Invariant mean value property and the associated integral equations

Abstract

In this paper we consider a class of integral equations associated with the invariant mean value property for M-harmonic functions. We have shown that nonconstant solutions of the integral equations are functions of unbounded variation and do not attain their supremum or infimum on [0, 1]. We also discuss in detail the behavior of the kernel of the corresponding integral operator and obtained certain growth estimates of the integral operator.