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.
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