A Sum Operator with Applications to Self-Improving Properties of Poincar� Inequalities in Metric Spaces

Abstract

We define a class of summation operators with applications to the self-improving nature of Poincare–Sobolev estimates, in fairly general quasimetric spaces of homogeneous type. We show that these sum operators play the familiar role of integral operators of potential type (e.g., Riesz fractional integrals) in deriving Poincare–Sobolev estimates in cases when representations of functions by such integral operators are not readily available. In particular, we derive norm estimates for sum operators and use these estimates to obtain improved Poincare–Sobolev results.