Continuous Linear Functionals On Hardy Spaces Associated With Ultraspherical Series

The purpose of this thesis is to study the generalized analytic functions on the unit disk associated with ultraspherical series and the measure |sinθ|^2λdθ(λ> 0). A function f on the unit disk is called to beλ-analytic if T_zf:= (?)f/(?)z-λf(z)-f(z)/z-z= 0. The main works include the following three parts:(i) The properties of the differential-reflection operators and the basis functions are studied, and the A-harmonic functions and the A-analytic functions are characterized by " power series " in terms of the ultraspherical polynomials;(ii) The H_λ^p estimates for the Cauchy kernel C(z,ω) and T_ωC(z,ω) are given;(iii) For p near to 1, i.e.2λ+1/2λ+2< p < 1, some representations of the continuous linear functional on the Hardy spaces associated with ultraspherical series are obtained.