Lipschitz Functions Associated With The Hankel Transform

The purpose of this thesis is to study theα-harmonic functions or the Poisson integrals associated with the Hankel transform and to give some L_α^p,q-mixed-norm estimates, Using which some characterizations are obtained for the Lipschitz-Hankel functions.It is hoped that these conclusions would be applied in studying the continuous linear functions on the Hardy spaces associated with the Hankel transform.The main works include the following three parts:(ⅰ) A mean value property forα-harmonic functions is proved;the L_α^p estimates for theα-harmonic functions and their partial derivatives are given;and it is verified that the conjugate Poisson integral preserves the increasing order in norm of the Poisson integral near the boundary; (ⅱ) some L_α^p,q-mixed-norm estimates of theα-harmonic functions and their partial derivatives are given;and the equivalent relations in L_α^p,q-mixed-norm among the partial derivatives of different orders are found;(ⅲ) Using the L_α^p,q-mixed-norm estimates of the partial derivatives of the Poisson integrals,some characterizations for the Lipschitz-Hankel functions are obtained.