TRANSFERENCE OF SCALE-INVARIANT ESTIMATES FROM LIPSCHITZ TO NONTANGENTIALLY ACCESSIBLE TO UNIFORMLY RECTIFIABLE DOMAINS

In relatively nice geometric settings, in particular, on Lipschitz domains, absolute continuity of elliptic measure with respect to the surface measure is equivalent to Carleson measure estimates, to square function estimates, and to ε-approximability, for solutions to the second-order divergence-form elliptic partial differential equations Lu = − div(A∇u) = 0. In more general situations, notably, in an open set Ω with a uniformly rectifiable boundary, absolute continuity of elliptic measure with respect to the surface measure may fail, already for the Laplacian. In the present paper, extending and clarifying our previous work (Duke Math J. 165:12 (2016), 2331–2389), we demonstrate that nonetheless, Carleson measure estimates, square function estimates, and ε-approximability remain valid in such Ω, for solutions of Lu = 0, provided that such solutions enjoy these properties in Lipschitz subdomains of Ω. Moreover, we establish a general real-variable transference principle, from Lipschitz to chord-arc domains, and from chord-arc to open sets with uniformly rectifiable boundary, that is not restricted to harmonic functions or even to solutions of elliptic equations. In particular, this allows one to deduce the first Carleson measure estimates and square function bounds for higher-order systems on open sets with uniformly rectifiable boundaries and to treat subsolutions and subharmonic functions.