estimates of Cauchy transform with function Ornstien –Uhlenbeck operator

Abstract

We study the Lebesgue space affine isoperimetric inequalities which deal with the inequalities for centroid and projection bodies . we prove the volume inequalities for subspaces of the Lebesgue space and duality of the Lebesgue space affine isoperimetric inequalities. We establish the functional calculus for Ornstein – Uhlenbeck operator and the Fourier integral operators on noncompact symmetric spaces of real rank equal one. The affine and sharp Sobolev inequality and isoperimetric inequalities split twice are considered . We show the weighted norm inequalities and the harmonic analysis of elliptic operators. We investigate the good and Calderion –Zygmund methods .We give some properties of the Cauchy transform on a bounded domain and show the best possible estimates of the second term in the spectral asymptotic of Cauchy transform . We discum the holomorphy of the spectral multipliers and bounded mean oscillation with atomic Hardy space of the Ornstein – Uhlenbeck operator