On Higher Riesz Transforms for Gaussian Measures

Abstract

Given an n × n matrix B, positive definite and symmetric, let L_B be the differential operator in Rⁿ given by $L_B=\frac{1}{2}\Delta -Bx\cdot \mbox{grad}_x.$ A class of higher Riesz transforms associated with L_B is defined by means of higher gradients of order k. It is shown that these transformations are bounded in the space $L^p,\ 1<p<\infty,$ with respect to the measure that makes L_B selfadjoint. The constants obtained are independent of the dimension n and depend only on k,p, and the number of different eigenvalues of the matrix B. The proof of the results uses an extension of the Littlewood-Paley-Stein theory of square functions to the vector-valued case and inequalities previously proved by one of the authors in the context of the Riesz transforms of order one.