Absolute continuity of degenerate elliptic measure

Abstract

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

arXiv commentarXiv admin note: text overlap with arXiv:2101.06064

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