Transference of scale-invariant estimates from Lipschitz to Non-tangentially accessible to Uniformly rectifiable domains

TL;DR

This paper extends key estimates for elliptic PDE solutions from Lipschitz domains to more general uniformly rectifiable domains using a transference principle, broadening applicability beyond classical settings.

Contribution

It introduces a transference principle that allows estimates from Lipschitz domains to be applied to uniformly rectifiable domains for elliptic PDEs and related functions.

Findings

Carleson measure estimates hold in uniformly rectifiable domains.

Square function estimates are valid beyond Lipschitz domains.

The transference principle applies to higher order systems and subharmonic functions.

Abstract

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.