# When does a plurisubharmonic function belongs to Sobolev space?

Let u be a plurisubharmonic function defined on the unit ball $$\mathbb{B}$$ of $$\mathbb{C}^{k}$$ such that $$u \ge 1$$.

Question : why the partial derivates $$\frac{\partial u}{\partial x_{i}}$$ (which are defined in the weak sense of distributions) of u belongs to $$L^{2}_{loc}$$? (Here I wright $$z_{i} = x_{i} + iy_{i}$$ the standards coordinates).

More generally, suppose u is such that the partial derivates $$\frac{\partial u}{\partial x_{i}}, \frac{\partial u}{\partial y_{i}}$$ of u belongs to $$L^{2}_{loc}$$ . Question : if $$(u_{j})$$ are PSH and decrease to u point wise, then why do the partial derivates $$\frac{\partial u}{\partial x_{i}}, \frac{\partial u}{\partial y_{i}}$$ of u belongs to $$L^{2}_{loc}$$ for j large enough and converge to $$\frac{\partial u}{\partial x_{i}}, \frac{\partial u}{\partial y_{i}}$$ in the $$L^{2}_{loc}$$ sense?

Here is what I've tried : Let us recall $$d = \partial + \bar{\partial}$$ and $$d^{c} = \frac{1}{2\pi i}(\partial - \bar{\partial})$$. The result is local so it's enough to show it on a relatively open set of $$\mathbb{B}$$. Since $$u \ge 1$$, $$u^{2}$$ is also plurisubharmonic. One can then approximate $$u$$ by a non-increasing sequence $$(u_{j})$$ of smooth plurisubharmonic functions in the $$L^{1}_{loc}$$ sens. We have $$dd^{c}u^{2}_{j} = 2u_{j}dd^{c}u_{j} + 2 du_{j} \wedge d^{c}u_{j}$$. Let $$l$$ be an index and let set $$T := dz_{1} \wedge d\bar{z}_{1} \wedge ... \wedge dz_{l-1} \wedge d\bar{z}_{l-1} \wedge dz_{l+1} \wedge d\bar{z}_{l+1} \wedge ... \wedge dz_{n} \wedge d\bar{z}_{n}$$. Then, for any compact $$K$$, $$Constant \times \int_{K}|\frac{\partial u_{j}}{\partial z_{l}}|^{2}dV = \int_{K}(dd^{c}u^{2}_{j} - 2u_{j}dd^{c}u_{j}) \wedge T$$ which is uniformly bounded in $$j$$ by the Chern–Levine–Nirenberg inequalities. Then, by compactness in $$L^{2}_{loc}$$, one can suppose $$(\frac{\partial u_{j}}{\partial z_{l}})$$ converge in $$L^{2}_{loc}$$. But as it already converges in the sense of distributions to $$\frac{\partial u}{\partial z_{l}}$$, it follows $$\frac{\partial u}{\partial z_{l}}$$ (and thus $$\frac{\partial u}{\partial x_{i}}, \frac{\partial u}{\partial y_{i}}$$) belongs to $$L^{2}_{loc}$$. What do you think?

2)If $$u$$ is bounded from below, then I can use the previous part and results about Monge-Ampère operators to conclude. But in general, I don't know.

I thank you in advance and wish you a good day.

• Is it possible to move this question on mathoverflow? I mean I'd like to ask it on mathoverflow even if it's already asked here. Jul 11, 2022 at 10:26
• A partial solution is given here : mathoverflow.net/questions/426454/… . Jul 25, 2022 at 8:48