Lp-dimension free boundedness for Riesz transforms associated to Hermite functions

Abstract

Riesz transforms associated to Hermite functions were introduced by S. Thangavelu, who proved that they are bounded operators on L^p(R^d ), 1 < p < ∞ . In this paper we give a different proof that allows us to show that the Lp-norms of these operators are bounded by a constant not depending on the dimension d. Moreover, we define Riesz transforms of higher order and free dimensional estimates of the L^p-bounds of these operators are obtained. In order to prove the mentioned results we give an extension of the Littlewood-Paley theory that we believe of independent interest.