An Identity with Applications to Harmonic Measure

In this note we use an elementary integral formula to give a new short proof of the theorem of B. E. J. Dahlberg [2] on the mutual absolute continuity of harmonic and surface measure on bounded Lipschitz domains. (See [4] for the relevant definitions.) The formula also provides a new proof of the so-called reverse Holder inequality (see [2]), and //-estimates for the Dirichlet problem (see [3]). Furthermore, we are able to treat domains whose boundaries are worse than Lipschitz. In particular, we show that if D C R is given locally (in the appropriate sense) by the graph of a function 0 with V0 £ L,p > n — 1, and D is regular for the Dirichlet problem, then harmonic measure and surface measure on dD are mutually absolutely continuous. The hypothesis of regularity is superfluous when n — 3, or under the stronger assumption V0 G BMO, but not necessarily otherwise. We wish to thank Professor E. M. Stein for many helpful conversations.