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Jones and Rosenblatt started the study of an ergodic transform which is analogous to the martingale transform. In this paper we present a unified treatment of the ergodic transforms associated to positive groups induced by nonsingular flows and to general means which include the usual averages, Cesàro-α averages and Abel means. We prove the boundedness in , 1 < p < ∞, of the maximal ergodic transforms assuming that the semigroup is Cesàro bounded in . For p = 1 we find that the maximal ergodic...
We characterize those homogeneous translation invariant symmetric non-local operators with positive maximum principle whose harmonic functions satisfy Harnack's inequality. We also estimate the corresponding semigroup and the potential kernel.
We develolp a new method to solve an evolution equation in a non-cylindrical domain, by reduction to an abstract evolution equation..
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