernel, which are new when Delta is not elliptic.

Riesz transforms on solvable extensions of stratified groups / Martini, A; Vallarino, M. - In: STUDIA MATHEMATICA. - ISSN 0039-3223. - STAMPA. - 259:2(2021), pp. 175-200. [10.4064/sm190927-4-1]

Riesz transforms on solvable extensions of stratified groups

Martini, A;Vallarino, M
2021

Abstract

Let G be the semidirect product of N and A, where N is a stratified group and A = R acts on N via automorphic dilations. Homogeneous sub-Laplacians on N and A can be lifted to left-invariant operators on G and their sum is a sub-Laplacian Delta on G. Here we prove weak type (1, 1), L-p-boundedness for p is an element of (1, 2] and H-1 -> L-1 boundedness of the Riesz transforms Y Delta^{-1/2} and Y Delta^{-1} Z, where Y and Z are any horizontal left-invariant vector fields on G, as well as the corresponding dual boundedness results. At the crux of the argument are large-time bounds for spatial derivatives of the heat kernel, which are new when Delta is not elliptic.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11583/2903472