Hardy-Littlewood maximal functions on some solvable Lie groups

Journal of the Australian Mathematical Society

G. Gaudry¹ 

[1] S. Giulini

关键词: primary 43 A 80;    22 E 30;    secondary 42 B 25;   

DOI  :  10.1017/S1446788700032286

学科分类：数学（综合）

来源: Cambridge University Press

【 摘 要 】

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 × A ≅ N × a, a being the Lie algebra of A. We consider a family of “rectangles” Br 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 {Br}. 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.

【 授权许可】

Unknown