Pointwise Ergodic Theorems for Radial Averages on the Heisenberg Group

✍ Scribed by Amos Nevo; Sundaram Thangavelu

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 423 KB
• Volume: 127
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1006/aima.1997.1641

✦ Synopsis

Let H=H n =C n _R denote the Heisenberg group, and let _ r denote the normalized Lebesgue measure on the sphere [(z, 0): |z| =r]. Let (X, B, m) be a standard Borel probability space on which H acts measurably and ergodically by measure preserving transformations, and let ?(_ r ) denote the operator canonically associated with _ r on L p (X ). We prove maximal and pointwise ergodic theorems in L p , for radial averages _ r on the Heisenberg group H n , n>1. The results are best possible for actions of the reduced Heisenberg group. The method of proof is to use the spectral theory of the Banach algebra of radial measures on the group and decay estimates for its characters to establish maximal inequalities using spectral methods, in particular Littlewood Paley Stein square-functions and analytic interpolation.

1997 Academic Press

DEFINITIONS AND STATEMENT OF RESULTS

1.1. Definitions

The present paper will establish maximal and pointwise ergodic theorems for radial averages on the Heisenberg group H=H n , the reduced Heisenberg group and the Heisenberg motion group. (Note: in what follows, the terms radial averages'' and spherical means'' are interchangeable.) We begin by recalling the relevant definitions (see [N1], [N2]).

Let (X, B, m) be a standard Borel probability space and let the locally compact second countable group G have a Borel measurable action on X, article no.