Riesz transform and integration by parts formulas for random variables

TL;DR

This paper develops estimates for the Riesz transform's L^p norms using integration by parts formulas, leading to regularity results for densities of Wiener space functionals and a semi-distance for boundary convergence.

Contribution

It introduces new estimates for the Riesz transform on Wiener space and links these to regularity and boundary behavior of densities of non-degenerate functionals.

Findings

Derived L^p norm estimates for the Riesz transform

Established regularity and density estimates for Wiener space functionals

Introduced a semi-distance characterizing boundary convergence

Abstract

We use integration by parts formulas to give estimates for the $L^p$ norm of the Riesz transform. This is motivated by the representation formula for conditional expectations of functionals on the Wiener space already given in Malliavin and Thalmaier. As a consequence, we obtain regularity and estimates for the density of non degenerated functionals on the Wiener space. We also give a semi-distance which characterizes the convergence to the boundary of the set of the strict positivity points for the density.