Growth properties for generalized Riesz potentials in central Herz-Morrey spaces (The structure of function spaces and its environment)

Abstract

Riesz decomposition theorem says that a superharmonic function on the punctured unit ball B_{0} is represented as the sum of a generalized potential and a harmonic function outside the origin. Our first aim in this note is to study growth properties near the origin for generalized Riesz potentials of functions in central Herz-Morrey spaces on B_{0}. We know another Riesz decomposition theorem which says that a superharmonic function on the unit ball B is represented as the sum of another generalized potential and a harmonic function on B . Our second aim in this note is to obtain growth properties near the boundary partial B for generalized Riesz potentials of functions in central Herz-Morrey spaces on B. A continuous function u on an open set $Omega$ is called monotone in the sense of Lebesgue [18] if for every relatively compact open set Gsubset $Omega$, displaystyle mathrm{m}{frac{mathrm{a}{G}mathrm{x}u=max upartial G and displaystyle mathrm{m}{frac{mathrm{i}{G}mathrm{n}u=min_{partial G}u. Harmonic functions on $Omega$ are monotone in $Omega$. More generally, solutions of elliptic partial differential equations of second order and weak solutions for variational problems may be monotone (see [15]). Our final aim in this note is concerned with growth properties for monotone Sobolev functions in central Herz-Morrey spaces.