The F. and M. Riesz theorem without connectivity
Abstract
A classical theorem of F. and M. Riesz states that for a simply connected domain in the complex plane with a rectifiable boundary, harmonic measure and arc length measure on the boundary are mutually absolutely continuous. On the other hand, an example of C. Bishop and P. Jones shows that the latter conclusion may fail, in the absence of some sort of connectivity hypothesis.
In this work, we nonetheless establish versions of the F. and M. Riesz theorem, in higher dimensions, in which other properties of har¬monic functions substitute for the absolute continuity of harmonic measure. These substitute properties are natural “proxies” for har¬monic measure estimates, in the sense that, in more topologically be¬nign settings, they are actually equivalent to a scale-invariant quanti¬tative version of absolute continuity.