Let N be a nilpotent simply connected Lie group, and A 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. Form the split extension S = N x A ≈ N x a, a being the Lie algebra of A. We consider a family of "rectangles" B_r in S, parameterized by r > 0, such that the measure of Br behaves asymptotically as a fixed power of r. One can construct the Hardy-Littlewood maximal function operator f —• Mf relative to left translates of the family {B_r}. We prove that M is of weak type (1,1). This complements a result of J.-O. Stromberg concerning maximal functions defined relative to hyperbolic balls in a symmetric space.

Hardy-Littlewood maximal functions on some solvable Lie groups

MANTERO, ANNA MARIA;GIULINI, SAVERIO;
1988-01-01

Abstract

Let N be a nilpotent simply connected Lie group, and A 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. Form the split extension S = N x A ≈ N x a, a being the Lie algebra of A. We consider a family of "rectangles" B_r in S, parameterized by r > 0, such that the measure of Br behaves asymptotically as a fixed power of r. One can construct the Hardy-Littlewood maximal function operator f —• Mf relative to left translates of the family {B_r}. We prove that M is of weak type (1,1). This complements a result of J.-O. Stromberg concerning maximal functions defined relative to hyperbolic balls in a symmetric space.
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11567/387283
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus ND
  • ???jsp.display-item.citation.isi??? ND
social impact