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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.

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  • $\begingroup$ 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. $\endgroup$
    – Analyse300
    Jul 11, 2022 at 10:26
  • $\begingroup$ A partial solution is given here : mathoverflow.net/questions/426454/… . $\endgroup$
    – Analyse300
    Jul 25, 2022 at 8:48

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