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 $\mathbb R^n$, 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 $\Gamma$ of dimension $d$ in $\mathbb R^n$, $d<n-1$, with a small Lipschitz constant. We construct a linear degenerate elliptic operator $L$ such that the corresponding harmonic measure $\omega_L$ is absolutely continuous with respect to the Hausdorff measure on $\Gamma$. More generally, we provide sufficient conditions on the matrix of coefficients of $L$ which guarantee the mutual absolute continuity of $\omega_L$ and the Hausdorff measure.