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Let be an -type group and be its harmonic extension. We study a left invariant Hardy–Littlewood maximal operator on , 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 is of weak type .
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 . La parte cruciale della nostra presentazione consisterà nell'introduzione di una nuova teoria di spazi di Hardy...
Let be the Lie group endowed with the Riemannian symmetric space structure. Let be a distinguished basis of left-invariant vector fields of the Lie algebra of and define the Laplacian . In this paper we consider the first order Riesz transforms and , for . We prove that the operators , but not the , are bounded from the Hardy space to . We also show that the second-order Riesz transforms are bounded from to , while the transforms and , for , are not.
We consider a complete connected noncompact Riemannian manifold M with bounded geometry and spectral gap. We prove that the imaginary powers of the Laplacian and the Riesz transform are bounded from the Hardy space X¹(M), introduced in previous work of the authors, to L¹(M).
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