Integral Operators With General Measurable Kernels Dominated by Maximal Operators
by
Mircea Martin
Baker University, Baldwin City, KS, USA

The celebrated Hardy-Littlewood-Sobolev fractional integral theorem and its companion, the Hardy-Littlewood-Wiener maximal theorem, point out that the Riesz potential operators and the Hardy-Littlewood maximal operator on an Euclidean space are continuous on the standard Lebesgue spaces with exponents greater than one.

Our goal is to generalize these classical results to the setting of measure spaces with or without a topological or group structure. The main results completely characterize the kernels with the property that corresponding integral operstors are dominated by maximal operators naturally associated with those kernels.

As a direct application, we will derive a general qualitative Hartogs-Rosenthal type theorem on uniform approximation by solutions of elliptic equations.

Date received: May 6, 2002