A Sum Operator withApplications toSelf-Improving Properties ofPoincaré Inequalities inMetric Spaces

We define a class of summation operators with applications to the self-improvingnature of Poincaré–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 Poincaré–Sobolev estimates in cases when representationsof functions by such integral operators are not readily available. In particular, we derive normestimates for sum operators and use these estimates to obtain improved Poincaré–Sobolev results.