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A maximal function on harmonic extensions of $H$-type groups

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A maximal function on harmonic extensions of H-type groups ANNALES MATHÉMATIQUES BLAISE PASCAL Maria Vallarino A maximal function on harmonic extensions of H-type groups Volume 13, no 1 (2006), p. 87-101. <http://ambp.cedram.org/item?id=AMBP_2006__13_1_87_0> © Annales mathématiques Blaise Pascal, 2006, tous droits réservés. L’accès aux articles de la revue « Annales mathématiques Blaise Pas- cal » (http://ambp.cedram.org/), implique l’accord avec les condi- tions générales d’utilisation (http://ambp.cedram.org/legal/). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Publication éditée par les laboratoires de mathématiques de l’université Blaise-Pascal, UMR 6620 du CNRS Clermont-Ferrand — France cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Annales mathématiques Blaise Pascal 13, 87-101 (2006) A maximal function on harmonic extensions of H-type groups Maria Vallarino Abstract Let N be anH-type group and S ' N×R+ be its harmonic extension. We study a left invariant Hardy–Littlewood maximal operatorMRρ on S, obtained by taking maximal averages with respect to the right Haar measure over left-translates of a family R of neighbourhoods of the identity. We prove that the maximal operator MRρ is of weak type (1, 1). 1. Introduction Let n be a Heisenberg type Lie algebra (briefly, an H-type Lie algebra) with inner product 〈·, ·〉 and corresponding norm | · | . We denote by N the connected and simply connected Lie group associated to n; N is called an H-type group. Let S be the one-dimensional extension of N obtained by letting A = R+ act on N by homogeneous dilations. Let H denote a vector in a acting on n with eigenvalues 1/2 and (possibly) 1; we extend the inner product on n to the algebra s = n ⊕ a by requiring n and a to be orthogonal and H u

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