Two Classes Of Boundary Value Problems In Clifford Analysis And The H(?)lder Continuity Of Pompeiu Operator T In Quaternionic Analysis

This paper mainly studies two classes of boundary valueproblems in Clifford analysis and the Ho|¨lder continuity of Pompeiu operator Tin quaternionic analysis by using complex analysis method. In Chapter one, weconsider the Plemelj's formula and a class of nonlinear boundary value problemsfor generalized regular function in Clifford analysis. Applying the method ofintegral equations and Schauder's fixed point theorem, we prove the existenceof solution for the above problem and obtain the integral representation ofthe solution. In Chapter two, similar to one complex variable theory, wediscuss some properties of k-regular function in Clifford analysis, such as itsrepresentation, Cauchy type integral, Plemelj's formula. Then we study itsRiemann boundary value problem, obtain the solvability of this problem andthe integral representation of the solution to this problem. In last Chapter,we deal with some properties of the Pompeiu operator T for non-homogeneousDirac equations in functional spaces L_p,v(Q) of quaternionic analysis. we showthat Pompeiu operator T is Ho|¨lder continuous in the space Q and discuss theproperty of the function T_Qf(z) at∞.