Boundary sets where harmonic functions may become infinite

Gardiner SJ, Hansen W. Boundary sets where harmonic functions may become infinite. Mathematische Annalen. 2002;323(1):41-54.This paper characterizes the subsets E of R-n which have the following property: there exists a harmonic function u on R-n x (0, + infinity) such that u(x, t) --> +infinity as t --> 0+ for each x in E. This problem has its roots in classical work of Lusin and Privalov, and the answer has been known for some time in the case where n = 1. However, the characterization turns out to be more delicate in higher dimensions