Singular behavior of conformal Martin kernels, and non-tangential limits of conformal mappings

The Martin boundary corresponding to Q-Laplacian operator (where Q is the Ahlfors regularity dimension of the space) was constructed in [HST]. In particular, it was shown in [HST] that if the domain is bounded, uniform, and has uniformly fat complement, the (conformal) Martin kernel functions in the conformal Martin boundary of the domain vanish Holder continuously at the metric boundary points of the domain that do not arise as accumulation points of the corresponding fundamental sequence of points in the domain. The aim of this note is to extend the study of these Martin kernel functions to Q-almost locally uniform domains and by exploring their behavior near the metric boundary points of the domain that are accumulation points of any fundamental sequence associated with the Martin kernel function. We show that the kernel function exhibits singular behavior near such boundary points, that is, they converge to infinity along quasihyperbolic geodesic curves terminating at such boundary points. We use this singular behavior of conformal Martin kernel functions to establish that conformal mappings between two bounded locally uniform domains whose complements are uniformly fat have non-tangential limits at every metric boundary of the domain of the mapping.