Endpoint estimates for first-order Riesz transforms associated to the Ornstein-Uhlenbeck operator

TL;DR

This paper investigates the boundedness of first-order Riesz transforms related to the Ornstein-Uhlenbeck operator on Gaussian spaces, revealing dimension-dependent endpoint estimates between Hardy, L^1, BMO, and L^ spaces.

Contribution

It characterizes which Riesz transforms are bounded at the endpoints on Gaussian Hardy and BMO spaces, highlighting the influence of ambient space dimension.

Findings

Endpoint boundedness depends on the space dimension.

Some Riesz transforms are bounded from H^1(g) to L^1(g).

Other transforms are bounded from L^(g) to BMO(g).

Abstract

In the setting of Euclidean space with the Gaussian measure g, we consider all first-order Riesz transforms associated to the infinitesimal generator of the Ornstein-Uhlenbeck semigroup. These operators are known to be bounded on L^p(g), for 1<p<\infty. We determine which of them are bounded from H^1(g) to L^1(g) and from L^\infty(g) to BMO(g). Here H^1(g) and BMO(g) are the spaces introduced in this setting by the first two authors. Surprisingly, we find that the results depend on the dimension of the ambient space.