Hörmander in *Analysis of linear partial differential equations II* proves Proposition 16.1.2 on page 304 to the effect that a sequence of subharmonic functions converging to another subharmonic function in the distributional sense necessarily converges in the $L^p_{loc}$ sense. Note that the proposition *assumes* the existence of a limiting subharmonic function. Is there a version of the Arzela-Ascoli theorem in this context that would guarantee the existence of a limit for a suitable subsequence, and *under what hypotheses*? I am mainly interested in the real 2-dimensional case.

As a follow-up, can one get control over the set of points where the limiting function is $-\infty$?

subharmonicfunctions. I wonder if in the 2-dimensional case one can get control over the set of points where the limiting function is $-\infty$. Does that ring a bell? $\endgroup$ – Mikhail Katz Mar 13 '17 at 17:48