The Non-tangential Boundary Value Of The Generalized Harmonic Function In The Dunkl Framework

The purpose of this thesis is to study the boundary behavior of the generalized harmon-ic functions in the one-dimensional Dunkl setting.One important conclusion in harmonic analysis is that,the existence of the non-tangential limit of a harmonic function at almost every point in a measurable set on the boundary,could be determined by the non-tangential boundedness of the function at every point in the set.This is the local Fatou theorem.An-other deeper assertion is that,a harmonic function has a non-tangential limit at almost every point in a measurable set on the boundary,if and only if its Lusin area integral is finite at almost every point in the set.The Laplacian in the Dunkl setting is a second-order operator containing a first-order term and a reflection term with variable coefficients.The topics in the thesis are on the char-acterization of the existence of non-tangential boundary values in local sense of the general-ized harmonic functions in the Dunkl setting,and on the relationship of the non-tangential boundary value problem of the generalized harmonic functions and the area integral in the Dunkl setting.The main results in the thesis are the following two conclusions:(i)If a generalized harmonic function in the upper half-plane in the Dunkl setting is non-tangentially bounded at every point in a measurable set on the boundary,then the function has a non-tangential limit at almost every point in the set.(ii)A area integral associated with the Dunkl operator is introduced,and it is proved that,if a generalized harmonic function in the upper half-plane in the Dunkl setting has a non-tangential limit at every point in a measurable set on the boundary,its area integral is finite at almost every point in the set.