Boundary Harnack principle for abstract supersolutions on uniform domains

Abstract

We develop an axiomatic approach to the boundary Harnack principle on diameter-uniform domains in proper doubling metric measure spaces. Under critical-density and upper-contraction assumptions, we establish a scale-invariant one-sided boundary comparison between nonnegative supersolutions and positive solutions and Hölder continuity of quotients of positive solutions. Our framework does not require linearity or rectifiable curves. Applications include classical elliptic equations, spectral fractional Laplacians, harmonic functions on fractals, and Pucci operators.