Is there an Arzela-Ascoli theorem for subharmonic functions? 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 $-\infty$?
 A: Yes, there is such an analog. Let $U$ be a family of subharmonic functions in a region $D$ subject to
the following conditions:
a) For every compact $K\subset D$ there is a constant $C(K)$ such that $u(z)\leq C(K)$ for $z\in K$ and $u\in U$. And
b) For every $u\in U$ we have $u(z_0)>-C$ for some $z_0\in D$ and some real $C$.
Then there is a subsequence which converges to a subharmonic function. The convergence is in $L^1_{\mathrm{loc}}$ and in $D'$ and in many other senses.
In particular, Laplacians converge in the weak topology of measures.
If you remove the condition b), one has to add the possibility that the subsequence converges to $-\infty$ (uniformly on compact subsets).
Condition b) can be modified in various ways. For example, you can make $z_0$ depending on $u$ but restricted to some fixed compact $K$. All this is true in any dimension.
All this is contained in Theorem 4.1.9 (Hormander, vol. I), or with more details
about modes of convergence, in his book Notions of convexity, Birkhauser, Boston, 1994.
