Applications of the theory of Hardy spaces to harmonic analysis on the Heisenberg group

Abstract

The aim of this paper is to prove some results about the (non-commutative) Fourier transform on the Heisenberg groups H n. These results are all consequences ofa"containing Banach space" embedding result for H p spaces (Theorems 2.1 and 2.2). Theorems 4.4 and 4.5 are of interest because they give estimates for the size of Fourier transforms of elements of L p and H p. These estimates differ from Hausdorff-Young phenomena in that they do not involve fractional powers of operators. Theorem 4.3 gives sufficient conditions for an operator valued function M(2) (not necessarily diagonal) to be a "Fourier multiplier" of H q to L 2. More generally, Theorems 3.2 and 3.3 give sufficient conditions for a tempered distribution to be a convolution operator from H q to L p where 0 < q < 1 and l <_p<_ oo(p4=q). We mention that some important ingredients of this paper are the atomic definition of H p, the work of Geller [5], and the dyadic decomposition of the identity introduced by de Michele and Mauceri [2]. Related material can be found in [9].