Elliptic Measures and Square Function Estimates on 1-Sided Chord-Arc Domains

Abstract In nice environments, such as Lipschitz or chord-arc domains, it is well-known that the solvability of Dirichlet problem for an elliptic operator in $$L^p$$ L p , some finite p equivalent to fact associated measure belongs Muckenhoupt class $$A_\infty $$ A ∞ . turn, any these conditions occurs if and only gradient every bounded null solution satisfies a Carleson estimate. This has been recently extended much rougher settings those 1-sided is, sets which are quantitatively open connected with boundary Ahlfors–David regular. this paper, we work same environment consider qualitative analog latter equivalence showing one can characterize absolute continuity surface respect terms finiteness almost everywhere truncated conical square function solution. As consequence our main result particularized Laplace previous results, show domain rectifiable harmonic function. addition, obtain two given operators $$L_1$$ 1 $$L_2$$ 2 property provided disagreement coefficients satisfy quadratic estimate cones vertex. Finally, case on either transpose its symmetric part corresponding upon assuming antisymmetric controlled oscillation