Singular Spherical Maximal Operators on a Class of Step Two Nilpotent Lie Groups

SINGULAR SPHERICAL MAXIMAL OPERATORS ONA CLASS OF TWO STEP NILPOTENT LIE GROUPSDetlef M uller

Singular Spherical Maximal Operators on a Class of Two Step Nilpotent Lie Groups

Let H be the Heisenberg group and let μt be the normalized surface measure for the sphere of radius t in R. Consider the maximal function defined by Mf = supt>0 |f ∗μt|. We prove for n ≥ 2 that M defines an operator bounded on L(H) provided that p > 2n/(2n − 1). This improves an earlier result by Nevo and Thangavelu, and the range for L boundedness is optimal. We also extend the result to a mor...

The Space of Bounded Spherical Functions on the Free Two Step Nilpotent Lie Group

Let N be a connected and simply connected 2-step nilpotent Lie group and K be a compact subgroup of Aut(N). We say that (K,N) is a Gelfand pair when the set of integrable K-invariant functions on N forms an abelian algebra under convolution. In this paper, we construct a one-to-one correspondence between the set ∆(K, N) of bounded spherical functions for such a Gelfand pair and a set A(K, N) of...

The Heat Kernel for H-type Groups

Theorem 1 gives an explicit formula for the heat kernel on an H -type group. Folland (2] has shown that for stratified nilpotent Lie groups the heat semigroup is a semigroup of kernel operators on LP, 1 5 p < oo and on Co. Cygan (1] has obtained formulas for heat kernels for any two step nilpotent simply connected Lie group. Cygan found the heat kernel for a free simply connected two step nilpo...

Computing Maximal Tori Using LiE and Mathematica

This paper describes an algorithm for computing maximal tori of the reductive centralizer of a nilpotent element of an exceptional complex symmetric space. It illustrates also a good example of the use of Computer Algebra Systems to help answer important questions in the field of pure mathematics. Such tori play a fundamental rôle in several problems such as: classification of nilpotent orbits ...