Let $u\_k$ be a sequence of subharmonic functions on an open set $X$ and $\psi\_\delta$ a family of standard mollifiers with compact support. Hörmander claims in *The Analysis of Linear Partial Differential Operators Vol I*, Theorem 4.1.9(b) that if $u\_k$ converge as distributions to a subharmonic $u$ then $v\_j \* \psi\_\delta(x) \to v \* \psi\_\delta$ "uniformly on compact sets if $\delta$ is small enough" (I assume this means that for every compact $K \subset X$ there exists such a $\delta$).


I can easily see why the convergence is true pointwise, but I don't know how to verify that it's uniform.


If you assume the conclusion of the theorem, namely that $u\_k \to u$ in $L^1\_\mathrm{loc}(X)$ then the result follows quickly from the contiuity of convolutions as a map $L^1 \times L^\infty \to C^0$, but this would of course be circular.



If subharmonic functions converge weakly to a subharmonic limit, why do their smoothings converge uniformly on compact sets?
More detail on weak convergence of subharmonic functions, including a proof of this statement, can be found in his other book:


Hormander, Notions of convexity, Theorems 3.2.12 and 3.2.13.


