Endpoint estimates for higher order Gaussian Riesz transforms

Abstract

We will show that, contrary to the behavior of the higher order Riesz transforms studied so far on the atomic Hardy space H1 (Rn, γ), associated with the Ornstein-Uhlenbeck operator with respect to the n-dimensional Gaussian measure γ, the new Gaussian Riesz transforms are bounded from H1 (Rn, γ) to L1 (Rn, γ), for any order and dimension n. We will also prove that the classical Gaussian Riesz transforms of higher order are bounded from an adequate subspace of H1 (Rn, γ) into L1 (Rn, γ), extending T. Bruno (2019) for the first order case.