Uniform coarse ellipticity, $A_2$ weights, and boundary regularity

Abstract

We study symmetric divergence form elliptic equations with measurable coefficients that may be degenerate and unbounded, under a uniform bound on the coarse ellipticity ratio over all ambient cubes and scales. We prove that uniform coarse ellipticity and classical $A_2$ weighted ellipticity define incomparable coefficient classes. For every admissible choice of coarse parameters, we construct a scalar weight with a logarithmically corrected, locally integrable singularity along a product Cantor set. Its codimension is chosen according to the upper coarse discount exponent, and the resulting coefficient satisfies the uniform coarse condition but is comparable to no $A_2$ weight. Conversely, an explicit $A_2$ weight fails the coarse condition for every admissible parameter choice. Under uniform coarse ellipticity we give a direct PDE treatment of boundary regularity in the approximable energy realization. Using coarse interior estimates, we prove boundary decay and solvability of the continuous Dirichlet problem in bounded NTA domains. An almost-positivity argument, with energy approximation and boundary testing justified, yields boundary Harnack and Hölder continuity of quotients of positive solutions with common zero boundary values. The argument requires no additional energy Sobolev inequality and avoids the construction of Green functions. Further examples distinguish uniform coarse control from finite coefficient moments and from coercivity on $\mathrm L^2(\mathrm{d} x)$. We also record interior Hölder estimates and continuity criteria under nonuniform control along nested scales.