On The Mean Convergence of Biharmonic Functions

Abstract

Let denote the unit circle in the complex plane. Given a function , one uses t usual (harmonic) Poisson kernel for the unit disk to define the Poisson integral of , namely . Here we consider the biharmonic Poisson kernel for the unit disk to define the notion of -integral of a given function ; this associated biharmonic function will be denoted by . We then consider the dilations for and . The main result of this paper indicates that the dilations are convergent to in the mean, or in the norm of .