Inequalities of Riesz-Laguerre Transforms and Symmetrization of Logarithmic Sobolev Inequalities

Abstract

We give jointly the needed relations between Riesz transforms and g- function for Laguerre expansions. We also stated and introduced aSobolev space for the Laguerre function system. Norm estimates for operators associated with the Ornstein-Uhlenbeck semigroup are considered. For anon-symmetric Ornstein-Uhlenbeck semigroup we used Riesz transforms .We characterizes higher order Riesz transforms, fractional , derivatives, and special Sobolev spaces for Laguerre expansions. We also show higher Riesz transforms and certain imaginary powers concerned to the harmonic oscillator. We give structure for Riesz transforms, g-functions, and multipliers for the Laguerre semigroup. We investigate an orthogonal expansions for Riesz transforms .We study Riesz transforms for multi-dimensional Laguerre function expansions and logarithmic Sobolev inequalities with spectral gaps. We also present isoperimetry and symmetrization for logarithmic Sobolev inequalities, in addition we determined the spectral multipliers of Laplace transform type for the Laguerre operator. We discuss and give the weak type estimates and inequalities for higher order Riesz– Laguerre transforms.