On the basis of the defination of K-integral,this essay deducts out the relation between K-analytic function and K-integral and the relation between K-analytic function and K-harmonic functions.

Moreover, it is also proved that solutions of the Dirichlet boundary value problems for above two classes of functions and the k-harmonic function uniquely exist.

A partial differential equations is introduced on the basis of the definitions of k-hypermonogenic function with vector value and the k-hyperbolically harmonic function,then the porperties of k-hypermonogenic function with vector value and their relations are discussed,at last a sufficient and necessary condition for the solvability of partial differential equations is obtained.

Integral Representation and Estimation of Harmonic Functions in Half-Plane;

The Dirichlet boundary value problem for harmonic function;

Then,we further obtain that each bounded weak solution is of sharp Hlder exponent with anyγ:0≤γ<k under the additional data regularity assumptions,where k is just as the local Hlder index of A-harmonic functions.

karp to prove that on such kind of 2 Mfd which curvature k≥-1r 2 log r outside a compact set there exist no nonconstant subharmonic functions which bounded f.

In this paper, applying Life theory of complex-functional, not only the space harmonic functions in polynomial form but also the spherical functions are obtained.

It is proved for harmonic functions an integral identity.