Dimension free \(L^P\) estimates for Riesz transforms associated with Laguerre function expansions of Hermite type.

*(English)*Zbl 1285.42023Dimension free \(L^p\) estimates for the classical Riesz transforms on \(\mathbb R^d\) were shown by E. M. Stein. Later, analogous results were proved for Riesz transforms defined in different settings.

In this paper the authors study dimension free \(L^p\) estimates for Riesz transforms associated with multi-dimensional Laguerre function expansions of Hermite type. They start with a brief overview of known results concerning dimension free \(L^p\) estimates for orthogonal expansions: the Hermite polynomial case, the Hermite function case, the Jacobi polynomial case and the Laguerre polynomial case.

The main result of the paper is to prove the dimension free \(L^p\) estimates for Riesz transforms naturally associated with multi-dimensional Laguerre expansions of Hermite type for the Laguerre type multi-index.

The authors use a technique, namely the method of \(g\)-functions. This technique, known as the Littlewood-Paley-Stein theory, occurred to be successful in treating the problem of dimensional free \(L^p\) estimates in several settings. In short, the main ingredient of this method consists in constructing appropriate \(g\)-functions defined in terms of some semigroups, that properly relate a function and its Riesz transform, and proving dimension free \(L^p\) bounds for these \(g\)-functions.

The authors define the relevant \(g\)-functions in terms of Poisson and modified Poisson semigroups and they state the corresponding \(L^p\) bounds. The line of arguments used by the authors follows some already proposed in a published paper.

In this paper the authors study dimension free \(L^p\) estimates for Riesz transforms associated with multi-dimensional Laguerre function expansions of Hermite type. They start with a brief overview of known results concerning dimension free \(L^p\) estimates for orthogonal expansions: the Hermite polynomial case, the Hermite function case, the Jacobi polynomial case and the Laguerre polynomial case.

The main result of the paper is to prove the dimension free \(L^p\) estimates for Riesz transforms naturally associated with multi-dimensional Laguerre expansions of Hermite type for the Laguerre type multi-index.

The authors use a technique, namely the method of \(g\)-functions. This technique, known as the Littlewood-Paley-Stein theory, occurred to be successful in treating the problem of dimensional free \(L^p\) estimates in several settings. In short, the main ingredient of this method consists in constructing appropriate \(g\)-functions defined in terms of some semigroups, that properly relate a function and its Riesz transform, and proving dimension free \(L^p\) bounds for these \(g\)-functions.

The authors define the relevant \(g\)-functions in terms of Poisson and modified Poisson semigroups and they state the corresponding \(L^p\) bounds. The line of arguments used by the authors follows some already proposed in a published paper.

Reviewer: María-José Cantero (Zaragoza)