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Dimension free $$L^P$$ estimates for Riesz transforms associated with Laguerre function expansions of Hermite type. (English) Zbl 1285.42023
Dimension 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.

##### MSC:
 42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B25 Maximal functions, Littlewood-Paley theory
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