Dahlberg's theorem in higher co-dimension

Abstract

In 1977 the celebrated theorem of B. Dahlberg established that the harmonic measure is absolutely continuous with respect to the Hausdorff measure on a Lipschitz graph of dimension n−1 in Rn, and later this result has been extended to more general non-tangentially accessible domains and beyond.In the present paper we prove the first analogue of Dahlberg's theorem in higher co-dimension, on a Lipschitz graph Γ of dimension d in Rn, d<n−1, with a small Lipschitz constant. We construct a linear degenerate elliptic operator L such that the corresponding harmonic measure ωL is absolutely continuous with respect to the Hausdorff measure on Γ. More generally, we provide sufficient conditions on the matrix of coefficients of L which guarantee the mutual absolute continuity of ωL and the Hausdorff measure.