Absolute continuity of degenerate elliptic measure

Abstract

Let $\Omega \subset R^{n+1}$ be an open set whose boundary may be composed of pieces of different dimensions. Assume that Ω satisfies the quantitative openness and connectedness, and there exist doubling measures m on Ω and μ on ∂Ω with appropriate size conditions. Let $Lu=-\mathrm{div}\left(A\nabla u\right)$ be a real (not necessarily symmetric) degenerate elliptic operator in Ω. Write ${\omega }_L$ for the associated degenerate elliptic measure. We establish the equivalence between the following properties: (i) ${\omega }_L\in A_{\infty }\left(\mu \right)$, (ii) the Dirichlet problem for L is solvable in $L^p\left(\mu \right)$ for some $p\in (1,\infty )$, (iii) every bounded null solution of L satisfies Carleson measure estimates with respect to μ, (iv) the conical square function is controlled by the non-tangential maximal function in $L^q\left(\mu \right)$ for all $q\in (0,\infty )$ for any null solution of L, and (v) the Dirichlet problem for L is solvable in $\mathrm{BMO}\left(\mu \right)$. On the other hand, we obtain a qualitative analogy of the previous equivalence. Indeed, we characterize the absolute continuity of ${\omega }_L$ with respect to μ in terms of local $L^2\left(\mu \right)$ estimates of the truncated conical square function for any bounded null solution of L. This is also equivalent to the finiteness μ-almost everywhere of the truncated conical square function for any bounded null solution of L.