  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
$\varepsilon$-approximability, for solutions to the second order divergence
form elliptic partial differential equations $ Lu= -{\rm div\,} (A \nabla
u)=0$. In more general situations, notably, in an open set $\Omega$ 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, the authors demonstrate that nonetheless, Carleson measure
estimates, square function estimates, and $\varepsilon$-approximability remain
valid in such $\Omega$, for solutions of $Lu=0$, provided that such solutions
enjoy these properties in Lipschitz subdomains of $\Omega$.
  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.
