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A maximal function on harmonic extensions of H -type groups

Maria Vallarino — 2006

Annales mathématiques Blaise Pascal

Let N be an H -type group and S N × + be its harmonic extension. We study a left invariant Hardy–Littlewood maximal operator M ρ on S , obtained by taking maximal averages with respect to the right Haar measure over left-translates of a family of neighbourhoods of the identity. We prove that the maximal operator M ρ is of weak type ( 1 , 1 ) .

Spazi di Hardy su gruppi a crescita esponenziale di volume

Maria Vallarino — 2013

Bollettino dell'Unione Matematica Italiana

Questa è una rassegna di alcuni risultati recenti su spazi di Hardy nel contesto di gruppi di Lie a crescita esponenziale di volume, che ho presentato nella conferenza da me tenuta a Bologna in occasione del XIX Congresso dell'Unione Matematica Italiana. Faremo un breve cenno alla teoria degli spazi di Hardy in ambito euclideo e al ruolo svolto da tali spazi nell'analisi armonica su n . La parte cruciale della nostra presentazione consisterà nell'introduzione di una nuova teoria di spazi di Hardy...

Boundedness from H 1 to L 1 of Riesz transforms on a Lie group of exponential growth

Peter SjögrenMaria Vallarino — 2008

Annales de l’institut Fourier

Let G be the Lie group 2 + endowed with the Riemannian symmetric space structure. Let X 0 , X 1 , X 2 be a distinguished basis of left-invariant vector fields of the Lie algebra of G and define the Laplacian Δ = - ( X 0 2 + X 1 2 + X 2 2 ) . In this paper we consider the first order Riesz transforms R i = X i Δ - 1 / 2 and S i = Δ - 1 / 2 X i , for i = 0 , 1 , 2 . We prove that the operators R i , but not the S i , are bounded from the Hardy space H 1 to L 1 . We also show that the second-order Riesz transforms T i j = X i Δ - 1 X j are bounded from H 1 to L 1 , while the transforms S i j = Δ - 1 X i X j and R i j = X i X j Δ - 1 , for i , j = 0 , 1 , 2 , are not.

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