Compactness properties of plurisubharmonic functions

Question

I'm quite interested in this topic, but the main text on Several Complex Variables say little of nothing about it. Here are my questions, and I'd be grateful of any reference or information.

Let $\Omega$ be an open subset of $\mathbb{C}^n$ and let $F$ be a family of plurisubharmonic functions on $\Omega.$ We may assume that these functions are continuous or smooth if we wish; also, I'm particularly interested in $n=2.$

Q1. In what convergence can we get a converging subsequence to a plurisubharmonic function?

Q2. Are there further special conditions on the family $F$ that ensure compactness in stronger sense?

Answer 1

First of all, I would say that there exists $\textit{one}$ good topology for psh functions, that is the $L^1_{loc}$ topology. One of the main result is the following one :

Let $(u_n)$ be a sequence of psh functions on a connected open subset $\Omega \subset \mathbb{C} ^n$ with $u_n \not \equiv -\infty $. We suppose that $(u_n)$ converges to $u$ psh, in the weak topology of distributions. Then $(u_n)$ is locally upper bounded and $u_n \to u$ in $L^p_{loc}(\Omega)$ for every $p\in[1,+\infty[$.

Other similar results are :

-every bounded subset of $Psh(\Omega) \cap L^1_{loc}(\Omega)$ is relatively compact;

-if $(u_n)$ is locally upper bounded on $\Omega$, then either $(u_n)$ converges locally uniformly to $-\infty$ on $\Omega$ (for the $L^1_{loc}$ topology), either there exists some subsequence converging to a psh function on $\Omega$ (in $ L^1_{loc}$ - or $L^p_{loc}, p\geq 1$, this is the same-).

As for the references, the one I prefer is "Notions of convexity" by Hörmander, 94, around section 3.2. The online book of Demailly is good too, but far less detailed about this topic.

Answer 2

I agree with you that there is a worldwide conspiracy about this. The main result is the following.

Suppose that a family of PS functions is uniformly bounded from above on every compact subset of the domain. Then from every sequence one can choose a subsequence which will converge either to a PS function or to identical $-\infty$.

In other words, if your family is bounded from below at one point, then the sequence will converge to a PS function.

What topology? Almost all reasonable topologies are equivalent here :-) You can use Schwarz distributions $D'$. This is especially convenient because the Laplace operator is continuous in $D'$. You can use $L^1_{loc}$, or $L^p_{loc}$ with any $p$ (with respect to the Lebesgue measure).

But much more can be said about this convergence, about the size of the set where $|u_n(z)-u(z)|>\epsilon$. It is really very small.

The reference is the 1-st volume of Hormander's Linear differential operators, where this is discussed for subharmonic functions. The PS functions are very carefully treated in the book of L. Ronkin, Functions of completely regular growth.

Answer 3

Besides the above-mentioned results in various topologies, the following is quite useful: Let $\Omega$ be an open subset of $\mathbb{C}^n$. If $\Omega$ is connected and $u_j, j \in \mathbb{N}$ is a decreasing sequence of PSH functions, then $u = \lim_{j \to \infty}u_j \in PSH(\Omega)$ or $u \equiv -\infty$. This is Theorem 2.9.14 (ii) in: M. Klimek: Pluripotential Theory, Clarendon Press 1991. It also appears as Proposition I.3 (iii) in the Appendix I by Russakovskii and Favorov to the Russian translation of P. Lelong, L. Gruman: Entire functions of several complex variables, Springer-Verlag 1986. The result mentioned by Alex Eremenko can also be found in the main body of Lelong and Gruman's book, as Theorem 1.27 (in a slightly more general form, for upper-semicontinuous regularization of the upper limit (in the sense of filters) of a locally bounded from above family of PSH functions). The conspiracy is vast indeed, but it seems to be limited to authors of texts in complex analysis.