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In the recent work [DFM1, DFM2] G. David, J. Feneuil, and the first authorhave launched a program devoted to an analogue of harmonic measure forlower-dimensional sets. A relevant class of partial differential equations,analogous to the class of elliptic PDEs in the classical context, is given bylinear degenerate equations with the degeneracy suitably depending on thedistance to the boundary.  The present paper continues this line of research and focuses on the criteriaof quantitative absolute continuity of the newly defined harmonic measure withrespect to the Hausdorff measure, $\omega\in A_\infty(\sigma)$, in terms ofsolvability of boundary value problems. The authors establish, in particular,square function estimates and solvability of the Dirichlet problem in BMO fordomains with lower-dimensional boundaries under the underlying assumption$\omega\in A_\infty(\sigma)$. More generally, it is proved that in all domainswith Ahlfors regular boundaries the BMO solvability of the Dirichlet problem isnecessary and sufficient for the absolute continuity of the harmonic measure.

Square function estimates, BMO Dirichlet problem, and absolute continuity of harmonic measure on lower-dimensional sets