Applications on powers and singular operators and estimates for oscillatory integral operators

Abstract

We give remarks on the construction of linear evolution equations. Logarithms and some examples of unbounded imaginary powers of operators are considered, and we determined the best constants in Young's inequality and its converse with its generalization to more than three functions. We show a hypercontractivity for multipliers on the orthogonal polynomials and estimates for oscillatory integrals and their extended domains. We studied the boundedness of oscillatory singular integrals on Lebesgue and Hardy spaces. We established the differentiability of fractional powers of self- adjoint operators. We show some applications of the purely imaginary powers of operators in Hilbert spaces. We present the Contractivity of the free semigroups and singular operators on Boson fields as forms on spaces of entire functions on Hilbert spaces with uniform estimates for oscillatory integral .operators