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Cubo (Temuco)

versión On-line ISSN 0719-0646

Cubo vol.19 no.2 Temuco jun. 2017

http://dx.doi.org/10.4067/S0719-06462017000200011 

Articles

On the hypercontractive property of the Dunkl-Ornstein-Uhlenbeck semigroup

Iris A. López1 

1 Universidad Simón Bolivar, Departamento de Matemáticas Puras y Aplicadas, Aptdo 89000. Caracas 1080-A. Venezuela. e-mail: iathamaica@usb.ve

Abstract:

The aim of this paper is to prove the hypercontractive propertie of the Dunkl-Ornstein-Uhlenbeck semigroup, . To this end, we prove that the Dunkl-Ornstein-Uhlenbeck differential operator Lk with k ≥ 0 and associated to the group, satisfies a curvature-dimension inequality, to be precise, a C(ρ, ∞)-inequality, with 0≤ρ≤1. As an application of this fact, we get a version of Meyer's multipliers theorem and by means of this theorem and fractional derivatives, we obtain a characterization of Dunkl-potential spaces.

Keywords and Phrases: Dunkl-Ornstein-Uhlenbeck operator; generalized Hermite polynomial; squared field operator; Meyer's multiplier theorem; Dunkl-potential space; fractional integral; fractional derivative.

Resumen:

El objetivo de este artículo es demostrar la propiedad hipercontractiva del semigrupo de Dunkl-Ornstein-Uhlenbeck, .. Para lograr esto, probamos que el operador diferencial de Dunkl-Ornstein-Uhlenbeck Lk con k ≥ 0 y asociado al grupo , satisface una desigualdad de curvatura-dimensión, para ser precisos, una C(ρ,∞)-desigualdad, con 0≤ρ≤1. Como una aplicación de este hecho, obtenemos una versión del teorema de multiplicadores de Meyer y a través de este teorema y derivadas fraccionales, obtenemos una caracterización de espacios Dunkl-potenciales.

2010 AMS Mathematics Subject Classification: 33C45, 6A33, 33C52.

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