J Austral Math Soc Ser A 45 pp78--82, 1988.

Hardy-Litlewood Maximal Functions on some Solvable Lie Groups

G. Gaudry, S. Giulini, A. Hulanicki and A. M. Mantero

Abstract

Let N be a nilpotent simply connected Lie group, and A be a commutative connected d-dimensional Lie group of automorphisms of N which correspond to semisimple endomorphisms of the Lie algebra of N with positive eigenvalues. From the split extension S = N × A @ N × a, a being the Lie algebra of A. We consider a family of "rectangles" B_r is S, parametrized by r > 0, such that the measure of B_r behaves asymptotically as a fixed power of r. One can construct the Hardy-Littlewood maximal function operator f ® M_f relative to left translates of teh family {B_r}. We prove that M is of weak type (1, 1). This complements a result of J.-O. Strömberg concerning maximal functions defined relative to hyperbolic balls in a symmetric space.

1980 AMS Subject Classification: primary 43A80, 22E30; secondary 42B25

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Authors

G. Gaudry

School of Mathematical Sciences, Flinders University of South Australia, Bedford Park, South Australia 5042, Australia.

S. Giulini

Dipartimento di Matematica, "F. Enriques", Università di Milano, Milano, Italy.

A. Hulanicki

Mathematics Institute, Wroclaw University, Wroclaw, Poland.

A. M. Mantero

Istituto di Matematiica, Università di Genova, Genova, Italy.