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# Modulus of Continuity of Solutions to Complex Monge-Ampère Equations on Stein Spaces

Guilherme C. Gonçalves

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Guilherme C. Gonçalves. Modulus of Continuity of Solutions to Complex Monge-Ampère Equations on Stein Spaces. 2024. hal-04852244

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# MODULUS OF CONTINUITY OF SOLUTIONS TO COMPLEX MONGE-AMPÈRE EQUATIONS ON STEIN SPACES

## GUILHERME CERQUEIRA GONÇALVES

**ABSTRACT.** In this paper, we study the modulus of continuity of solutions to Dirichlet problems for complex Monge-Ampère equations with $L^p$ densities on Stein spaces with isolated singularities. In particular, we prove such solutions are Hölder continuous outside singular points if the boundary data is Hölder continuous.

## CONTENTS

<table><tr><td>Introduction</td><td>1</td></tr><tr><td>1. Preliminaries</td><td>2</td></tr><tr><td>2. Construction of barriers</td><td>5</td></tr><tr><td>3. Regularization of the solution</td><td>7</td></tr><tr><td>4. L<sup>1</sup> Estimate and general case</td><td>10</td></tr><tr><td>References</td><td>13</td></tr></table>

## INTRODUCTION

Let $X$ be a complex space that is reduced, locally irreducible, and of complex dimension $n \ge 1$ with an isolated singularity, $X_{\text{sing}}$. Equip $X$ with a Hermitian metric whose fundamental form is $\beta$, a positive (1, 1)-form, and let $d_{\beta}(\cdot, \cdot)$ be the distance induced by $\beta$. Let $\Omega$ be a bounded, strongly pseudoconvex open subset of $X$, such that $X_{\text{sing}} \subset \Omega$, and fix $\rho$ as a smooth, strictly plurisubharmonic defining function, i.e., $\Omega = \{\rho < 0\}$, hence $\Omega$ is a Stein space. Given $\phi \in C^0(\partial\Omega)$ and $f \in L^p(\Omega, \beta^n)$ with $p > 1$, we consider the Dirichlet problem:

$$MA(\Omega, \phi, f) : \begin{cases} (dd^c u)^n = f \beta^n & \text{in } \Omega, \\ u = \phi & \text{on } \partial\Omega, \end{cases}$$

where $d = \partial + \bar{\partial}$ and $d^c = i(\bar{\partial} - \partial)$. This Dirichlet problem for the complex Monge-Ampère operator has been intensively studied in domains of $\mathbb{C}^n$ (see [22] for a historical account).

The study of complex Monge-Ampère equations is key to understanding canonical metrics in Kähler manifolds. This significance extends to mildly singular varieties as well. Interactions with birational geometry, such as the Minimal Model Program, make it enticing across fields to comprehend these equations.

In this paper, our objective is to explore the impact of boundary data and singularity types on the modulus of continuity of solutions. In particular, we prove that if $\phi$ is Hölder continuous, then so is the solution of $MA(\Omega, \phi, f)$ outside the singular point.

*Date*: November 10, 2024.
Regarding applications of the Hölder regularity of solutions to complex Monge-Ampère equations, we refer to [18] for some geometric consequences of the Hölder continuity of Kähler-Einstein potentials and to [14] for more details about applications in complex dynamics.

Several works have focused on the Hölder continuity of solutions, both on domains of $C^n$ (see [1, 6, 7, 20]) and on compact complex manifolds (see [12, 13, 23, 24]).

However, there are only a few literature dealing with this problem on domains with singularities. Recently, it was proved in [17] that the solution $u = u(\Omega, \phi, f)$ for $MA(\Omega, \phi, f)$ is continuous and unique in $\bar{\Omega}$. Further smoothness properties have been provided in [9, 16, 21] on the regular part of domains that contain an isolated singularity in the case of smooth boundary data.

Our main result is the following:

**Theorem A.** Assume that $\phi \in C^0(\partial\Omega)$ and $f \in L^p(\Omega, \beta^n)$ with $p > 1$. Then the unique solution $u \in PSH(\Omega) \cap C^0(\bar{\Omega})$ to $MA(\Omega, \phi, f)$ has the following modulus of continuity at $x \in \Omega$:

$$ \omega_{u,x}(t) \le C_x \max\{\omega_{\phi}(t^{1/2}), t^{\frac{1}{nq+1}}\} $$

for some constant $C_x > 0$ such that $C_x \to +\infty$ as $x \to X_{sing}$, where $\omega_{u,x}$ is the modulus of continuity of $u$ at the point $x$ and $\frac{1}{p} + \frac{1}{q} = 1$.

In particular, Theorem A implies:

**Corollary B.** When $\phi \in C^{0,\alpha}(\partial\Omega)$, $0 < \alpha \le 1$, and $f \in L^p(\Omega, \beta^n)$ with $p > 1$, the unique solution $u \in PSH(\Omega) \cap C^0(\bar{\Omega})$ to $MA(\Omega, \phi, f)$ is $\alpha^*$-Hölder continuous outside the singular point, for $\alpha^* < \min\{\frac{\alpha}{2}, \frac{1}{nq+1}\}$.

**Remark.** This Hölder exponent<fnref n="1" /> is the same for smooth domains [7, Theorem 1.2].

**Remark.** One can similarly treat the case when $X$ has finitely many isolated singularities.

**Acknowledgements.** The author would like to first thank his PhD advisor, Vincent Guedj, for his support, guidance and comments. The author also thanks Ahmed Zeriahi and Chung-Ming Pan for useful discussions and references. The author would like to thank the referee for useful comments and corrections. This work received support from the University Research School EUR-MINT (State support managed by the National Research Agency for Future Investments program bearing the reference ANR-18-EURE-0023).

# 1. PRELIMINARIES

## 1.1. Complex analysis on Stein spaces.
Throughout this paper, we let $X$ be a reduced, locally irreducible complex analytic space of pure dimension $n \ge 1$. We denote by $X_{reg}$ the complex manifold of regular points of $X$ and $X_{sing} := X \setminus X_{reg}$ the set of singular points, which is an analytic subset of $X$ with complex codimension $\ge 1$.

By definition, for each point $x_0 \in X$, there exists a neighborhood $U$ of $x_0$ and a local embedding $j: U \to \mathbb{C}^N$ onto an analytic subset of $\mathbb{C}^N$ for some $N \ge 1$. These local embeddings allow us to define the spaces of smooth forms of given degree on $X$ as smooth forms on $X_{reg}$ that are locally on $X$ restrictions of an ambient form on $\mathbb{C}^N$. Other differential notions and operators, such as holomorphic and plurisubharmonic functions, can also be defined in this way. (See [11] for more details). Two different notions can be defined for plurisubharmonicity:

<footnote n="1">¹Relative to the induced distance of the chosen $\beta$.</footnote>
**Definition 1.1.** Let $u: X \to \mathbb{R} \cup \{-\infty\}$ be a given function.

(1) We say that $u$ is plurisubharmonic (psh for short) on $X$ if it is locally the restriction of a plurisubharmonic function in a local embedding of $X$.

(2) We say that $u$ is weakly plurisubharmonic on $X$ if $u$ is locally bounded from above on $X$, and its restriction to the complex manifold $X_{\text{reg}}$ is plurisubharmonic.

Fornæss and Narasimhan [15, Theorem 5.3.1] proved that $u$ is plurisubharmonic on $X$ if and only if, for any analytic disc $h: \mathbb{D} \to X$, the function $u \circ h$ is subharmonic or identically $-\infty$. If $u$ is weakly plurisubharmonic on $X$, then $u$ is plurisubharmonic on $X_{\text{reg}}$, and thus upper semi-continuous on $X_{\text{reg}}$. It is natural to extend $u$ to $X$ using the following formula:

$$ (1.1) \qquad u^*(x) := \limsup_{X_{\text{reg}} \ni y \to x} u(y), \quad x \in X. $$

The function $u^*$ is upper semi-continuous, locally integrable on $X$, and satisfies $dd^c u^* \ge 0$ in the sense of currents on $X$ [11, Théorème 1.7]. The two notions are equivalent when $X$ is locally irreducible, as shown in [11, Théorèm 1.10]:

**Theorem 1.2.** Let $X$ be a locally irreducible analytic space and $u: X \to \mathbb{R} \cup \{-\infty\}$ be a weakly plurisubharmonic function on $X$. Then the function $u^*$ defined by (1.1) is psh on $X$.

Note that since $u$ is plurisubharmonic on $X_{\text{reg}}$, we have $u^* = u$ on $X_{\text{reg}}$. Then $u^*$ is the upper semi-continuous extension of $u|_{X_{\text{reg}}}$ to $X$. With this, we have the following:

**Corollary 1.3.** Let $\mathcal{U} \subset \text{PSH}(X)$ be a non-empty family of plurisubharmonic functions which is locally bounded from above on $X$. Then its upper envelope

$$ \mathcal{U} := \sup\{u; u \in \mathcal{U}\} $$

is a well-defined Borel function whose upper semi-continuous regularization $\mathcal{U}^*$ is psh on $X$.

Recall that from [15, Theorem 6.1] $X$ is Stein if it admits a $C^2$ strongly plurisubharmonic exhaustion. We will use the following definition:

**Definition 1.4.** A domain $\Omega \in X$ is strongly pseudoconvex if it admits a negative smooth, strongly plurisubharmonic defining function, i.e., a strongly plurisubharmonic function $\rho$ in a neighborhood $\Omega'$ of $\bar{\Omega}$ such that $\Omega := \{x \in \Omega'; \rho(x) < 0\}$ and for any $c < 0$,

$$ \Omega_c := \{x \in \Omega'; \rho(x) < c\} \in \Omega $$

is relatively compact.

We denote by $\text{PSH}(X)$ the set of plurisubharmonic functions on $X$.

On complex spaces, the complex Monge-Ampère operator has been defined and studied in [2] and [11]. In this setting, if $u \in \text{PSH}(X) \cap L_{\text{loc}}^{\infty}(X)$, the Monge-Ampère measure $(dd^c u)^n$ is well defined on the regular part $X_{\text{reg}}$ and can be extended to $X$ as a Borel measure with zero mass on $X_{\text{sing}}$. This notion extends the foundational work of pluripotential theory by Bedford-Taylor [3, 4].

Consequently, several standard properties of the complex Monge-Ampère operator acting on $\text{PSH}(X) \cap L_{\text{loc}}^{\infty}(X)$ extend to this setting (see [2, 11]). In particular, we have the following comparison principle [2, Theorem 4.3]:

**Proposition 1.5 (Comparison principle).** Let $\Omega \in X$ be a relatively compact open set and $u, v \in \text{PSH}(\Omega) \cap L^{\infty}(\Omega)$. Assume that $\liminf_{x \to \zeta} (u(x) - v(x)) \ge 0$ for any $\zeta \in \partial\Omega$. Then

$$ \int_{\{u < v\}} (dd^c v)^n \le \int_{\{u < v\}} (dd^c u)^n. $$

In particular, if $(dd^c u)^n \le (dd^c v)^n$ weakly on $\Omega$, then $v \le u$ on $\Omega$.
1.2. **Notations.** We fix the following notations throughout this paper:

* $X$ is a complex space that is reduced and locally irreducible of complex dimension $n \ge 1$ with an isolated singularity $X_{\text{sing}}$.

* $\beta$ is a fixed smooth positive (1,1)-form on $X$, the fundamental form of a Hermitian metric on $X$.

* $d_{\tau}$ is the distance induced by a fundamental form $\tau$ on some manifold $(Y, \tau)$. When no confusion can occur $d_{\beta}$ will be written as $d(\cdot, \cdot)$. Also, for any $x \in X$ we set $B_r(x) \subseteq X$ as the ball of radius $r$ centered in $x$ for the specified metric, if no metric is specified then the reference is $d_{\beta}$.

* $\Omega \subseteq X$ is a bounded, strongly pseudoconvex open subset of $X$ and fix $\rho$ smooth strictly psh defining function, i.e., $\Omega = \{\rho < 0\}$, that will be the Stein space where we work on.

* $0 \le f \in L^p(\Omega, \beta^n)$, $p > 1$ and $1/p + 1/q = 1$ and $\phi \in C^0(\partial\Omega)$.

* $u(\Omega, \phi, f)$ is the solution to the Dirichlet problem:

$$MA(\Omega, \phi, f) : \begin{cases} (dd^c u)^n = f \beta^n & \text{in } \Omega, \\ u = \phi & \text{on } \partial \Omega. \end{cases}$$

* $\omega_{g,x}$ is the modulus of continuity of some function $g$ at the point $x$.

* $\lambda(x) = d(x, X_{\text{sing}})$ the distance of a point $x$ to the singular point. The point $x$ will be omitted when there is no chance of confusion.

* $\mathbb{B}_{2n}$ is the volume of unit ball in $\mathbb{C}^n$. $dV$ is the standard euclidian volume form in $\mathbb{C}^n$. $d\mathbb{C}^n$ is the standard euclidian distance in $\mathbb{C}^n$.

1.3. **Useful Results.** We need the stability estimate from [17, Proposition 1.8]:

**Theorem 1.6.** Let $\varphi, \psi$ be two bounded plurisubharmonic functions in $\Omega$ and $(dd^c\varphi)^n = f\beta^n$ in $\Omega$. Fix $0 \le \gamma < \frac{1}{nq+1}$. Then there exists a uniform constant $C = C(\gamma, ||f||_{L^p(\Omega)}) > 0$ such that:

$$\sup_{\Omega} (\psi - \varphi) \le \sup_{\partial \Omega} (\psi - \varphi)_+^* + C(||(\psi - \varphi)_+||_{L^1(\Omega, \beta^n)})^{\gamma}$$

where $(\psi - \varphi)_+ := \max(\psi - \varphi, 0)$ and $w^*$ is the upper semi-continuous extension of a bounded function $w$ on $\Omega$ to the boundary, i.e., $w^*(\xi) := \limsup_{z \to \xi} w(z)$.

The local theory [1, Theorems 3.3, 3.4 and Lemma 3.5] will be used in the proof of Lemma 4.1. Here reformulated to:

**Theorem 1.7.** Let $\hat{\Omega} \subseteq \mathbb{C}^n$ strongly pseudoconvex domain, $0 \le \hat{f} \in L^p(\hat{\Omega})$; $p > 1$ and $\varepsilon > 0$ small enough. For $\hat{u} \in \text{PSH}(\hat{\Omega}) \cap C^0(\hat{\Omega})$ such that $(dd^c\hat{u})^n = \hat{f}dV$ in $\hat{\Omega}$. We have

$$||\Lambda_{\delta/2}\hat{u} - \hat{u}||_{L^1(\hat{\Omega}_{\delta})} \le C\delta^{1-\varepsilon}$$

for some constant $C = C(n, \varepsilon, \hat{\Omega}, ||\hat{u}||_{L^{\infty}(\hat{\Omega})}) > 0$, where $\hat{\Omega}_{\delta} := \{z \in \hat{\Omega} | d\mathbb{C}^n(z, \partial\hat{\Omega}) > \delta\}$ and for $z \in \hat{\Omega}_{\delta}$, $\Lambda_{\delta}\hat{u}(z) = \frac{1}{\mathbb{B}_{2n}\delta^{2n}} \int_{|z-\zeta|\le\delta} \hat{u}(\zeta)dV(\zeta)$ is the mean volume regularizing function in $\mathbb{C}^n$.

Moreover, if $\Delta\hat{u}$ has finite mass in $\hat{\Omega}$ then one can get $\delta^2$ instead of $\delta^{1-\varepsilon}$.

We will also use, in the proof of Lemma 2.3, the following classical result from singular Riemannian foliations in [25, Lemma 1], here reformulated to:
**Lemma 1.8.** Let $(M, g)$ be a smooth, connected, complete manifold with Riemannian metric $g$ and induced distance function $d$. Fix $f \in C^2(M)$ such that $g(\nabla f, \nabla f) = 1$. Suppose $[a, b] \subset f(M) \subset \mathbb{R}$ contains no critical points of $f$. Then for any $x \in f^{-1}(a)$ and $y \in f^{-1}(b)$ we have:

$$d(x, f^{-1}(b)) = d(f^{-1}(a), y) = \int_a^b df = b - a.$$

## 2. CONSTRUCTION OF BARRIERS

In this section, we develop barrier functions, which allow us to control the modulus of continuity of solutions to Dirichlet problems close to the boundary. The barriers we will construct are described in the following definition:

**Definition 2.1.** Let $u = u(\Omega, \phi, f)$. We call barrier functions for the problem $MA(\Omega, \phi, f)$ two functions $v, w \in PSH(\Omega) \cap C^0(\bar{\Omega})$, such that:

$$ (1) \quad v(\xi) = \phi(\xi) = -w(\xi), \forall \xi \in \partial\Omega, $$

$$ (2) \quad v(z) \le u(z) \le -w(z), \forall z \in \Omega, $$

The construction of barriers depends on the behavior around the boundary. In Lemma 2.2 we construct barriers for the case where $\phi = 0$ and $f$ is bounded near $\partial\Omega$ (its idea coming from [20, Lemma 2.2]). Next, in Lemma 2.3 we treat the case where $f = 0$ (idea from [6, Proposition 4.4]). The construction of barriers for the general problem $MA(\Omega, \phi, f)$ is postponed to Section 4 because one needs to understand the regularity of **solutions** to the particular problems dealt with in this section.

### 2.1. Densities bounded near the boundary.

**Lemma 2.2.** Assume that $\hat{f}$ is bounded near $\partial\Omega$ and set $u_0 := u(\Omega, 0, \hat{f})$. There exist $b_{\hat{f}}, w \in PSH(\Omega) \cap C^{0,1}(\Omega)$ such that:

$$ (1) \quad b_{\hat{f}}(\xi) = 0 = -w(\xi), \forall \xi \in \partial\Omega, $$

$$ (2) \quad b_{\hat{f}} \le u_0 \le -w \text{ in } \Omega. $$

*Proof.* Since $\hat{f}$ is bounded near $\partial\Omega$, there exists a compact $K \subset \Omega$ such that $0 \le \hat{f} \le M$ on $\Omega \setminus K$. Also, there exist $A_1, A_2 > 0$ large enough such that $(dd^c(A_1\rho))^n \ge M\beta^n \ge \hat{f}\beta^n$, on $\Omega \setminus K$ and $A_2\rho \le m \le u_0$ on a neighborhood of $K$, for $m := \min_{\bar{\Omega}} u_0$. Then, by taking the maximum of both constants, we have $A > 0$ large enough such that both conditions are satisfied for $b_{\hat{f}} := A\rho$, which is smooth plurisubharmonic on $\Omega$. As we also have $b_{\hat{f}} \le u_0$ on $\partial(\Omega \setminus K)$, by the comparison principle we get $b_{\hat{f}} \le u_0$ in $\Omega \setminus K$. By construction, we have $b_{\hat{f}} \le u_0$ on a neighborhood of $K$; hence, with the above argument we get $b_{\hat{f}} \le u_0$ in $\Omega$ and $b_{\hat{f}}(\partial\Omega) = \{0\}$, thus is a lower-barrier. For an upper barrier, take $w = 0$ in $\Omega$ as by the maximum principle $u_0 \le 0$ and it is zero on the boundary. $\square$

### 2.2. Zero density.

**Lemma 2.3.** Let $u_\phi := u(\Omega, \phi, 0)$. There exists a barrier $h_\phi \in PSH(\Omega) \cap C^0(\bar{\Omega})$ such that:

$$ (1) \quad h_\phi(\xi) = \phi(\xi) = -h_{-\phi}(\xi), \forall \xi \in \partial\Omega, $$

$$ (2) \quad h_\phi \le u_\phi, \le -h_{-\phi} \text{ in } \Omega, $$

$$ (3) \quad \omega_{h_\phi}(t) \le C\omega_\phi(t^{1/2}). $$
Proof. We set $h_\phi \in PSH(\Omega) \cap C^0(\bar{\Omega})$ such that:

$$ egin{cases} \omega_{h_\phi}(t) \le C\omega_\phi(t^{1/2}) & \text{in } \Omega, \\ h_\phi = \phi & \text{on } \partial\Omega. \end{cases} $$

With such $h_\phi \in PSH(\Omega)$ we have $(dd^c h_\phi)^n \ge 0 = (dd^c u_\phi)^n$. Hence, by the comparison principle, we get $h_\phi \le u_\phi$. Similarly, we get $-h_{-\phi} \le -u_\phi$; hence $h_{-\phi} \ge u_\phi$. Now we start the construction of such $h_\phi$:

Step 1: For fixed $\xi \in \partial\Omega$, construct a function $h_\xi$ close to $\xi$.

Fix $\xi \in \partial\Omega$. We want to find a function $h_\xi \in PSH(\Omega) \cap C^0(\bar{\Omega})$ such that:

$$ egin{cases} \omega_{h_\xi}(t) \le C\omega_\phi(t^{1/2}) & , \\ h_\xi(x) \le \phi(x) & \text{for any } x \in \partial\Omega, \\ h_\xi(\xi) = \phi(\xi) & . \end{cases} $$

In this first step, we will construct $h_\xi$ above just around $\xi$. Take $B > 0$ large enough such that: $g(x) := B\rho(x) - (d_\beta(x, \xi))^2$ is in $PSH(\Omega)$.

Consider $\bar{\omega}_\phi$, the minimal concave majorant of $\omega_\phi$, and define $\chi(t) := -\bar{\omega}_\phi((-t)^{1/2})$, which is a convex non-decreasing function on $[-d^2, 0]$, where $d = \text{diam}(\Omega)$, the diameter of $\Omega$.

Fix $r > 0$ sufficiently small such that $|g(x)| \le d^2$ in $B_r(\xi) \cap \Omega$ and define for $x \in B_r(\xi) \cap \bar{\Omega}$: $\tilde{g}(x) = \chi(g(x)) + \phi(\xi) = -\bar{\omega}_\phi[\left(\left(d_\beta(x, \xi)\right)^2 - B\rho(x)\right)^{1/2}] + \phi(\xi)$. Note that $\tilde{g}$ is a continuous psh function on $B_r(\xi) \cap \Omega$.

We have $\forall x \in \partial\Omega \cap B_r(\xi)$, $\phi(\xi) - \bar{\omega}_\phi(d_\beta(x, \xi)) \le \phi(x)$. However, by our construction, $\tilde{g}(x) = \phi(\xi) - \bar{\omega}_\phi(d_\beta(x, \xi))$ for any $x \in \partial\Omega \cap B_r(\xi)$. Hence, $\tilde{g}(x) \le \phi(x), \forall x \in \partial\Omega \cap B_r(\xi)$ and $\tilde{g}(\xi) = \phi(\xi)$. Using the subadditivity of $\bar{\omega}_\phi$ and also the fact that $\forall t, \lambda > 0; \omega_\phi(\lambda t) \le \bar{\omega}_\phi(\lambda t) \le (1 + \lambda)\omega_\phi(t)$ (see [6, Lemma 4.1]) we get:

$$ egin{align*} \omega_{\tilde{g}}(t) &= \sup_{d_\beta(x,y) \le t} |\tilde{g}(x) - \tilde{g}(y)| \\ &\le \sup_{d_\beta(x,y) \le t} \bar{\omega}_\phi\left[\left(d_\beta(x, \xi)\right)^2 - (d_\beta(y, \xi))^2 - B(\rho(x) - \rho(y))\right]^{1/2} \\ &\le \sup_{d_\beta(x,y) \le t} \bar{\omega}_\phi\left[(2d + B_1)^{1/2} (d_\beta(x, y))^{1/2}\right] \\ &\le (1 + (2d + B_1)^{1/2}) \sup_{d_\beta(x,y) \le t} \omega_\phi\left[(d_\beta(x, y))^{1/2}\right] \\ &\le (1 + (2d + B_1)^{1/2}) \sup_{d_\beta(x,y) \le t} \omega_\phi\left[(t)^{1/2}\right]. \end{align*} $$

From the second line to the third, there are the following arguments: the first term goes as $d_1^2 - d_2^2 = (d_1 - d_2)(d_1 + d_2)$. Then, the first difference of distances can be bounded by $d_\beta(x, y)$ by triangular inequality. Also, the second term can be bounded by $2d$. Finally, the last term comes from the fact that, close to the boundary, we have that $|\rho(x) - \rho(y)|$ behaves like $d_\beta(x, y)$, by Lemma 1.8 up to renormalization of the gradient close to the boundary.

Step 2: Extending our local $\tilde{g}$ to all $\bar{\Omega}$ as $h_\xi$.

Recall $\xi \in \partial\Omega$, fix $0 < r_1 < r$ and $\gamma_1 > \frac{d}{r_1}$ such that $\forall x \in \bar{\Omega} \cap \partial B_{r_1}(\xi)$:

$$ -\gamma_1 \bar{\omega}_\phi\left[\left(d_\beta(x, \xi)\right)^2 - B\rho(x)\right]^{1/2} = \gamma_1(\tilde{g}(x) - \phi(\xi)) \le \inf_{\partial\Omega} \phi - \sup_{\partial\Omega} \phi. $$

Set $\gamma_2 = \inf_{\partial\Omega} \phi$. Then:
$$ (2.1) \qquad \gamma_1(\tilde{g}(x) - \phi(\xi)) + \phi(\xi) \le \gamma_2 $$

for $x \in \partial B_{r_1}(\xi) \cap \bar{\Omega}$. Consider:

$$ h_{\xi}(x) := \begin{cases} \max\{\gamma_1(\tilde{g}(x) - \phi(\xi)) + \phi(\xi), \gamma_2\} & \text{for } x \in B_{r_1}(\xi) \cap \bar{\Omega}, \\ \gamma_2 & \text{for } x \in \bar{\Omega} \setminus B_{r_1}(\xi). \end{cases} $$

It follows from inequality (2.1) that $h_\xi$ is a psh function on $\Omega$, continuous on $\bar{\Omega}$ and such that $h_\xi(x) \le \phi(x), \forall x \in \partial\Omega$ because on $\partial\Omega \cap B_{r_1}(\xi)$:

$$ \gamma_1(\tilde{g}(x) - \phi(\xi)) + \phi(\xi) = -\gamma_1\bar{\omega}_\phi[d_\beta(x, \xi)] + \phi(\xi) \le -\bar{\omega}_\phi[d_\beta(x, \xi)] + \phi(\xi) \le \phi(x). $$

Then finally, we have that $h_\xi$ is a barrier relative to the point $\xi$, chosen generically, that is: for each $\xi \in \partial\Omega, \exists h_\xi \in S(\Omega, \phi, 0)$; $h_\xi(\xi) = \phi(\xi)$ and $\omega_{h_\xi}(t) \le C\omega_\phi(t^{1/2})$; for $C = (1 + (2d + B_1)^{1/2})$ and $S(\Omega, \phi, 0)$ is the set of subsolutions for the Dirichlet problem $MA(\Omega, \phi, 0)$.

**Step 3: Take envelope in $\xi$ to obtain the desired barrier $h_\phi$**.

Now set $h_\phi(x) = \sup\{h_\xi(x)|\xi \in \partial\Omega\}$, which is a sup over a compact family; hence $h_\phi^*$ is a psh function. Note that $0 \le \omega_{h_\phi}(t) \le C\omega_\phi(t^{1/2})$; then $\omega_{h_\phi}(t) \to 0$ as $t \to 0$. This implies $h_\phi \in C^0(\bar{\Omega})$ and $h_\phi = h_\phi^* \in PSH(\Omega)$. (Hence $h_\phi \in S(\Omega, \phi, 0)$.) By construction, $h_\phi = \phi$ on $\partial\Omega$. Thus, we have the desired lower-barrier for $MA(\Omega, \phi, 0)$ with modulus of continuity $\omega_\phi(t^{1/2})$. $\square$

### 3. REGULARIZATION OF THE SOLUTION

This section provides an appropriate regularization scheme of the solution $u = u(\Omega, \phi, f)$ to $MA(\Omega, \phi, f)$, following ideas from [12, 20].

#### 3.1. Defining regularization.
First, we define the set $\Omega_\delta := \{x \in \Omega \mid d_\beta(x, \partial\Omega) > \delta\}$ for $0 < \delta \le \delta_0$ where $\delta_0$ is fixed such that $\Omega_{\delta_0} \ne \emptyset$.

We consider $\pi : \tilde{X} \to X$ a resolution of singularities of $X$. We will denote objects on $\tilde{X}$ with a tilde. For example, $\tilde{\Omega} = \{\pi^*\rho := \tilde{\rho} < 0\}$. It is known that $\tilde{\Omega}$ is a Kähler manifold. Fix a Hermitian form $\tau$ on $\tilde{X}$ such that $\pi^*d \le d_\tau$, such as $\tau := \tilde{C}\pi^*\beta + \varepsilon\theta$ for some constants $\tilde{C} > 1$ big enough and $0 < \varepsilon$ small enough, with some smooth closed (1,1)-form $\theta$ Assume for simplicity<fnref n="2" /> that $\tau$ and $\beta$ closed.

Consider for each $\tilde{x} \in \tilde{X}$ the exponential map $\exp_{\tilde{x}} : T_{\tilde{x}}\tilde{X} \ni \xi \mapsto \exp_{\tilde{x}}(\xi) \in \tilde{X}$ defined by $\exp_{\tilde{x}}(\xi) = \sigma(1)$ with $\sigma$ being the geodesic such that $\sigma(0) = \tilde{x}$ and initial velocity $\sigma'(0) = \xi$. Let $\tilde{u} := \pi^*u$ be the pull-back of the solution to $MA(\Omega, \phi, f)$. Define its $\delta$-regularization $\eta_\delta\tilde{u}$ as in [10] by:

$$ \eta_\delta \tilde{u}(\tilde{x}) = \frac{1}{\delta^{2n}} \int_{\xi \in T_{\tilde{x}} \tilde{X}} \tilde{u} (\exp_{\tilde{x}}(\xi)) \eta \left( \frac{|\xi|^2}{\delta^2} \right) dV_\tau(\xi), \quad \delta > 0 \text{ and } \tilde{x} \in \tilde{\Omega}_\delta. $$

Here $\eta$ is a smoothing kernel, $|\xi|^2_\tau$ stands for $\sum_{i,j=1}^n g_{i\bar{j}}(\tilde{x})\xi_i\bar{\xi}_j$, and $dV_\tau(\xi)$ is the induced measure $\frac{1}{2^n n!} (dd^c|\xi|^2_\tau)^n$.

It is known that:

$$ \exp : T\tilde{X} \to \tilde{X}, \quad T\tilde{X} \ni (\tilde{x}, \xi) \mapsto \exp_{\tilde{x}}(\xi) \in \tilde{X}, \xi \in T_{\tilde{x}}\tilde{X} $$

has the following properties:

(1) $\exp$ is a $C^\infty$ mapping;

<footnote n="2">²By [12, Page 624] and [24, Page 2036], taking $\beta$ and $\tau$ not closed will change the exponential map but not the estimates.</footnote>
$$ (2) \quad \forall \tilde{x} \in \tilde{X}, \exp_{\tilde{x}}(0) = \tilde{x} \text{ and } D_{\xi} \exp(0) = \mathrm{Id}_{T_{\tilde{x}}\tilde{X}}. $$

One can formally extend $\eta_{\delta}\tilde{u}$ as a function on $\tilde{\Omega}_{\delta} \times \mathbb{C}$ by putting $U(\tilde{x}, w) := \eta_{\delta}\tilde{u}(\tilde{x})$ for $w \in \mathbb{C}$ with $|w| = \delta$. Coupling the estimate of the hessian of $U(\tilde{x}, w)$ with Kiselman's minimum principle, one gets [5, Lemma 1.12]:

**Lemma 3.1.** For a bounded psh function $\tilde{u}$ on the Kähler manifold $(\tilde{\Omega} \subset \tilde{X}, \tau)$ we let $U(\tilde{x}, w)$ be its regularization defined as above. Define the Kiselman-Legendre transform at level $c$ by

$$ \tilde{u}_{c,\delta}(\tilde{x}) := \inf_{0 \le t \le \delta} [U(\tilde{x}, t) + Kt^2 - K\delta^2 - c \log(t/\delta)] $$

for $\tilde{x} \in \tilde{\Omega}_{\delta}$ and some positive constant $K$ depending on the curvature of $(\tilde{\Omega}, \tau)$ such that the function $U(\tilde{x}, t) + Kt^2$ is increasing for $t \in (0, \delta_1)$ for some $0 < \delta_1 \le \delta_0$ small enough. Also one has the following estimate for the complex hessian:

$$ dd^c \tilde{u}_{c,\delta} \ge -(Ac + K\delta) \tau $$

where $A$ is a lower bound of the negative part of the bisectional curvature of $(\tilde{\Omega}, \tau)$.

**3.2. Correcting the positivity.** Before extending to the boundary as in [20], we need to correct the positivity of our regularizing function. The construction above creates a well behaved regularizing function on compact subsets. However, we require plurisubharmonicity to use the stability estimate.

As in [19, Section 3.2.4], since $X$ has an isolated singularity, we get the following argument: The exceptional divisor $E$ of the resolution has the property that there exist positive rational numbers $(b_i)_{i \in I}$ such that $-\sum_{i \in I} b_i E_i$ is $\pi$-ample, where each $E_i$ is an irreducible component of $E$. Then, on each $O_{\tilde{X}}(E_i)$ pick a section $s_i$ cutting out $E_i$ and choose an appropriate smooth Hermitian metric $h_i$ such that

$$ \rho' := B_1(\pi^*(B_0\rho) + \sum_{i \in I} b_i \log |s_i|_{h_i}^2) $$

is strictly psh in $\tilde{\Omega}$ for some $B_0 > 1$ big enough and choose $B_1 > 0$ such that $dd^c \rho' \ge \tau$. Assume without loss of generality $\tilde{u} \ge 0$. Now, assuming<fnref n="3" /> the existence of barriers as in Definition 2.1, we fix:

$$ (3.1) \qquad \tilde{u}_{\delta} := \tilde{u}_{c,\delta}; \quad Ac := H(\delta) - K\delta $$

for $H: \mathbb{R}^+ \to \mathbb{R}^+$ continuous, increasing function such that $\omega_H(\delta) \ge \omega_v(\delta) \ge O(\delta)$ and $H(0) = 0$, for $v$ a barrier from Definition 2.1 and $A, K$ from Lemma 3.1. Take $\delta < \delta_2 \le \delta_1$ for $\delta_2$ small enough such that the $c$ above is positive. We have control on its loss of positivity, hence:

$$ \hat{u}_{\delta} := \tilde{u}_{\delta} + (Ac + K\delta)\rho' = \tilde{u}_{\delta} + H(\delta)\rho'. $$

We have $dd^c \hat{u}_{\delta} = dd^c \tilde{u}_{\delta} + (Ac + K\delta)dd^c \rho'$. By the definition of $\rho'$ and Lemma 3.1 we get $\hat{u}_{\delta} \in PSH(\tilde{\Omega}_{\delta})$. Now, we set $\tilde{u}_{\delta} := \pi_* \hat{u}_{\delta}$, that is for any $x \in \pi^{-1}(\Omega_{\delta})$; $\tilde{u}_{\delta}(\pi(x)) = \hat{u}_{\delta}(x)$, which is well defined on all $\Omega_{\delta}$ by the following arguments: since on $E = \pi^{-1}(0)$ the $\hat{u}_{\delta}$ is constantly $-\infty$ by construction, and on the outside of $E$ we have that $\pi$ is an isomorphism. Moreover $\pi^{-1}(\Omega_{\delta}) \in \tilde{\Omega}_{\delta}$, because $d \le d_{\tau}$ by definition of $\tau$.

<footnote n="3">³One should keep in mind that Section 2 already garanties the existence of such barriers, and a control in their modululs of continuity, for some specific Dirichlet problems.</footnote>
### 3.3. Extending regularization to all $\Omega$.
Here, we will assume the existence of barriers as in Definition 2.1 to control $u$ $\delta$-close to the boundary. The extension will be, for some constant $C > 0$:

$$ \bar{u}_{\delta} = \begin{cases} \max\{\check{u}_{\delta}, u + 4CH(\delta)\} & \text{in } \Omega_{2\delta}, \\ u + 4CH(\delta) & \text{in } \Omega \setminus \Omega_{2\delta}. \end{cases} $$

Notice that because $\check{u}_\delta$ goes to $-\infty$ at $X_{\text{sing}}$, then close to the singularity (inside some ball $B_\mu(X_{\text{sing}})$, for $\mu > 0$ small enough) $u + 4CH(\delta)$ will eventually be the greater term inside the max above, making $\bar{u}_\delta$ continuous and psh in $\Omega$. For the gluing to be in $\text{PSH}(\Omega)$ we need that $\check{u}_\delta(x) \le u(x) + 4CH(\delta), \forall x \in \partial\Omega_{2\delta}$. The barriers will give us this last inequality through the following arguments:

#### 3.3.1. Behaviour at the boundary.
Fix some $\lambda_0 > 0$ small enough such that $\overline{B_{\lambda_0}(X_{\text{sing}})} \in \Omega_{\delta_0}$.

**Lemma 3.2.** Assume $\text{MA}(\Omega, \phi, f)$ admits barrier functions $v, w$. The solution $u = u(\Omega, \phi, f)$ satisfies, for some constant $C' > 0$:

$$ |u(x) - u(\xi)| \le C'\omega_v(d(x, \xi)), \forall x \in \Omega \setminus B_{\lambda_0}(X_{\text{sing}}); \forall \xi \in \partial\Omega. $$

*Proof.* From Definition 2.1:

$$ v(x) - v(\xi) \le u(x) - \phi(\xi) \le -(w(x) - w(\xi)), \forall x \in \Omega \setminus B_{\lambda_0}(X_{\text{sing}}); \forall \xi \in \partial\Omega. $$

By the modulus of continuity of the barriers we get, for some constant $C' > 0$:

$$ |u(x) - u(\xi)| \le C'\omega_v(d(x, \xi)), \forall x \in \Omega \setminus B_{\lambda_0}(X_{\text{sing}}); \forall \xi \in \partial\Omega. $$

□

#### 3.3.2. Behaviour near the boundary.

**Lemma 3.3.** Assume $\text{MA}(\Omega, \phi, f)$ admits barrier functions $v, w$. Take $r_0 > r > 0$ where $\overline{B_{r_0}(\xi)} \cap \Omega \subset \Omega \setminus B_{\lambda_0}(X_{\text{sing}}); \forall \xi \in \partial\Omega$. For any $\xi \in \partial\Omega$, Then the solution $u$ satisfies the following property:

$$ |u(x_1) - u(x_2)| \le 2C'\omega_v(r) $$

for some constant $C' > 0$ and $\forall x_1, x_2 \in \overline{B_r(\xi)} \cap \Omega$.

*Proof.* Fix $r > 0$ and an arbitrary $\xi \in \partial\Omega$. Take any two points $x_1, x_2 \in \overline{B_r(\xi)} \cap \Omega$. Using the triangular inequality we get:

$$ \begin{aligned} |u(x_1) - u(x_2)| &\le |u(x_1) - u(\xi)| + |u(\xi) - u(x_2)| \\ &\le C\omega_v(d(x_1, \xi)) + C\omega_v(d(\xi, x_2)) \\ &\le 2C\omega_v(r). \end{aligned} $$

The second line comes from Lemma 3.2. □

#### 3.3.3. Extension of regularization.
By construction, the value of $u_\delta := \pi_*\tilde{u}_\delta$ is under control relative to the supremum on a ball of radius $\delta$ for the points away from the singularity, where the resolution is an isomorphism. To prove the inequality necessary for the glueing process, assuming $\text{MA}(\Omega, \phi, f)$ admits barrier function $v, w$, we use an argument by contradiction:

*Proof.* Assume by contradiction that $\exists x_0 \in \partial\Omega_{2\delta}$ such that $\check{u}_\delta(x_0) > u(x_0) + 4CH(\delta)$. since $u_\delta(x_0) \le \max_{\overline{B_\delta(x_0)}} u$, then $\exists x^* \in \overline{B_\delta(x_0)}$ such that $u(x^*) + H(\delta)\pi_*\rho'(x_0) > u(x_0) + 4CH(\delta)$. The points on $\partial\Omega_{2\delta}$ have distance $2\delta$ to $\partial\Omega$, then take $\xi_0 \in \partial\Omega$ such that $d(\xi_0, x_0) = 2\delta$. We use the Lemma 3.3 for $r = 4\delta$, $x_1 = x_0$ and $x_2 = x^*$, for $\delta < \min\{r_0/2, \delta_2\}$. Then:
$$|u(x_0) - u(x^*)| \le 2C'\omega_v(4\delta).$$

By assumption $|u(x^*) - u(x_0)| > 4CH(\delta) - H(\delta)\pi_*\rho'(x_0) \ge 8C'H(\delta)$. For $C \ge 2C' + \inf_{\partial\Omega_\delta} \pi_*\rho' + \sup_{\Omega} \pi_*\rho' > 0$. Then we get: $2C'(\omega_v(4\delta) - 4H(\delta)) > 0$, which is a contradiction for any $\omega_v(\delta) \gtrsim \delta$, as $\omega_H(\delta) \gtrsim \omega_v(\delta)$. □

Hence, the global extension $\bar{u}_\delta$ is continuous and psh as we wanted to construct.

### 4. $L^1$ ESTIMATE AND GENERAL CASE

We will use the local theory developed in [20], and then refined in [1,7], to obtain the $L^1$ estimate of the regularizing function using a Laplacian estimate. With this we will obtain the desired regularity of solutions to the Dirichlet problems we already have barriers for, from Section 2, and then treat the general case.

#### 4.1. $L^1$ Laplacian estimate.
Here, we calculate the estimate that dictates the behavior away from the boundary, derived from the local theory using a Laplacian estimate. First, we compare the local regularizing function with the one constructed in Section 3:

**Lemma 4.1.** Let $\varepsilon > 0$ be small enough, and $u \in \text{PSH}(\Omega) \cap C^0(\bar{\Omega})$ with $(dd^c u)^n = f\beta^n$ in $\Omega$. Then for any sufficiently small $\delta > 0$ and chart on the resolution $(U \in \tilde{\Omega}_{\delta/2}, \psi)$ we have:

$$\eta_{\delta/2} \tilde{u} \le C_1 \Lambda_{\delta/2} \tilde{u} + C_2 \delta^2$$

for some constants $C_1, C_2 > 0$ and $\Lambda_\delta \tilde{u}(\tilde{x}) := \Lambda_\delta(\tilde{u} \circ \psi^{-1})(\psi(\tilde{x}))$ for any $\tilde{x} \in U$.

Moreover, we have $||\eta_\delta \tilde{u} - \tilde{u}||_{L^1(\tilde{\Omega}_{2\delta})} \le \tilde{C}\delta^{1-\varepsilon}$, for some constant $\tilde{C} > 0$. Also, if $\Delta \tilde{u}$ has finite mass in $\tilde{\Omega}$ then one can get $\delta^2$ instead of $\delta^{1-\varepsilon}$.

*Proof.* For a fixed chart $(U \in \tilde{\Omega}_{\delta/2}, \psi)$ we have from the proof of [12, Lemma 2.3]:

$$\eta_{\delta/2} \tilde{u}(\tilde{x}) = \frac{1}{(\delta/2)^{2n}} \int_{\tilde{y} \in \tilde{X}} \tilde{u}(\tilde{y}) \eta \left( \frac{|\log_{\tilde{x}} \tilde{y}|}{(\delta/2)^2} \right) dV_{\tau}(\log_{\tilde{x}} \tilde{y})$$

where $\tilde{y} \mapsto \xi = \log_{\tilde{x}} \tilde{y}$ is the inverse function of $\xi \mapsto \tilde{y} = \exp_{\tilde{x}}(\xi)$. By the proof of [12, Lemma 2.4] we have:

$$dV_{\tau}(\log_{\tilde{x}} \tilde{y}) = \bigwedge_{j=1}^{n} \frac{i}{2}(dz_j - dw_j) \wedge (d\bar{z}_j - d\bar{w}_j) + O(d_{\tau}(\tilde{y}, \tilde{x})^2),$$

where $(w, z) \mapsto (\tilde{x}, \log_{\tilde{x}}(\tilde{y}))$ represent local coordinates on a neighborhood of the zero section in $TU$, while $(\tilde{x}, \log_{\tilde{x}}(\tilde{y})) \mapsto (\tilde{x}, \tilde{y})$ is a diffeomorphism from that neighborhood onto the diagonal in $U \times U$. The $O(\cdot)$ term depends only on the curvature. By definition, coupled with the above equality:

$$\eta_{\delta/2} \tilde{u}(\tilde{y}) - \Lambda_{\delta/2} \tilde{u}(\tilde{y}) \le C_1 \Lambda_{\delta/2} \tilde{u}(\tilde{y}) + C_2 \delta^2,$$

for any $\tilde{y} \in U$ and some constants $C_1, C_2 > 0$. Moreover, taking $\sup \eta = 1$ one can choose $C_1 = 0$, consequently applying Theorem 1.7, one gets:

$$||\eta_{\delta/2} \tilde{u} - \tilde{u}||_{L^1(U)} \le C_3 \delta^{1-\varepsilon},$$

for some constant $C_3 > 0$ for the chart $(U, \psi)$. Then, by compactness of $\tilde{\Omega}_\delta$, the desired estimate is achieved for some constant $\tilde{C} > 0$:

$$||\eta_\delta \tilde{u} - \tilde{u}||_{L^1(\tilde{\Omega}_{2\delta})} \le \tilde{C}\delta^{1-\varepsilon}.$$

**Remark 4.2.** The proof doesn't involve the boundary values of $u$. This is clear by the proof of [1, Theorems 3.3, 3.4 and Lemma 3.5], nor it involves the barrier functions.
4.2. Main estimate. Theorem 4.3 below gives us, provided we have appropriate barrier functions, our main estimate Theorem A and also Corollary B:

**Theorem 4.3.** Assume $MA(\Omega, \phi, f)$ admits barriers $v, w$. The unique solution $u = u(\Omega, \phi, f)$ which belongs to $PSH(\Omega) \cap C^0(\bar{\Omega})$ to $MA(\Omega, \phi, f)$, satisfies for any $x \in \Omega$:

$$ \omega_{u,x}(t) \leq C_x \omega_H(t) $$

for some constants $C_x > 0$ such that $C_x \to +\infty$ as $x \to X_{sing}$, $0 \le \gamma < \frac{1}{(nq+1)}$ and $\omega_H(t) \gtrsim \max\{\omega_\phi(t^{1/2}), t^\gamma\}$ or $\max\{\omega_\phi(t^{1/2}), t^{2\gamma}\}$ if $\Delta \tilde{u}$ has finite mass in $\Omega$.

*Proof.* Fix $\gamma < \frac{1}{(nq+1)}$ and $\varepsilon > 0$ sufficiently small that $\gamma' := \frac{\gamma}{(1-\varepsilon)} < \frac{1}{(nq+1)}$. Applying Theorem 1.6 for $\gamma'$:

$$
\begin{aligned}
\sup_{x \in \Omega} (\bar{u}_{\delta}(x) - u(x) - 4CH(\delta)) &\le \sup_{x \in \partial \Omega} (\bar{u}_{\delta}(x) - u(x) - 4CH(\delta))^* + \bar{C} \|\bar{u}_{\delta} - u - 4CH(\delta)\|_{L^1(\Omega)}^{\gamma'} \\
&\le \bar{C} \|\bar{u}_{\delta} - u - 4CH(\delta)\|_{L^1(\Omega_{2\delta} \setminus B_{\mu}(X_{\text{sing}}))}^{\gamma'} \\
&\le \bar{C} \|\tilde{u}_{\delta} - u - 4CH(\delta)\|_{L^1(\Omega_{2\delta} \setminus B_{\mu}(X_{\text{sing}}))}^{\gamma'} \\
&\le \bar{C} \|\pi_* \tilde{u}_{\delta} - u\|_{L^1(\Omega_{2\delta} \setminus B_{\mu}(X_{\text{sing}}))}^{\gamma'} \\
&\le \bar{C} \|\pi_* \eta_{\delta} \tilde{u} - u\|_{L^1(\Omega_{2\delta} \setminus B_{\mu}(X_{\text{sing}}))}^{\gamma'} \\
&\le \bar{C} \|\eta_{\delta} \tilde{u} - \tilde{u}\|_{L^1(\widetilde{\Omega}_{2\delta} \setminus \pi^{-1}[B_{\mu}(X_{\text{sing}})])}^{\gamma'} \\
&\le \bar{C} (\tilde{C}^{\gamma'} \delta^{(1-\varepsilon)\gamma'}) \\
&\le C_0 \delta^{\gamma}
\end{aligned}
$$

Close to $\partial\Omega$ and $X_{\text{sing}}$ the terms are controlled by construction. From lines 3 to 4, we use the fact that $H(\delta)\rho' - 4CH(\delta) \le 0$. Remember that outside the singularity/divisors $\pi$ is an isomorphism. The second to last passage is achieved by applying Lemma 4.1. Hence, we get:

$$
\begin{aligned}
C_0 \delta^{\gamma} + 4CH(\delta) &\ge \sup_{x \in \Omega} (\bar{u}_{\delta}(x) - u(x)) \\
&\ge \bar{u}_{\delta}(x) - u(x), \quad \forall x \in \Omega \\
&\ge \pi_* \tilde{u}_{\delta}(x) - u(x) + H(\delta) \pi_* \rho'(x), \quad \forall x \in \Omega_{\delta} \\
&\ge \tilde{u}_{\delta}(\tilde{x}) - \tilde{u}(\tilde{x}) + H(\delta) M \log(d_{\tau}(\tilde{x}, E)), \quad \forall \tilde{x} \in \widetilde{\Omega}_{\delta}
\end{aligned}
$$

As $\rho'(\tilde{x}) \ge M \log(d_{\tau}(\tilde{x}, E))$, for any $\tilde{x} \in \widetilde{\Omega}$ and some constant $M > 0$. Then, for $S(\tilde{\lambda}) := M(-\log(d_{\tau}(\tilde{x}, E)))^4$:

$$ (4.1) \qquad \tilde{u}_{\delta}(\tilde{x}) - \tilde{u}(\tilde{x}) \le (C_1 + S(\tilde{\lambda}))H(\delta), \quad \forall \tilde{x} \in \widetilde{\Omega}_{\delta} $$

Following the proof of [12, Theorem D], we get a uniform lower bound on the parameter $t = t(\tilde{x})$ that realizes the infimum in the Kiselman-Legendre transform for $\tilde{u}_{\delta}$ at a fixed $\tilde{x} \in \widetilde{\Omega}_{\delta}$:

$$ \tilde{u}_{\delta}(\tilde{x}) - \tilde{u}(\tilde{x}) = \eta_{\delta} \tilde{u}(\tilde{x}) + Kt^2 - \tilde{u}(\tilde{x}) - K\delta^2 - c \log(t/\delta) \le (C_1 + S(\tilde{\lambda}))H(\delta) $$

As $t \mapsto \eta_t \tilde{u} + Kt^2$ is increasing, we have $\eta_t \tilde{u} + Kt^2 - \tilde{u} \ge 0$; thus:

$$ c \log(t/\delta) \ge -(C_2 + S(\tilde{\lambda}))H(\delta). $$

<footnote n="4">⁴The notation $\tilde{\lambda}(\tilde{x})$ in $\tilde{X}$ is the analogous of $\lambda(x)$ in $X$.</footnote>
Since $c = A^{-1}H(\delta) - A^{-1}K\delta = H(\delta)(A^{-1} - A^{-1}K\frac{\delta}{H(\delta)})$, we get the bound $c \ge \frac{A^{-1}H(\delta)}{2}$, by choosing $\delta \le \delta_3 := \min\{r_0/2, \delta_0, \delta_1, \delta_2, \varpi^{-1}(1/2K)\}$ for $\varpi(\delta) = \frac{\delta}{H(\delta)}$, because $\omega_H(\delta) \ge \delta$.
Then:

$$ \delta \ge t(\tilde{x}) \ge \delta\kappa(\tilde{x}), $$

where

$$ (4.2) \qquad \kappa(\tilde{x}) := e^{-2A(C_2+S(\tilde{\lambda}))} = e^{-2AC'_2} \cdot (\tilde{\lambda}(\tilde{x}))^{2AM} $$

Finally, using that $t \mapsto \eta_t\tilde{u} + Kt^2$ is increasing, $t(\tilde{x}) \ge \delta\kappa(\tilde{x})$ and the inequality (4.1), for every $\tilde{x} \in \tilde{\Omega}_\delta$ we get:

$$ (4.3) \qquad \eta_{\delta\kappa(\tilde{x})} \tilde{u}(\tilde{x}) - \tilde{u}(\tilde{x}) - K\delta^2 \le \tilde{u}_\delta(\tilde{x}) - \tilde{u}(\tilde{x}) \le (C_1 + S(\tilde{\lambda}))H(\delta) $$

which leaves us with:

$$ (4.4) \qquad \eta_\delta \tilde{u}(\tilde{x}) - \tilde{u}(\tilde{x}) \le (C_3 + S(\tilde{\lambda})) H(\delta/\kappa(\tilde{\lambda})) \le C_{\tilde{\lambda}} \omega_H(\delta), \quad \forall \tilde{x} \in \tilde{\Omega}_\delta $$

by the subadditivity of $\omega_H$ we get: $C_{\tilde{\lambda}} = \frac{(C_3+S(\tilde{\lambda}))}{\kappa(\tilde{\lambda})} = \frac{[C_3+M'(-\log(\tilde{\lambda}(\tilde{x})))]}{(\tilde{\lambda}(\tilde{x}))^{2AM}}$.

Fix $\tilde{x} \in \tilde{\Omega} \setminus E$, then there exists $\delta' \le \min\{1/2, \delta_3, \lambda(x)/2, d(x, \partial\Omega)/2\}$. Now fix any $0 < \delta < \delta'$, hence $\tilde{x} \in \tilde{\Omega}_\delta$. Applying [26, Theorem 3.4] for $\tilde{\Omega}_\delta$, $\tilde{u}$ and inequality (4.4):

$$ (4.5) \qquad |\tilde{u}(\tilde{x}) - \tilde{u}(\tilde{y})| \le D_0(C_{\tilde{\lambda}(\tilde{x})} + C_{\tilde{\lambda}(\tilde{y})})\omega_H(d_\tau(\tilde{x}, \tilde{y})) $$

for any $\tilde{y} \in \tilde{\Omega}_\delta$ and some constant $D_0 > 0$. Then, one can pass the right-hand side of (4.5) to $\Omega_\delta$ and substitute $\tilde{\lambda}$ by $\lambda$, remember that $\tau := \tilde{C}\pi^*\beta + \varepsilon\theta$ for some constants $\tilde{C} > 1$ big enough and $0 < \varepsilon$ small enough, with some smooth closed (1,1)-form $\theta$ so that $\pi^*d \le d_\tau$. Hence:

$$ (4.6) \qquad |u(x) - u(y)| \le (C_{\lambda(x)} + C_{\lambda(y)})\omega_H(d_\tau(\tilde{x}, \tilde{y})) $$

for any $\tilde{y} \in \tilde{\Omega}_\delta$ and some constant $C_\lambda > 0$ that goes to $+\infty$ as $\lambda \to X_{\text{sing}}$. Lastly, notice that we have the comparison $\tau_{\tilde{x}} \le K_{\lambda(x)}\pi^*\beta_{\tilde{x}}$ for some constant $K_\lambda > 0$ that goes to $+\infty$ as $\lambda \to X_{\text{sing}}$. Now, by the subadditivity of $\omega_H$ we get:

$$ (4.7) \qquad |u(x) - u(y)| \le (C'_{\lambda(x)})\omega_H(d(x, y)) \le (C'_{\lambda(x)})\omega_H(\delta) $$

for any $y \in B_\delta(x)$ and some constant $C'_\lambda > 0$ that goes to $+\infty$ as $\lambda \to X_{\text{sing}}$. $\square$

<figure><img src="image_3.png" /></figure>

**Remark 4.4.** One can notice that the proof of [26, Theorem 3.4] follows even if $\tilde{\Omega}_\delta$ have a boundary because $\tilde{u}$ is defined on all $\tilde{\Omega}$.

**Corollary 4.5.** Let $\hat{f} \in L^p(\Omega, \beta^n)$, $p > 1$ with $\hat{f}$ is bounded near $\partial\Omega$ and $0 \le \gamma < \frac{1}{(nq+1)}$. Then $u(\Omega, \phi, 0)$, $u(\Omega, 0, \hat{f})$ have modulus of continuity $C_x \max\{\omega_\phi(t^{1/2}), t^\gamma\}$ and $C_x t^{2\gamma}$ respectively, for $x \in \tilde{\Omega}$ and some constant $C_x > 0$ that goes to $+\infty$ as $x \to X_{\text{sing}}$.

*Proof.* Since we already have barriers for the problems $MA(\Omega, \phi, 0)$ and $MA(\Omega, 0, \hat{f})$, we can choose $H$ to have modulus of continuity $\max\{\omega_\phi(t^{1/2}), t^\gamma\}$, since their respective barriers have better regularity then that. Then from Theorem 4.3 the solution $u(\Omega, \phi, 0)$ will have modulus of continuity $C_x \max\{\omega_\phi(t^{1/2}), t^\gamma\}$. Similarly, we get that $u(\Omega, 0, \hat{f})$ will have modulus of continuity $C_x t^\gamma$, because the boundary data is zero. One can improve the regularity to $C_x t^{2\gamma}$ by first noticing that the barriers for $MA(\Omega, 0, \hat{f})$ in Lemma 2.2 have Laplacians with
uniformly bounded mass in $\Omega$, using Chern-Levine-Nirenberg inequalities<fnref n="5" /> ([8], [11, Theoreme 2.2], see [22, Theorem 3.9] for a more modern presentation) coupled with them being bounded and psh on a neighborhood of $\bar{\Omega}$. Now, if $\beta$ is Kähler then by the comparison principle, we get that the solution has Laplacian with uniformly bounded mass. If $\beta$ is not then it suffices to notice that in a neighborhood of $\Omega$ it is bigger/smaller then a Kähler form, such as $dd^c(A\rho)$ for some constant $A > 0$ big/small enough. Lastly, as this argument is the same in $\Omega$ or $\tilde{\Omega}$, therefore we get the regularity $t^{2\gamma}$ in Lemma 4.1. □

**4.3. The general case.** Now we construct the barrier functions for the solution of the general problem $u := u(\Omega, \phi, f)$:

**Proposition 4.6.** Let $u = u(\Omega, \phi, f)$. Then there exists two barrier functions $v, w \in PSH(\Omega) \cap C^0(\bar{\Omega})$, such that:

$$ (1) \quad v(\xi) = \phi(\xi) = -w(\xi), \quad \forall \xi \in \partial\Omega, $$

$$ (2) \quad v(z) \le u(z) \le -w(z), \quad \forall z \in \Omega, $$

$$ (3) \quad \omega_{v,x}(t), \omega_{w,x}(t) \le C_x \max\{\omega_{\phi}(t^{1/2}), t^{\gamma}\} $$

for some constant $C_x > 0$ that goes to $+\infty$ as $x$ approaches the singular point.

*Proof.* The upper-barrier will be $w = u(\Omega, -\phi, 0)$. Since $-w = u(\Omega, \phi, 0)$, $-w = \phi$ in $\partial\Omega$ and $0 = (dd^c(-w))^n \le (dd^c u)^n$, by the comparison principle we get $-w \ge u$ in $\Omega$ and by the Corollary 4.5 we know the modulus of continuity of $w$.

For the lower-barrier, we take a bigger pseudoconvex domain $\Omega \in \tilde{\Omega} \in X$ and extend trivially by zero the density $f$ to $\tilde{\Omega}$, name it $\tilde{f}$. Note that $\tilde{f}$ is bounded near $\partial\tilde{\Omega}$. The lower-barrier will be $v := u(\tilde{\Omega}, 0, \tilde{f})|_{\tilde{\Omega}} + u(\Omega, \phi - u(\tilde{\Omega}, 0, \tilde{f})|_{\partial\tilde{\Omega}}, 0)$. By construction $v|_{\partial\tilde{\Omega}} = \phi$ and $(dd^c v)^n \ge f\beta^n + 0$ in $\tilde{\Omega}$, by the comparison principle $v \le u$ in $\tilde{\Omega}$ and by Corollary 4.5 we know the modulus of continuity of $v$. □

Now that we have barriers for the general case $MA(\Omega, \phi, f)$ we can prove Theorem A and Corollary B:

*Proof.* By choosing $\omega_H(t) = \max\{\omega_\phi(t^{1/2}), t^\gamma\}$ on Theorem 4.3 one proves Theorem A, this choice of $H$ is possible because of the modulus of continuity of the barrier functions constructed in Proposition 4.6. Corollary B follows directly as for it $\omega_\phi(t) = t^\alpha$. □

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GUILHERME CERQUEIRA GONÇALVES, INSTITUT DE MATHÉMATIQUES DE TOULOUSE; UMR 5219, UNIVERSITÉ DE TOULOUSE; CNRS, UPS, 118 ROUTE DE NARBONNE, F-31062 TOULOUSE CEDEX 9, FRANCE

Email address: guicerqg@gmail.com / guilherme.cerqueira_goncalves@math.univ-toulouse.fr

URL: https://guicerqmath.github.io/