October 16, 2025
Let \((X,\omega)\) be a compact Hermitian manifold of dimension \(n\). We derive an \(L^\infty\)-estimate for bounded solutions to the complex \(m\)-th Hessian equations on \(X\), assuming a positive right-hand side in the Orlicz space \(L^{\frac{n}{m}}(\log L)^n(h\circ\log \circ \log L)^n\), where the associated weight satisfies Kołodziej’s Condition. Building upon this estimate, we then establish the existence of continuous solutions to the complex Hessian equation under the prescribed assumptions.
Let \((X,\omega)\) be a compact complex Hermitian manifold of dimension \(n\) equipped with a Hermitian metric \(\omega\). Fix \(0< m \le n\) and let \(dV_X\) be a smooth volume form. The problem of interest in this paper is to find a constant \(c>0\) and continuous \((\omega,m)\)-subharmonic solutions to the following complex \(m\)-Hessian equation \[ \label{Equation:32Hessian} (\omega+d d^c u)^m \wedge \omega^{n-m} = c e^{\lambda u} f dV_X, \lambda\geq 0,\tag{1}\] under the assumption that the densities \(f^{n/m}\) belong to an Orlicz space \(L^{\chi}\) with the weight \(\chi\) satisfying Kołodziej’s condition, which will be explained later.
When \(m=1\), this simplifies to the Poisson equation, while when \(m=n\), it reduces to the Monge–Ampère equation. Yau’s solution [1] of the Calabi Conjecture has motivated many subsequent studies on Monge–Ampère equations over the past fifty years. As presented by Błocki [2], using the simplifications due to Kazdan and to Aubin–Bourguignon, Yau’s a priori estimate can be extended to cases where the densities \(f\) belong to \(L^p\) for \(p>n\). One of the breakthrough was achieved by Kołodziej [3], who applied pluripotential theory to obtain an \(L^{\infty}\)-estimate for Monge–Ampère equations when \(f\) belongs to some Orlicz space, with the associated weight specifically satisfying Condition (K). As remarked by Kołodziej [3], such an a priori estimate is "almost sharp".
This integrability condition on densities \(f\) can be extended in the context of complex Hessian equations. We say that a convex weight \(\chi\) satisfies Kołodziej’s condition (Condition (K)) if it is increasing and \[\chi(t) \underset{t \to +\infty}{\sim} t (\log (t+1))^n \left(h \circ \log \circ \log (t+3)\right)^n,\] where the function \(h:\mathbb{R}^+ \rightarrow \mathbb{R}^+\) satisfying \(\int^{+\infty} h^{-1}(t)dt <+\infty.\) In this paper, we focus on the case where \(f^{n/m} \in L^{\chi}\). We refer the reader to Section 1 for more detailed discussions on Orlicz spaces and Condition (K).
The study of weak solutions to complex Monge–Ampère equations plays an important role in the analysis of Ricci flows (see [4]). On compact Kähler manifolds, one can study the modulus of continuity for solutions to complex Monge–Ampère equations. Hölder regularity for such equations with densities in \(L^p\) was established in [5], and has been further generalized to the case where the densities belong to certain Orlicz spaces, in [6] via a PDE approach, and in [7] using pluripotential theory.
To prove the existence of continuous solution to the complex Monge–Ampère equation, a key ingredient is the \(L^{\infty}\)-estimate. Various techniques have been developed to obtain \(L^{\infty}\)-estimate for Monge–Ampère equations on compact manifolds, including Yau’s [1] original proof, which employs Moser’s iteration scheme. Later, Kołodziej [3] developed \(L^{\infty}\)-estimates using properties of capacity and established an optimal integrability condition for the density \(f\) (see also [8], [9] for related generalizations). Inspired by the work of Chen–Cheng [10] on constant scalar curvature Kähler metrics, a very different approach was introduced by Guo–Phong–Tong [11], by comparing the given equation to an auxiliary complex Monge–Ampère equation. This PDE technique has since been generalized in [12], [13]. An efficient approach was developed by Guedj–Lu [14], [15], using properties of envelopes, which can be adapted to the case where \(f \in L \log^p L\) with \(p>n\), as noted in [15]. More recently, they generalized this result to Orlicz spaces satisfying Condition (K) on compact Kähler manifolds in [16], using a new approach that reduces Kołodziej’s criterion to Yau’s theorem.
On the other hand, complex \(m\)-Hessian equations are natural generalizations of Monge–Ampère equations. Błocki [17] established potential theory for \(m\)-Hessian equations in domains of \(\mathbb{C}^n\). Later, Dinew–Kołodziej [18] obtained a priori estimates and proved a stability result for weak solutions of the complex Hessian equation in domains of \(\mathbb{C}^n\) and compact Kähler manifolds. The modulus of continuity of solutions to complex Hessian equations has been studied in [19], [20] with densities \(f\in L^p,p>n/m\), and has been extended recently in [21] to the cases where \(f \in L^{n/m}\log^p(L),p>2n\). On compact Hermitian manifolds, Kołodziej and Nguyen proved the existence of weak solutions with a nonnegative right-hand side belonging to \(L^p\), \(p>n/m\), in [22], while the corresponding result for Monge–Ampère equations was established in [23].
The purpose of this paper is to extend the above results to prove the continuity of bounded solutions to the complex Hessian equation. We begin by establishing a subsolution lemma for the Monge–Ampère equation, generalizing Lemma 2.1 of [14] to the case where \(f \in L^{\chi}\), with \(\chi\) satisfying Condition (K). The lemma is stated as follows:
:::: Lemma A Lemma A 1 (Subsolution Lemma).
Assume \(\chi\) satisfies Condition (K). Then there exist a uniform constant \(s=s(\chi)>0\) such that for all \(0\leq g\in L^{\chi}\), we can find \(\psi\in \mathcal{PSH}(X,\omega)\), \(-1\leq \psi \leq 0\), satisfying \[(\omega +dd^c\psi)^n \geq s(\chi) \frac{g}{\|g\|_{\chi}} dV_X.\] ::::
We obtain the Subsolution Lemma by gluing the solutions to the Dirichlet problem in local context: \[\label{Prob:Dirch} \begin{align} (dd^cu)^n &= \frac{f }{ \| f\|_{\chi} } dV_X \text{, in }\Omega;\\ u &= 0 \text{, on }\partial \Omega. \end{align}\tag{2}\]
Compared with the case where the densities \(f \in L^p\), working with Orlicz norms is more involved, as it requires controlling the Orlicz norm of the solution \(u\) associated to the convex conjugate of \(\chi\). To simplify the argument, we adapt an approach from the recent article by Guedj–Lu [16] to derive a local \(L^{\infty}\)-estimate for problem 2 .
As a consequence of Lemma A, we obtain a uniform estimate for the complex Hessian equation on a Hermitian manifold, where the densities \(f\) satisfy Kołodziej’s optimal condition.
:::: Theorem B Theorem B 1 (\(L^{\infty}\)-estimate).
Let \(\varphi\) be a bounded \((\omega,m)\)-subharmonic function satisfying \[ \label{ean:Hess} (\omega+ dd^c\varphi)^m \wedge \omega^{n-m} \le f d V_X,\qquad{(1)}\] where \(0\le f^{n/m} \in L^{\chi}\) with the weight \(\chi\) satisfying Condition (K). Then \[\mathrm{Osc}_X (\varphi) \le C,\] where \(C\) depends on \(\omega, n, m\) and \(\left\| f^{\frac{n}{m} }\right\|_{\chi}\). ::::
By generalizing an idea in [24], we demonstrate that the \(L^{\infty}\)-estimate for complex Hessian equations can be reduced to the corresponding estimate for an associated Monge–Ampère equation: \[(\omega + dd^c\varphi)^n \le f dV_X,\] where \(f \in L^{\chi}\), and \(\chi\) satisfies Condition (K). To obtain the \(L^{\infty}\)-estimate for the Monge–Ampère equation, our main tool is the domination principle. The key novelty of our approach lies in constructing an appropriate subsolution. Our approach avoids potential computational difficulties associated with Orlicz norms and presents a different method compared to Kołodziej’s original proof.
Once the \(L^{\infty}\)-estimate is obtained, we can derive the existence and regularity of bounded solutions. This leads us to the second main result of the study:
:::: Theorem C Theorem C 1.
Let \(0 \le f^{n/m} \in L^{\chi}\), with the weight function \(\chi\) satisfying Condition (K). Then there exist a constant \(c >0\) and a solution \(u \in \mathcal{SH}_m(X, \omega)\cap \mathcal{C}^0(X)\) satisfying the complex Hessian equation \[(\omega+d d^c u)^m \wedge \omega^{n-m} = c e^{\lambda u} f dV_X,\; \lambda\geq 0.\] Moreover, all bounded solutions to the above complex Hessian equation are continuous. ::::
To prove the first statement of Theorem C, an essential component is to establish a stability theorem (Theorem 24). For \(f \in L^p\), with \(p>n/m\), such a stability result was proven in [22]. In the case where \(f\in L^{\chi}\) and the weight function \(\chi\) satisfies Condition (K), the continuity of solutions to Monge–Ampère equations was studied in [25] by adapting the method in [26]. We point out that our approach to the stability estimate is fundamentally different and remains of interest in the case where the densities \(f\in L^p\), as discussed in Remark 25. Our main tool will be the Domination Principle and the subsolution (Lemma A). Having established the stability estimate (Theorem 24), we approximate the density \(f\) by the sequence of bounded functions \(\min(f,j)\). We show that the corresponding solutions \(u_j\) converge uniformly to an \((\omega,m)\)-subharmonic function \(u\) solving the equation 1 , thereby obtaining a continuous solution. The continuity of all bounded solutions follows from the Domination Principle (Proposition 8).
The paper is organized as follows. Section 1 gives preliminary material on \((\omega,m)\)-subharmonic functions, as well as key concepts of Orlicz spaces. A local \(L^{\infty}\)-estimate is established, and Lemma A and Theorem B are proved in Section 2. In section 3, we establish a stability estimate and subsequently investigate the continuity of bounded solutions, leading to the proof of Theorem C.
Acknowledgment 1. The author thanks her supervisor, Hoang-Chinh Lu, for suggesting the problem, valuable discussions during the preparation of the work, and useful comments that improved the presentation. This research is part of a PhD program funded by the PhD scholarship CSC-202308070110, and is partially supported by the projects Centre Henri Lebesgue ANR-11-LABX-0020-01 and the KRIS project of Fondation Charles Defforey.
In the whole paper, we let \((X,\omega)\) denote a compact Hermitian manifold of complex dimension \(n\in \mathbb{N}^{*}\), equipped with a Hermitian form \(\omega\). We denote by \(dV_X\) a smooth volume form on \(X\). We use differential operators \(d = \partial + \bar{\partial}\), and \(d^c = i (\bar{\partial}- \partial)\), so that \(dd^c= 2i \partial \bar{\partial}\).
We recall the definition of \((\omega,m)\)-subharmonic functions following [27] and briefly review some results related to both \(\omega\)-plurisubharmonic and \((\omega,m)\)-subharmonic functions that will be used later.
Fix an integer \(1 \le m \le n\). Fix \(\Omega\) an open set in \(\mathbb{C}^n\). Given a hermitian metric \(\omega\) on \(\Omega\), a real \((1,1)\)-form \(\alpha\) on \(X\) is called \(m\)-positive with respect to \(\omega\) if at all points in \(X\), \[\alpha^{k} \wedge \omega^{n-k} \ge 0, \forall k =1, \cdots, m.\] A \(\mathcal{C}^2(\Omega)\) function \(u : \Omega \rightarrow \mathbb{R}\) is called harmonic with respect to \(\omega\) if \(dd^cu \wedge \omega^{n-1}=0\) at all points in \(\Omega\).
Definition 1. A function \(u:\Omega \rightarrow \{ -\infty\} \cup \mathbb{R}\) is subharmonic with respect to \(\omega\) if:
\(u\) is upper semicontinuous and \(u \in L^1_{loc}(\Omega)\);
for every relatively compact open set \(D \Subset \Omega\) and every function \(h\in \mathcal{C}^0(D)\) that is harmonic with respect to \(\omega\) on \(D\) the following implication holds: \[u \le h \text{ on }\partial D \implies u \le h \text{ in }D.\]
Definition 2. A function \(\varphi : \Omega \rightarrow \{ -\infty\} \cup \mathbb{R}\) is quasi-subharmonic with respect to \(\omega\) if locally \(\varphi = u + \rho\), where \(u\) is subharmonic with respect to \(\omega\) and \(\rho\) is smooth.
A function \(\varphi\) is \(\alpha\)-subharmonic with respect to \(\omega\) if \(\varphi\) is quasi-subharmonic with respect to \(\omega\) and \((\alpha + dd^c\varphi) \wedge \omega^{n-1}\ge 0\) in the sense of distributions.
The positive cone \(\Gamma_m(\omega)\) associated with the metric \(\omega\) is defined by \[\left\{ \gamma \text{ real }(1,1) \text{-form}: \gamma^k\wedge \omega^{n-k}>0, k=1,\cdots,m \right\}.\] It follows from Gårding’s inequality[28] that if \(\gamma_0, \gamma_1, \cdots, \gamma_{m-1}\in \Gamma_{m}(\omega)\), then \[\gamma_0 \wedge\gamma_1\wedge \cdots \wedge\gamma_{m-1} \wedge\omega^{n-m} >0.\] One can write \(\widetilde{\omega}= \gamma_1\wedge \cdots \wedge\gamma_{m-1} \wedge\omega\), and it is a strictly positive \((n-1,n-1)\)-form on \(\Omega\).
Definition 3. A function \(\varphi: \Omega \rightarrow [ -\infty, + \infty)\) is called \((\alpha,m)\)-subharmonic with respect to \(\omega\) if \(\varphi\) is \(\alpha\)-subharmonic with respect to \(\widetilde{\omega}\) in \(\Omega\) for all \(\widetilde{\omega}\) of the form \(\widetilde{\omega}^{n-1}= \gamma_1\wedge \cdots \wedge\gamma_{m-1} \wedge\omega\), where \(\gamma_1, \cdots,\gamma_{m-1} \in \Gamma_m(\omega)\).
A function \(u : X \rightarrow \mathbb{R} \cup \{- \infty \}\) is called \((\alpha,m)\)-subharmonic on \(X\) if \(u\) is \((\alpha,m)\)-subharmonic on each local chart \(U\) of \(X\).
The set of all locally integrable functions on \(U\) which are \((\alpha,m)\)-subharmonic with respect to \(\omega\) in \(U\) is denoted by \(\mathcal{SH}_{\omega,m}(U,\alpha)\). In many contexts, such as [24], the forms \(\omega\) and \(\alpha\) may differ.
However, in this paper, we focus on the set \(\mathcal{SH}_{\omega,m}(U,\omega)\). To simplify the notations, we denote by \(\mathcal{SH}_m(U,\omega)\) the set of all \((\omega,m)\)-subharmonic functions with respect to \(\omega\) in \(U\). The set of all \(\omega\)-plurisubharmonic functions on \(U\) is denoted by \(\mathcal{PSH}(U, \omega)=\mathcal{SH}_n(U,\omega)\).
For notational convenience, we denote \(\omega_{u} \mathrel{\vcenter{:}}= \omega + dd^cu\).
Remark 4. By Gårding’s inequality[28], if \(u \in \mathcal{C}^2(X)\), then \(u\) is \((\omega,m)\)-subharmonic with respect to \(\omega\) on \(X\) if and only if the associated form \(\omega _u\) belongs to the closure of \(\Gamma_m(\omega)\).
It is well known that when \(n=m\), the set \(\mathcal{PSH}(X,\omega)\) is a closed subset of \(L^r(X)\), for all \(r \ge 1\). For \((\omega,m)\)-subharmonic functions, the following \(L^1\)-compactness result was proved in [22].
Lemma 5. Let \(u\in \mathcal{SH}_m(X,\omega)\) be normalized by \(\sup_X u =0\). Then there exists a uniform constant \(A>0\) depending only on \(X, \omega\) such that \[\int_X |u|\omega^n \le A.\]
Analogous to the Monge–Ampère operators, the complex Hessian operators also satisfy a domination principle. The domination principle for Hessian operator associated with continuous \((\omega,m)\)-subharmonic functions is established by Guedj–Lu in [24]. Following the demonstration that the Hessian operators for bounded functions are well-defined, we show the domination principle associated with bounded \((\omega,m)\)-sh functions here.
Lemma 6 (Maximum principle). Assume that \(u\) and \(v\) are bounded \((\omega,m)\)-subharmonic functions. Then \[(\omega + dd^c(\max(u,v)))^m \wedge \omega^{n-m} \ge \mathbb{1}_{\{ u >v \} } (\omega+dd^cu)^m \wedge \omega^{n-m} + \mathbb{1}_{\{ v\ge u \} } (\omega+dd^cv)^m \wedge \omega^{n-m}.\] In particular, if \(u \le v\), then, \[\mathbb{1}_{\{ v= u \} } (\omega+dd^cu)^m \wedge \omega^{n-m} \le \mathbb{1}_{\{ v= u \} } (\omega+dd^cv)^m \wedge \omega^{n-m}\]
In the case where \(u\) and \(v\) are continuous, the above lemma is proven in [24]. When \(u\) is bounded, thanks to the continuity of complex Hessian operators ([29]), we can approximate \(u\) by a sequence of decreasing smooth \((\omega,m)\)-subharmonic functions.
The comparison principle for \(u,v \in \mathcal{SH}_m(\Omega) \cap L^{\infty}(\Omega)\) is introduced in [29].
Proposition 7 (Comparison principle). Let \(u,v\) be bounded \(m\)-subharmonic functions with respect to \(\omega\) and such that \(\liminf_{z \rightarrow \partial\Omega}u(z)-v(z) \ge0\). Assume \[(dd^cv)^m \wedge \omega^{n-m}\ge(dd^cu)^m \wedge \omega^{n-m} \text{ in } \Omega.\] Then it follows that \(u \ge v\) in \(\Omega\).
Proof. See [29]. ◻
We will need a global version of domination principle:
Proposition 8 (Domination principle). Assume that \(u, v \in \mathcal{PSH}(X, \omega) \cap L^{\infty}(X)\) satisfy \[1_{\{u<v\}}\left(\omega+d d^c u\right)^m \wedge \omega^{n-m} \leq c 1_{\{u<v\}}\left(\omega+d d^c v\right)^m \wedge \omega^{n-m},\] for some constant \(0 \le c<1\). Then \(u \ge v\).
The case in which \(u,v\) are continuous has been treated in [24]. In the case where \(u,v \in L^{\infty}\), our proof is essentially analogous to the one developed for the Monge–Ampère operators in [30].
Proof. we may assume \(v \geq 1\). We fix a constant \(a\) such that \(c<a^m<1\), and we prove that \(u \geq a v\). The result then follows by letting \(a \rightarrow 1\). Assume it is not the case and let \(\{x_j\}\) be a sequence converging to \(x_0 \in X\) with \[\lim _{j \rightarrow+\infty}\left(u\left(x_j\right)-a v\left(x_j\right)\right)=m_a:=\inf _X(u-a v)<0.\] Let \(B\) be a small neighborhood of \(x_0\) and fix a smooth strictly plurisubharmonic function \(\rho\) such that \(\rho=0\) on \(\partial B,|\rho| \leq v\), and \(d d^c \rho \leq \omega\). One can take a holomorphic coordinate chart around \(x_0\) which is biholomorphic to the unit ball and \(\rho(x)=b\left(|x|^2-1\right)\), for a small constant \(b\). Observe that \(\rho<0\) in \(B\). Consider now \(\varphi=a v-(1-a) \rho\). A direct computation then shows that \(\omega+d d^c \varphi \geq 0\) in \(B\). Also, by construction \(\varphi \leq v\) and so \[\{u<\varphi\} \cap B \subset\{u<a v-(1-a) \rho\} \subset\{u<v\}.\] Hence by assumption and using the fact that \(\omega+d d^c(-\rho) \geq 0\), we get \[\begin{align} \mathbb{1}_{\{u<\varphi\}}\left(\omega+d d^c u\right)^m \wedge\omega^{n-m} \leq & c \mathbb{1}_{\{u<\varphi\}}\left(\omega+d d^c v\right)^m \wedge \omega^{n-m}\\ \leq& a^m \mathbb{1}_{\{u<\varphi\}}\left(\omega+d d^c v\right)^m\wedge \omega^{n-m}\\ \leq & \mathbb{1}_{\{u<\varphi\}}\left(\omega+d d^c \varphi\right)^m \wedge \omega^{n-m}. \end{align}\] From this and the maximum principle (Lemma 6) we infer \[\left(\omega+d d^c u\right)^m \wedge \omega^{n-m} \leq\left(\omega+d d^c \max (u, \varphi)\right)^m \wedge \omega^{n-m} \text{ in } B.\] It thus follows from the comparison principle (Proposition 7) that \[\inf _B(u-\max (u, \varphi)) \geq \liminf _{x \rightarrow \partial B}(u-\max (u, \varphi))(x)\] Evaluating this inequality at \(x_j\) and letting \(j \rightarrow+\infty\), we arrive at \[\lim _{j \rightarrow+\infty} \min \left(u\left(x_j\right)-a v\left(x_j\right)+(1-a) \rho\left(x_j\right), 0\right) \geq \liminf _{x \rightarrow \partial B} \min \left(u\left(x\right)-a v\left(x\right), 0\right).\] Since by construction, \[\lim _{j \rightarrow+\infty} u\left(x_j\right)-a v\left(x_j\right)+(1-a) \rho\left(x_j\right) = m_a - (1-a)b,\] we infer that \[m_a - (1-a)b\ge \lim _{x \rightarrow \partial B}\left(u\left(x\right)-a v\left(x\right)\right) \ge m_a,\] which leads to a contradiction. ◻
As a direct consequence of the domination principle, we have
Corollary 9. Fix \(\lambda>0\). If \(e^{-\lambda v} \omega_v^m \wedge \omega^{n-m} \ge e^{-\lambda u} \omega_u^m \wedge \omega^{n-m}\), then \(u \ge v\).
Proof. Fix \(C > 0\). On the set \(\{ u < v-C \}\) we have \[\omega_u^m \wedge \omega^{n-m} \le e^{\lambda (u-v)} \omega_v^m \wedge \omega^{n-m} \le e^{-\lambda C} \omega_v^m \wedge \omega^{n-m} .\] It follows from Proposition 8 that \(u \ge v - C\). Since this holds for all \(C >0\), we conclude that \(u \ge v\). ◻
Corollary 10. Assume \(u, v\) are bounded \((\omega,m)\)-subharmonic functions on \(X\) such that \[\left(\omega+d d^c u\right)^m \wedge \omega^{n-m} \leq c\left(\omega+d d^c v\right)^m \wedge \omega^{n-m}\] for some positive constant \(c\). Then \(c \geq 1\).
In Section 2, we study the Monge–Ampère equation in a local context. Here, we recall some essential definitions of the Cegrell classes [31], [32].
Fix a bounded hyperconvex domain \(\Omega \subset \mathbb{C}^n\). Let \(\mathcal{T}(\Omega)\) denote the set of bounded plurisubharmonic functions \(u\) in \(\Omega\) such that \(\lim_{z \rightarrow \zeta } u(z) =0\) for every \(\zeta \in \partial \Omega\), and \(\int_{\Omega}(dd^cu)^n < +\infty\).
A function \(u\) belongs to \(\mathcal{F}(\Omega)\) iff there exists a sequence \(u_j \in \mathcal{T}(\Omega)\) decreasing to \(u\) in all of \(\Omega\), which satisfies \(\sup_{j} \int_{\Omega}(dd^cu_j)^n < +\infty\).
As demonstrated by Cegrell [31], [32], the complex Monge–Ampère operator can be defined for functions belonging to the class \(\mathcal{F}(\Omega)\) even when these functions are unbounded. The Dirichlet problem in \(\mathcal{F}(\Omega)\) was solved in [32]:
Theorem 11. Assume that \(\mu\) is a positive measure on \(\Omega\). If \(\mu(\Omega) < +\infty\) and \(\mu\) vanishes on all pluripolar sets, then there exists a unique function \(u \in \mathcal{F}(\Omega)\) such that \((dd^cu)^n = \mu\).
Proof. See [32]. ◻
We recall some essential facts about Orlicz spaces that will be used throughout, following [33]. Let \(\mu\) be a probability measure on \(X\). A weight \(\chi : \mathbb{R}^{*} \to \mathbb{R}^* \cup \{+\infty\}\) is called normalized Young weight if it is convex, lower semi-continuous, non-trivial satisfying the normalizing condition \[\chi(0)= 0 \text{ and } 1\in\partial\chi(1).\] Recall that \(\partial \chi(l) \subset \mathbb{R}\) denotes the set of subgradients of \(\chi\) at \(l\), which means that \(a \in \partial \chi(t)\) if and only if \(\chi(t) + a b \leq \chi(t + b)\) for all \(b \in \mathbb{R}\). Its conjugate convex weight \(\chi^{*}\) is the Legendre transform of \(\chi\): \[\chi^{*}(h) = \sup_{t\in \mathbb{R}^*} \{ ht - \chi(t)\}.\] One can easily verify that the following propositions hold:
Proposition 12. Let \(\chi\) be a normalized Young weight. Then
\(\chi^*\) is also a normalized Young weight;
(Fenchel–Moreau theorem). The biconjugate weight \(\chi^{**}\) is also a normalized Young weight, and \(\chi^{**}= \chi\).
One can define a function space with respect to the Young function. Let \(L^{\chi}(\mu)\) denote the space of measurable functions defined as follows: \[L^{\chi}(\mu) \mathrel{\vcenter{:}}= \left\{ f: X \rightarrow \mathbb{R}^{*}\cup \{+\infty\}: \exists s >0, \int_X \chi(s f) < +\infty \right\}.\] One may also define the norm (also known as Luxembourg norm) on \(L^{\chi}(\mu)\). \[\| f \|_{\chi,\mu} \mathrel{\vcenter{:}}= \inf \left\{ r>0 : \int_X \chi \left(\frac{f}{r} \right) \le \chi(1) \right\}.\] We remark that \((L^{\chi}(\mu), \| \cdot\|_{\chi,\mu})\) constitutes a normed space and is complete, hence a Banach space. In the remainder of the paper, we fix \(\mu = dV_X\), a smooth probability measure, and write \((L^{\chi}, \| \cdot\|_{\chi})\) for simplicity.
In [3], S.Kołodziej developped a technique using pluripotential theory that enables one to bound the solution to the Monge–Ampère equation by the Orlicz norm, \(\| \cdot\|_{\chi}\), of the right-hand side, associated with a weaker integrable condition. We thus introduce the Condition (K).
Definition 13 (Condition (K)). A weight \(\chi: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}\) satisfies condition (K) if it is convex increasing and \[\chi(t) \underset{t \to +\infty}{\sim} t^{n/m} \left(\log(1+t)\right)^{n} \left(h \circ \log \circ \log(1+t)\right)^{n} \text{, where } \int^{+\infty} \frac{dt}{h(t)} <+\infty.\]
Example 14. We discuss and present a few typical examples:
When \(\chi(t)= \frac{t^p}{p}\) for \(p>1\), its convex conjugate is given by \(\chi^{*}(t ) = \frac{t^q}{q}\), where \(q = \frac{p}{p-1}\) is the Hölder conjugate of \(p\). The Orlicz norm \(\|\cdot\|_{\chi}\) here is the usual \(L^p\) norm.
The weight \(\chi\) defined in the Condition (K) above is a Young function hence induce an Orlicz space. In particular, the weight \(\chi(t)= t \log^p(1+t)\) with \(p>n\) is a Young function. One can check that as \(t\rightarrow +\infty\), the conjugate weight grows like \(\chi^{*}(t) = t^{1-\frac{1}{p}} \exp (t^{\frac{1}{p}})\).
We will focus on the Orlicz space \((L^{\chi},\| \cdot \|_{\chi})\) in section 3. It is important to note that in such an Orlicz space, the Hölder inequality (or Hölder–Young inequality) holds.
Proposition 15 (Hölder–Young inequality). For \(f \in L^{\chi}\) and \(g \in L^{\chi^{*}}\), we have \[\int_X |fg| dV_X \le \|f \|_{\chi} \| g\|_{\chi^{*}}.\]
Proof. See [33]. ◻
We will need the following additive Young inequality (also known as Fenchel–Young inequality).
Proposition 16. For every \(a,b \in \mathbb{R}^+\), it holds that \[ab \le \chi(a)+\chi^*(b),\] with equality if and only if \(b \in \partial \chi(a)\).
In particular, let \(f,g : X \rightarrow \mathbb{R}^+\) be two measurable functions. Then \[f \left((\chi^*)^{-1} \circ g \right)\le \chi\circ f + g.\]
Given a Lebesgue measurable function \(l: X \to \mathbb{R}\), the \(\omega\)-plurisubharmonic envelope of \(l\) is defined by \[P_{\omega} (l ) \mathrel{\vcenter{:}}= \sup\{ u \in \mathcal{PSH}(X,\omega) : u \le l\}.\] The \((\omega,m)\)-subharmonic envelope of \(l\) is defined by \[P_{\omega,m} (l) \mathrel{\vcenter{:}}= \sup\{ u \in \mathcal{SH}_m(X,\omega) : u \le l\}.\] We now summarize the basic properties of these envelopes, following [24].
Proposition 17. If \(l : X \to \mathbb{R}\) is continuous, then
The complex Monge–Ampère measure \((\omega + dd^cP_{\omega}(l))^n\) is concentrated on the contact set \(\{P_{\omega}(l)=l \}\).
The complex Hessian measure \((\omega + dd^cP_{\omega,m}(l))^m \wedge \omega^{n-m}\) is concentrated on the contact set \(\{P_{\omega,m}(l)=l \}\).
If \(l\) is \(\mathcal{C}^{1,1}\)-smooth, then \(P_{\omega}(l )\) is \(\mathcal{C}^{1,1}\)-smooth and \[(\omega + dd^cP_{\omega}(l ))^n = \mathbb{1}_{\{P_{\omega} (l)=l\} } (\omega + dd^cl)^n.\]
If \(l\) is \(\mathcal{C}^{1,1}\)-smooth, then \(P_{\omega,m}(l)\) is \(\mathcal{C}^{1,1}\)-smooth and \[(\omega + dd^cP_{\omega,m}(l))^m \wedge \omega^{n-m} = \mathbb{1}_{\{P_{\omega,m} (l)=l\} } (\omega + dd^cl)^m \wedge \omega^{n-m}.\]
For detailed proofs and additional discussion, we refer the reader to [34]–[39]. As a consequence of the preceding results, we will make use of the following statements in the sequel.
Lemma 18. Fix \(\lambda >0\). Let \(u,v \in \mathcal{SH}_m(X,\omega) \cap L^{\infty}(X)\).
If \((\omega + dd^cu)^m \wedge \omega^{n-m} \ge e^{\lambda u}fdV_X\) and \((\omega + dd^cv)^m \wedge \omega^{n-m} \ge e^{\lambda u }gdV_X\), then \[(\omega+dd^c(\max(u,v)))^m \wedge \omega^{n-m} \ge e^{\lambda \max(u,v) }\min(f,g)dV_X.\]
If \((\omega + dd^cu)^m \wedge \omega^{n-m} \le e^{\lambda u}fdV_X\), and \((\omega + dd^cv)^m \wedge \omega^{n-m}\le e^{\lambda v}gdV_X\), then \[(\omega+dd^c(P_{\omega,m}(\min(u,v)))^m \wedge \omega^{n-m} \le e^{\lambda P_{\omega,m}(\min(u,v) )}\max(f,g)dV_X.\]
The proof follows the same strategy as that for the Monge–Ampère measures presented in [37].
Proof. Let \(\psi \mathrel{\vcenter{:}}= \max(u,v)\). As a consequence of the maximum principle (Lemma 6), we have \[\begin{align} (\omega + dd^c\psi)^m \wedge \omega^{n-m} \ge & \mathbb{1}_{\{ u \ge v\} } (\omega + dd^cu)^m \wedge \omega^{n-m}+\mathbb{1}_{\{ v >u\} }(\omega + dd^cv)^m \wedge \omega^{n-m} \\ \ge & \mathbb{1}_{\{ u \ge v\} } e^{\lambda u}fdV_X +\mathbb{1}_{\{ v >u\} }e^{\lambda v}gdV_X \\ = & \mathbb{1}_{\{ u \ge v\} } e^{\lambda \psi}fdV_X +\mathbb{1}_{\{ v >u\} }e^{\lambda \psi}gdV_X \ge e^{\lambda \psi} \min(f,g) dV_X. \end{align}\] Set \(\varphi \mathrel{\vcenter{:}}= P_{\omega,m}(\min(u,v))\). We prove the second statement. It follows from the proposition 17 that the complex Hessian measure \((\omega+dd^c\varphi)^m \wedge \omega^{n-m}\) has support in \[\left\{ P_{\omega,m}(\min(u,v)) = \min(u,v)\right\} = \{ P_{\omega,m}(\min(u,v)) = u <v\} \cup \{ P_{\omega,m}(\min(u,v)) = v\}.\] We therefore obtain that \[\label{equa:maxp} (\omega + dd^c\varphi)^m \wedge \omega^{n-m} \le \mathbb{1}_{ \{ \varphi = u <v\}} (\omega + dd^c\varphi)^m \wedge \omega^{n-m} + \mathbb{1}_{\{ \varphi = v\} } (\omega + dd^c\varphi)^m \wedge \omega^{n-m}.\tag{3}\] Moreover, since \(\varphi \le u\) and \(\varphi \le v\), the maximum principle (Lemma 6) yields \[\mathbb{1}_{ \{ \varphi = u\}} (\omega + dd^c\varphi)^m \wedge \omega^{n-m} \le \mathbb{1}_{ \{ \varphi = u\}} (\omega + dd^cu)^m \wedge \omega^{n-m},\] and \[\mathbb{1}_{ \{ \varphi = v\}} (\omega + dd^c\varphi)^m \wedge \omega^{n-m} \le \mathbb{1}_{ \{ \varphi = v\}} (\omega + dd^cv)^m \wedge \omega^{n-m}.\] Together with 3 , we deduce that \[\begin{align} (\omega + dd^c\varphi)^m \wedge \omega^{n-m} \le &\mathbb{1}_{ \{ \varphi = u <v\}} (\omega + dd^cu)^m \wedge \omega^{n-m} + \mathbb{1}_{\{ \varphi = v\} } (\omega + dd^cv)^m \wedge \omega^{n-m}\\ \le& \mathbb{1}_{ \{ \varphi = u <v\}} e^{\lambda u}fdV_X + \mathbb{1}_{\{ \varphi = v\} } e^{\lambda v }g dV_X\\ \le& \mathbb{1}_{ \{ \varphi = u <v\}} e^{\lambda \varphi}fdV_X + \mathbb{1}_{\{ \varphi = v\} } e^{\lambda \varphi }g dV_X \le e^{\lambda \varphi} \max(f,g) dV_X. \end{align}\] ◻
In this section, we aim to establish a uniform \(L^{\infty}\)-estimate for solutions \(\varphi \in \mathcal{PSH}(X,\omega) \cap L^{\infty}(X)\) to the Monge–Ampère equation: \[ \label{equa:globMA} (\omega + dd^c\varphi)^n \le f d V_X,\tag{4}\] assuming \(f \in L^{\chi}\), and the associated weight function \(\chi\) satisfies Condition (K). We then generalize it to the complex Hessian equation.
For the sake of convenience, we fix \[\chi(t) = t \log^n (t+1) (h \circ\log \circ \log (t+3))^n.\] We fix a constant \(C_0 >0\) such that \(\|f \|_{\chi} \le C_0 \int_X \chi(f)dV_X\) for all \(f \in L^{\chi}\).
Let \(\Omega\) be the unit ball in \(\mathbb{C}^n\). We define \(g \mathrel{\vcenter{:}}= \frac{f }{ \| f\|_{\chi} }\) to normalize \(f\). We now consider the following Dirichlet problem to find a plurisubharmonic solution \(u\): \[ \label{equation:locMA} \begin{align} (dd^cu)^n &= g dV_X \text{, in }\Omega;\\ u &= 0 \text{, on }\partial \Omega. \end{align}\tag{5}\] We now establish a local uniform estimate for \(u\). We start by establishing a useful result, which, for Kähler manifold, was shown in [40] and [16]. We present here a local version. Its proof is simpler than that of the global case, as plurisubharmonicity is preserved under addition.
Lemma 19. Assume that \(f \in L^p(\Omega, dV_X)\) for some \(p >1\), and that \(u\) is normalized so that \(\sup_{\Omega}u =0\) and \(u=0\) on \(\partial \Omega\). Suppose that there exists a bounded function \(v \in PSH(\Omega)\) normalized so that \(\sup_{\Omega} v=0\) and \(v=0\) on \(\partial \Omega\), such that \[(dd^cu)^n \le (dd^cv)^n + fdV_X,\] then \(u \ge -C\), where \(C=C(n,p, \|f\|_p, \|v\|_{\infty})\).
Proof. Let \(\rho \in \mathcal{PSH}(\Omega)\) be the solution to the Monge–Ampère equation \[(dd^c\rho)^n =f dV_X\text{ in }\Omega; \; \rho=0 \text{ on }\partial\Omega.\] The uniform bound for \(\rho\) can be deduced from the results in [3], [14]. Consider now \(\psi = v + \rho\). It is sufficient to show that \(\psi \le u\). Indeed, \(\psi\) satisfies \[(dd^c\psi)^n \ge (dd^cv)^n + (dd^c\rho)^n = (dd^cv)^n + f dV_X \ge (dd^cu)^n.\] It thus follows from the Comparison Principle (Proposition 7) that \(u \ge \psi\) on \(\Omega\). ◻
We are now prepared to prove a local uniform estimate.
Lemma 20 (Local uniform estimate for Monge–Ampère equations). Assume that \(u\) is a bounded plurisubharmonic solution to the Dirichlet problem 5 . Then \[\mathrm{Osc}_{\Omega} u \le C,\] where \(C=C(n, \Omega)\).
Proof. Let \(v \in \mathcal{F}(\Omega)\) be such that \[(dd^cv)^n = \chi \left( g \right) dV_X \text{, in }\Omega;\; v=0 \text{, on } \partial \Omega.\] The existence of \(v\in \mathcal{F}(\Omega)\) is ensured by Theorem 11. Let \(\alpha>0\) be fixed (to be specified later). On the set \(\left\{ \log \left( g+3 \right) < - \alpha v \right\}\) we have \[(dd^cu)^n \le (e^{- \alpha v} -3 ) dV_X .\] We are going to show that \(e^{-\alpha v }\in L^2\). To see this, observe that \[\label{equ:estlocv} \int_{\Omega} (dd^cv)^n = \int_{\Omega} \chi\left( g\right) dV_X \le \chi(1).\tag{6}\] We now choose \(\alpha = \frac{1}{\chi(1) }\). It follows from [41] that \[\int_{\Omega} e^{-2 \alpha v } \le C_0(n,\Omega).\] We proceed to bound \(\mathbb{1}_{\left\{\log \left( g +3 \right) \ge - \alpha v \right\}} (dd^cu)^n\). Let \(\Theta : \mathbb{R}^- \rightarrow \mathbb{R}^-\) be a convex increasing function. Then \(\Theta \circ v\) satisfies \[\Theta \circ v \in \mathcal{PSH}(\Omega) \text{, and } ( dd^c(\Theta \circ v ))^n=\left( \Theta''(v) dv \wedge d^c v + \Theta'(v) dd^cv\right)^n \ge (\Theta'\circ v)^n(dd^cv)^n.\] On the set \(\left\{\log \left( g +3\right) \ge -\alpha v \right\}\), we deduce that \[\label{equ:Thev} ( dd^c(\Theta \circ v ))^n \ge (\Theta'\circ v)^n(dd^cv)^n \ge \left(\Theta'\left( \frac{-\log \left( g +3\right)}{\alpha}\right)\right)^n \chi \left( g \right) dV_X.\tag{7}\] We aim to construct a uniformly bounded function \(\Theta\) such that, for every \(t >1\), the following inequality is satisfied: \[\label{Formule:32Conlam} \left(\Theta'\left(\frac{-t}{\alpha}\right)\right)^n t^n (h \circ \log t)^n \ge 2 .\tag{8}\] To see this, let \(x = \log t\) and let \[b(x)= - \Theta \left(\frac{-e^x }{\alpha} \right).\] Then the condition 8 is equivalent to \(b'(x)h(x) \ge \frac{2^{1/n}}{\alpha}\), for all \(x>0\). We define \(b\) by \[b(x) \mathrel{\vcenter{:}}= \int_0^{x}\frac{2^{1/n}}{\alpha h(s)} ds.\] Since \(h^{-1}\) is integrable, the function \(b(x)\) remains bounded, which in turn implies that \(\Theta\) is uniformly bounded, as claimed. Taking \(t=\log \left( g +3\right)\) in 8 yields that \[\label{formule:Theta} \left(\Theta'\left(\frac{-\log \left(g +3\right)}{\alpha}\right)\right)^n \left(\log \left( g +3\right) \right)^n \left(h \circ \log \log \left( g +3\right) \right)^n \ge 2.\tag{9}\] Without loss of generality, we assume that there exist \(k_0 >0\) such that \[\chi(k)\ge \frac{1}{2}k\log^n (k+3) (h \circ\log \circ \log (k+3))^n, \;\forall k \ge k_0.\] From 7 and 9 , it follows that on the set \(\{\log \left( g +3 \right) \ge -\alpha v \} \cap \{ g \ge k_0\}\), \[( dd^c(\Theta \circ v ))^n \ge g dV_X = (dd^cu)^n.\] On the other hand, on the set \(\{\log \left( g +3 \right) \ge -\alpha v \} \cap \{ g < k_0\}\), we deduce that \[(dd^cu)^n < k_0 dV_X.\] From the preceding arguments, we arrive at the result that \[(dd^cu)^n \le (e^{ -\alpha v} +k_0 ) dV_X + \left( dd^c(\Theta \circ v )\right)^n.\] It follows from Lemma 19 that \(\mathrm{Osc}_{\Omega} u \le C(n, \Omega )\). ◻
We now provide a proof of the global \(L^{\infty}\) bound for the solution to the Monge–Ampère equations on Hermitian manifolds, that only relies on local solutions. The strategy is to construct a bounded subsolution by solving equations on each small ball, and then to show that the uniform boundedness of these subsolutions implies a uniform bound for the solution, using domination principle. This method is adapted from the work of Guedj–Lu, in [14]. We start by establishing a subsolution lemma.
Lemma 21 (Subsolution Lemma). Assume \(\chi\) satisfies Condition (K). Then there exist a uniform constant \(s=s(\chi)>0\) such that for all \(0\leq g\in L^{\chi}\), we can find \(\psi\in \mathcal{PSH}(X,\omega)\), \(-1\leq \psi \leq 0\), with \[(\omega +dd^c\psi)^n \geq s(\chi) \frac{g}{\|g\|_{\chi}} dV_X.\]
Proof. Cover \(X\) by balls, \(X \subset \cup_{j=1}^N B_j\). Choose \(B_j'=\{\rho_j <0 \}\), where \(\rho_j: X \rightarrow \mathbb{R}\) are smooth functions such that \(\omega+ dd^c\rho_j>0\) in a neighborhood of \(B_j'\), and such that \(B_j \Subset B_j^{'}\). In \(B_j'\), let \(v_j \in \mathcal{PSH}(B_j')\) be solution to Monge–Ampère equation \[(dd^cv_j)^n = \frac{g}{\|g\|_{\chi} } dV_X \text{ in }B_j'; \; v_j = -1 \text{ on }\partial \Omega.\] The local \(L^{\infty}\)-estimate (Lemma 20) implies \(\mathrm{Osc}_{B_j'}v_j \le C_j'\). We define \(\psi_j \mathrel{\vcenter{:}}= \max (v_j, \xi_j \rho_j)\), where \(\xi_j >1\) for all \(j\). Thus, \(\psi_j\) coincides with \(v_j\) in \(B_j\), and coincides with \(\xi_j \rho_j\) in \(X \setminus B_j'\) and in the neighborhood of \(\partial B_j'\). Moreover, \(\psi_j\) is \(\delta^{-1}\omega\)-plurisubharmonic for some uniform constant \(\delta>0\) in \(B_j\). Taking \(\psi\mathrel{\vcenter{:}}= \frac{1}{N}\sum_{j=1}^N \delta \psi_j\), and taking \(0 <s(\chi) \le \frac{\delta^n}{N^n}\), we infer that in \(B_j\), \[\omega_{\psi} ^n \ge \frac{\delta^n}{N^n}(\omega+dd^c\psi_j)^n \ge s(\chi) (dd^cv_j)^n dV_X = s(\chi) \frac{g }{\|g\|_{\chi} }dV_X.\] Therefore, \(\psi\) is the subsolution that we are looking for. ◻
Theorem 22 (A priori estimates for Monge–Ampère equations). Let \(\chi\) be as above, and let \(f \in L^{\chi}\) with \(f \ge 0\). Suppose that \(\varphi\in \mathcal{PSH}(X,\omega)\cap L^{\infty}(X)\) satisfies \[(\omega+dd^c \varphi)^n \leq fdV_X.\] Let \(s\mathrel{\vcenter{:}}= s(\chi)\) be the constant in Lemma 21 and assume that \(A>0\) satisfies \[\|\mathbb{1}_{\{f >A\} }f \|_{\chi}\leq \frac{s(\chi)}{4}.\] Then \(Osc_X(\varphi)\leq C\), where \(C\) depends on \(A\), and \(\chi\).
Proof. We consider the function \[g = \frac{\mathbb{1}_{\{f >A\} }f }{\|\mathbb{1}_{\{f >A\} }f \|_{\chi}}.\] Applying the Subsolution Lemma (Lemma 21) to \(g\) yields that there exists subsolution \(\psi \in \mathcal{PSH}(X,\omega)\), such that \[(\omega + dd^c\psi)^n \ge s g dV_X.\] The dominated convergence theorem implies that \[\lim_{t \rightarrow + \infty}\int_X \chi(\mathbb{1}_{\{ f>t\} }f ) dV_X = \int_X \lim_{t \rightarrow + \infty} \chi \left(\mathbb{1}_{\{ f>t\} }f \right) dV_X =0.\] We can thus choose \(A\) sufficiently large, depending only on \(f\) and \(s\), such that \[\| \mathbb{1}_{\{ f>A\} } f\|_{\chi} \le \frac{s}{4} .\] It follows that \[(\omega + dd^c\psi )^n \ge 4 \mathbb{1}_{\{ f>A\} } (\omega + dd^c\varphi)^n.\] Therefore, \[(\omega + dd^c\varphi)^n \le \frac{1}{4 }(\omega + dd^c\psi)^n + A dV_X.\] Fix \(B>0\) large enough. It follows from [14] that there exist \(\rho \in \mathcal{PSH}(X, \omega), -1 \le \rho \le 0\) such that \[(\omega + dd^c\rho )^n \ge 4 A \mathbb{1}_{\{\varphi < -B\} } dV_X.\] Normalize \(\varphi\) such that \(\sup_X \varphi =0\). Then on the set \(\{ \varphi < \frac{\psi + \rho}{2}-B\} \subset \{ \varphi <-B\}\), we have \[(\omega + dd^c\varphi)^n \le \frac{1}{4 } (\omega + dd^c\psi)^n + \frac{1}{4}(\omega+ dd^c\rho)^n \le \frac{1}{2} \left(\omega + dd^c\frac{\psi + \rho}{2}\right)^n.\] We conclude from the domination principle (Proposition 8) that \(\varphi \ge \frac{ \psi + \rho}{2}-B \ge -1-B\). ◻
The uniform bound for the Hessian equation follows the method of [24], which assumes \(f \in L^p\) with \(p > n/m\). A generalization to Orlicz spaces is outlined in Remark 3.2 therein. It has also been shown in the recent article [16] that this approach can be adapted in certain general complex geometric PDEs. We provide a more detailed presentation in the proof of Theorem B here.
The following lemma will be used to obtain a uniform estimate for bounded solutions.
Lemma 23. Assume \(f \in L^1(dV_X)\). Let \(\varphi \in \mathcal{SH}_m(X,\omega) \cap L^{\infty}(X)\) be such that \[(\omega + dd^c\varphi)^{m}\wedge \omega^{n-m} = f \omega^n.\] Then, we have \((\omega+ dd^cu)^n \le f^{n/m} \omega^n\), where \(u = P_{\omega}(\varphi)\).
Proof. The proof proceeds in two steps. We first treat the case where the density \(f\) is bounded. In the general case, we approximate a given \(f\in L^{1}\) by a sequence of bounded densities. Normalize the solution \(\varphi\) such that \(\sup_X \varphi=0\).
Step 1. Assume that \(f\) is bounded. Let \(\{f_j\}\) be a sequence of smooth, uniformly bounded functions such that \(f_j \to f\) in \(L^{2n}\). It follows from [22] that there exists a decreasing sequence of \((\omega,m)\)-subharmonic functions \(\{ \psi_j\}\) such that \(\psi_j \to \varphi\). We consider the following complex Hessian equation: \[(\omega + dd^c\varphi_j)^m \wedge \omega^{n-m} = e^{\varphi_j - \psi_j} f_j \omega^n.\] The existence of smooth, \((\omega,m)\)-subharmonic solution \(\varphi_j\) follows from [42]. Then, from [22] we deduce that \(\varphi_j\) converges uniformly to \(\widetilde{\varphi}\in \mathcal{SH}_m(X,\omega) \cap \mathcal{C}^0(X)\), which is the solution to \[(\omega + dd^c\widetilde{\varphi})^m \wedge \omega^{n-m} = e^{\widetilde{\varphi} - \varphi} f \omega^n.\] By the domination principle (Proposition 8), we obtain that \(\widetilde{\varphi} = \varphi\). Taking \(u_j = P_{\omega}(\varphi_j)\), it follows from Proposition 17 that \(u_j \in \mathcal{C}^{1,1}(X)\). The mixed Monge–Ampère inequality [43] implies that \[(\omega + dd^cu_j)^n \le e^{\frac{n(\varphi_j - \psi_j)}{m}} f_j^{n/m} \omega^n.\] Since \(u_j\) converges uniformly to \(u\), passing to the limit, we conclude that \[(\omega + dd^cu)^n \le f^{n/m} \omega^n.\]
Step 2. Assume now that \(f \in L^{1}(dV_X)\). We consider the equation given by \[(\omega + dd^c\varphi_j)^m \wedge \omega^{n-m} = e^{\varphi_j - \varphi} \min(f,j) \omega^n.\] Theorem 3.19 of [23] ensures the existence of a bounded solution \(\varphi_j\), satisfying \(\varphi_j \ge \varphi\). The domination principle yields that \(\varphi_j\) is decreasing to a \((\omega,m)\)-subharmonic function \(\widetilde{\varphi}\). It follows from the domination principle again that \(\widetilde{\varphi}=\varphi\). Taking \(u_j \mathrel{\vcenter{:}}= P_{\omega}(\varphi_j)\), then \(u_j\) decreases to \(u\). Moreover, Step 1 yields \[(\omega + dd^cu_j)^n \le e^{n/m(\varphi_j - \varphi)} f_j^{n/m} \omega^n.\] Applying the continuity of Monge–Ampère operator along monotone sequences [44], [45], we obtain \[(\omega + dd^cu)^n \le f^{n/m} \omega^n,\] as required. ◻
Proof of Theorem B. Normalize \(\varphi\) such that \(\sup_X \varphi=0\). Let \(u=P_{\omega} \left( \varphi\right)\) and let \(\mathcal{C}\) denote the contact set \(\mathcal{C} \mathrel{\vcenter{:}}= \{u=\varphi\}\). We remark that \(u \in \mathcal{PSH}(X, \omega) \subset \mathcal{SH}_m(X, \omega)\). Therefore, it follows from the mixed Monge–Ampère inequality [43] and proposition 17 that \[\mathbb{1}_{\mathcal{C}} (\omega +dd^cu)^m \wedge \omega^{n-m} \le \mathbb{1}_{\mathcal{C}} (\omega + dd^c\varphi)^m \wedge \omega^{n-m} \le f dV_X\] We remark that \(h_X \mathrel{\vcenter{:}}= \omega^n / d V_X \ge \delta_0\) is positive on X. Lemma 23 ensures us to take \(g \mathrel{\vcenter{:}}= \left(\omega+d d^c u\right)^n / d V_X\). Then, the mixed Monge–Ampère inequality yields that \[\delta_0^{1-m/n} g^{m/n} \le h_X^{1-m/n} g^{m/n} \le f,\] hence \(g \le C_0 f^{n/m}\), so that \(g \in L^{\chi_0}\), where \(\chi_0= t \log^n(1+t) [h \circ \log \circ \log (3+t)]^n\) satisfies Condition (K). It follows from Theorem 22 that \[\mathrm{Osc}_X(u) \le A.\] It suffices to bound \(\sup_X u\) from below. Set \[v^{-}_A(\omega) \mathrel{\vcenter{:}}= \inf \left\{ \int_X (\omega + dd^c\varphi)^n: \varphi\in L^{\infty}(X) \cap \mathcal{PSH}(X, \omega), -A \le \varphi \le 0\right\}.\] Recall that \(v_A^{-}(\omega)\) is strictly positive by [37]. We denote by \((\chi_0^{*})^{-1}\) the inverse of \(\chi_0^{*}\). Then it follows from additive Hölder–Young inequality (Proposition 16) that \[\begin{align} v^{-}_A(\omega)(\chi_0^{*})^{-1}\left(-\sup_X u\right) & \le \int_X (\chi_0^{*})^{-1}(-u)\left(\omega+d d^c u\right)^n \\ &= \int_X (\chi_0^{*})^{-1}(-u)g d V_X \\ &\le \int_X \chi_0(g) d V_X + \int_X (-u) d V_X. \end{align}\] We remark that \(\int_{X} (-u)dV_X\) is uniformly bounded from above by Lemma 5. Thus we obtain a uniform lower bound of \(\sup_X u\), which yields together with \(\mathrm{Osc}_X u \le A\) that \(|u|\) has a uniform bound, and since \(u \le \varphi\), the solution \(\varphi\) is also uniformly bounded. ◻
Theorem 24. Let \(u,v \in \mathcal{SH}_m(X,\omega)\cap L^{\infty}(X)\) be such that \(\sup_X u =0\) and \(v \le 0\). Assume that \(u \in \mathcal{SH}_m(X,\omega)\) is a solution to the complex Hessian equation \[\label{eqn:32stab} \omega_u^m \wedge \omega^{n-m}= e^{\lambda u} f \omega^n,\qquad{(2)}\] where \(\lambda\ge0\) and \(f^{n/m} \in L^{\chi}\) with \(\chi\) satisfying Condition (K). Then, \[\sup_X (v-u)_{+} \le C \| (v-u)_+f ^{n/m}\|_{\chi}^{\frac{1}{n+1}},\] where the positive constant \(C\) depends only on \(n,m\), \(\omega\), \(\|f^{\frac{n}{m} }\|_{\chi}\), and \(\sup_X|v|\).
Proof. We consider only the case \(\lambda =0\); when \(\lambda >0\), the \(L^{\infty}\)-estimate implies that \(e^{\lambda u}f \le f\), allowing the problem to be reduced to the case \(\lambda=0\). We now fix \(\epsilon>0\), to be chosen later, and define \(g_{\epsilon} \mathrel{\vcenter{:}}= \mathbb{1}_{\{u <v -\epsilon \} } f^{n/m}\).
By the subsolution lemma (Lemma 21), there exist an uniform constant \(s \mathrel{\vcenter{:}}= s(\chi)\), and \(-1 \le \psi \le 0\) such that \[(\omega+dd^c \psi)^n \geq s \frac{g_{\epsilon}dV_X}{\|g_{\epsilon}\|_{\chi} } .\] It follows from mixed Monge–Ampère inequality [43] that \[(\omega+dd^c \psi)^{m} \wedge \omega^{n-m} \geq s^{m/n} \frac{\mathbb{1}_{\{u <v - \epsilon \} } f}{\|g_{\epsilon}\|_{\chi}^{m/n}}.\] We set \(\varphi \mathrel{\vcenter{:}}= (1- \delta)v + \delta \psi\), where \(0<\delta<1\) is to be specified later. Then there exists a positive constant \(C_2\), depending only on \(\| \psi \|_{\infty}\) and \(\sup_X|v|\) such that \[\{\varphi > u + C_2\delta +\epsilon\} \subset \{u < v - \epsilon\}.\] From the \((\omega,m)\)-subharmonicity of \(v\) and \(\psi\), and applying [23], we derive that \[\begin{align} \mathbb{1}_{\{\varphi > u + C_2 \delta + \epsilon\}} \omega_{\varphi}^m \wedge \omega^{n-m} =& \mathbb{1}_{\{\varphi > u + C_2 \delta + \epsilon\}} \left[ (1-\delta)\omega_v +\delta \omega_{\psi}\right]^m \wedge \omega^{n-m} \\ =& \mathbb{1}_{\{\varphi > u + C_2 \delta + \epsilon\}} \sum_{k=0}^m \binom{m}{k} \left[(1-\delta)\omega_v\right]^k \wedge (\delta \omega_{\psi})^{m-k} \wedge \omega^{n-m}\\ \ge & \mathbb{1}_{\{ \varphi > u + C_2 \delta + \epsilon\}} \delta^{m} (\omega_{\psi})^m \wedge \omega^{n-m} . \end{align}\] If \(\|g_{\epsilon}\|_{\chi}\) is not small, then the stability property follows trivially by adjusting \(C\). Hence, we can assume that \(\|g_{\epsilon}\|_{\chi} < \frac{s}{2^{n/m}}\). Choosing \(\delta = \frac{2^{1/m} \|g_{\epsilon}\|_{\chi}^{1/n} }{s^{1/n}}\), we obtain that \[\begin{align} \mathbb{1}_{\{ \varphi > u + C_2 \delta + \epsilon\}} \delta^{m} (\omega_{\psi})^m \wedge \omega^{n-m} \ge& 2 \mathbb{1} _{\{ \varphi > u + C_2 \delta + \epsilon\}}f \omega^n \\ =& 2 \mathbb{1} _{\{ \varphi > u + C_2 \delta + \epsilon\}} \omega_u^m \wedge \omega^{n-m}. \end{align}\] It follows from domination principle (Proposition 8) that \[\varphi \le u + C_2\delta +\epsilon.\] Hence, \[\label{formule:stabes} \begin{align} (v-u )_{+}\le & C_3 \left(\frac{2^{1/m}}{s^{1/n}}\right)\|g_{\epsilon}\|_{\chi}^{1/n} + \epsilon\\ \le & C_4 \|g_{\epsilon}\|_{\chi}^{1/n} + \epsilon\\ \le & C_4 ( \epsilon)^{-1/n} \|(v-u)_{+} f^{n/m} \|^{1/n}_{\chi} +\epsilon. \end{align}\tag{10}\] Optimizing the right hand side by choosing \[\epsilon = \left(\frac{ \|C_4(v-u)_{+} f^{n/m} \|^{1/n}_{\chi}}{n} \right)^{\frac{n}{n+1}},\] we obtain \[(v-u)_{+} \le C \left( \|(v-u)_{+} f^{n/m} \|_{\chi}\right)^{\frac{1}{n+1}}.\] The conclusion follows by taking the supremum of the left-hand side. ◻
Remark 25. When \(\chi(t)= t^p/p\), with \(p >1\), the Orlicz norm \(\|\cdot\|_{\chi}\) is simply the Lebesgue \(L^p\)-norm. Assume \(f^{n/m} \in L^{p}\), for \(p>1\). Let \(\chi_q(t)= t^q/q\) for some \(1<q<p\). Then \(\psi \in \mathcal{PSH}(X,\omega)\) is a subsolution such that \[(\omega + dd^c\psi)^{n} \ge s'g \omega^n, \text{ where } g =\frac{\mathbb{1}_{\{u < v- B\epsilon \}} f^{n/m}}{\| \mathbb{1}_{\{u < v- B\epsilon \}} f^{n/m}\|_{q} } .\] Following the same procedure and applying the Chebyshev inequality to \[(v-u )_{+} \le C_4 \|(v-u )_{+} g_{\epsilon}\|_q^{1/n} +\epsilon,\]we obtain that \[(v-u )_{+} \le \frac{C'}{( \epsilon)^{\frac{p-q}{p q n}}} \|(v-u )_{+}\|^{\frac{p-q}{pqn}}_{1} \|f^{n/m}\|^{1/n}_p + \epsilon.\] Choosing an optimal \(\epsilon = C'_1 \|(v-u )_{+}\|^{\frac{p-q}{ (npq+p-q)}}_{1} \|f^{n/m}\|^{\frac{pq}{(npq+p-q)}}_p\), we deduce a stability estimate \[(v-u )_{+} \le C^{''} \|(v-u )_{+}\|_1^{\frac{p-q}{(npq+p-q)}} \left\|f^{\frac{n}{m} }\right\|^{\frac{pq}{(npq+p-q)}}_p,\] which slightly generalize the stability estimate in [22].
The following result is derived by applying the above theorem twice.
Corollary 26. Suppose that \(u,v \in \mathcal{SH}_m(X,\omega)\), normalized such that \(\sup_X u=0\), \(\sup_X v =0\), satisfy the complex Hessian equations: \[\omega_u^m \wedge \omega^{n-m} = e^{\lambda u}f \omega^n, \omega_v^m \wedge \omega^{n-m} = e^{\lambda v}g \omega^n,\] where \(0 < f^{\frac{n}{m} },g^{\frac{n}{m} } \in L^{\chi}\), and \(\lambda\ge 0\). Then, \[\| u-v\|_{\infty} \le C\left( \| (v-u)_+f^{\frac{n}{m} }\|_{\chi}^{\frac{1}{n+1}} + \| (u-v)_+g^{\frac{n}{m} }\|_{\chi}^{\frac{1}{n+1}} \right),\] where the constant \(C\) depends on \(n, m, \omega\), \(\|f^{\frac{n}{m} }\|_{\chi}\) and \(\|g^{\frac{n}{m} }\|_{\chi}\).
This subsection is devoted to the proof of Theorem C. We approximate \(f\) by positive densities \[f_j= \min (f,j).\] It follows from the main theorem of [22] that for each \(j\), there exist a continuous solution \(u_j\) and a constant \(c_j>0\) such that \[(\omega + dd^cu_j)^m \wedge \omega^{n-m} = c_j e^{\lambda u_j}f_j dV_X, \;\lambda\ge 0.\] We normalized \(u_j\) such that \(\sup_Xu_j=0\). Our first step is to bound the coefficients \(c_j\) from above and below. This requires an inequality involving mixed Hessian measures. It’s well known that if \({\varphi}_j\in \mathcal{SH}_m(X,\omega)\cap \mathcal{C}^{\infty}(X)\) and \((\omega+dd^c{\varphi}_j)^m \wedge \omega^{n-m} = f_j dV_X\), then Gårding’s inequality [28] holds: \[(\omega +dd^c{\varphi}_1)\wedge\cdots \wedge (\omega+ dd^c{\varphi}_m)\wedge \omega^{n-m} \ge (f_1\cdots f_m)^{1/m}dV_X.\] In the case where \(\varphi_j\)’s are not smooth, we can apply an approximation argument to obtain the following inequality.
Proposition 27. Let \(u \in \mathcal{SH}_m(X,\omega)\) be bounded functions such that \[(\omega + dd^c{\varphi})^m \wedge \omega^{n-m} \ge f\omega^n.\] Then we have that \((\omega + dd^c{\varphi}) \wedge \omega^{n-1} \ge f^{1/m} \omega^n\).
Proof. Let \(\{g_j\}\) be a sequence of bounded, positive and smooth functions converging to \(f\). Let \(v_j\) be a sequence of smooth, \((\omega,m)\)-subharmonic functions converging to \({\varphi}\). We consider the following Hessian equation: \[(\omega + dd^c{\varphi}_j)^m \wedge \omega^{n-m} = e^{{\varphi}_j - v_j}g_j \omega^n.\] The existence of smooth solution \({\varphi}_j\) follows from [42]. Applying Gårding’s inequality [28] yields \[(\omega + dd^c{\varphi}_j) \wedge \omega^{n-1} \ge e^{\frac{{\varphi}_j - v_j}{m}}g_j^{1/m} \omega^n.\] Moreover, we deduce from [22] that \({\varphi}_j\) converges uniformly to \(\widetilde{{\varphi}} \in \mathcal{SH}_m(X,\omega)\). Passing to the limit as \(j \to +\infty\), we obtain \[(\omega + dd^c\widetilde{u} )^m \wedge \omega^{n-m} = e^{\widetilde{{\varphi}} -{\varphi}}f \omega^n.\] The domination principle (Proposition 8) then implies \(\widetilde{{\varphi}}={\varphi}\). Consequently, we conclude that \[(\omega +dd^c{\varphi}) \wedge \omega^{n-1} \ge f^{1/m} \omega^n.\] ◻
We now turn to obtaining uniform bounds for \(c_j\).
Lemma 28. The sequence of constant \(\{ c_j\}\) is uniformly bounded. In particular, the Orlicz norm \(\| c_j f_j\|_{\chi}\) is uniformly bounded from above.
Proof. We first bound \(c_j\) from above. Fix \(\delta_0 >0\) such that \(\delta_0 \le \omega^n /dV_X\). Proposition 27 implies that for each \(j\), \[(\omega + dd^cu_j) \wedge \omega^{n-1} \ge c_j^{1/m} f_j^{1/m} \omega^n \ge \delta_0 c_j^{1/m} f_j^{1/m} dV_X.\] Thus, by taking the integral of both sides: \[\delta_0 c_j^{1/m} \int_X f_j^{1/m} dV_X \le \int_X(\omega+ dd^cu_j) \wedge \omega^{n-1}.\] Note that \(u_j \in \mathcal{SH}_m(X, \omega) \subset \mathcal{SH}_1(X, \omega)\), and the sequence \((u_j)\) is relatively compact in \(L^1\). Moreover, we have \(\int_X f_j^{1/m} \omega^n \rightarrow \int_X f^{1/m} \omega^n >0\) as \(j \rightarrow +\infty\). It follows that there exist \(j_0\) large enough such that for each \(j > j_0\), the following inequality holds: \[c_j^{1/m} \le \frac{2}{\delta_0 \int_X f^{1/m} dV_X} \int_X (\omega+ dd^cu_j)\wedge \omega^{n-1}.\] It suffices to bound the right hand side. Indeed, it follows from Stokes’ theorem that \[\int_X (\omega+ dd^cu_j)\wedge \omega^{n-1} = \int_X \omega^n + \int_X u_j dd^c(\omega^{n-1}) \le \int_X \omega^n + B\int_X |u_j |\omega^{n}.\] By the \(L^1\)-compactness result for \((\omega,1)\)-subharmonic functions (Lemma 5), we conclude that the right hand side above is bounded. Thus \(c_j\) is bounded from above.
We next show that \(\{c_j\}\) is bounded away from \(0\). Let \(g_j= f_j^{n/m}\in L^{\chi}\). By the subsolution Lemma (Lemma 21), there exists a uniform constant \(\delta>0\), such that \[(\omega+dd^cv_j)^n \ge \delta_j g_j dV_X.\] Then, it follows from mixed Monge–Ampère inequality [43] that \[\omega_{v_j}^m \wedge \omega^{n-m} \ge \delta_j^{m/n} g_j^{m/n} \omega^n \ge \delta_j^{m/n} f_j \omega^n.\] The domination principle (Proposition 8) yields that \[\delta_j^{m/n} \le c_j.\] We thus obtain a uniform lower bound for \(c_j\)’s, since \(\delta_j\)’s are strictly positive, uniform constants. The uniqueness of \(c_j\) for each \(f_j\) follows directly from domination principle (Proposition 8). ◻
Uniform bound of \(u_j\). It follows from Theorem B that \(\{ u_j\}\) have uniform bound: \[-C \le u_j \le 0.\] By extracting, we can assume that \(\{ u_j \}\) converges in \(L^1(X)\) to a bounded \((\omega,m)\)-subharmonic function \(u\), as \(j \rightarrow +\infty\).
Existence of solutions. We now show that, by construction, the limit \(u\) is a solution to the Hessian equation \[(\omega+dd^cu)^m \wedge \omega^{n-m} = ce^{\lambda u}f dV_X, \lambda \ge 0.\] We first consider the simpler case in which \(\lambda>0\).
Theorem 29. Let \(\lambda>0\). Then \(u \in \mathcal{SH}_m(X,\omega) \cap L^{\infty}(X)\) is the unique solution such that \[(\omega + dd^cu )^m \wedge \omega^{n-m} = e^{\lambda u} f dV_X.\] Moreover \(u\) is uniformly bounded.
Proof. To simplify the notation, we assume that \(\lambda=1\). Then each \(u_j\) satisfies \[(\omega+ dd^cu_j)^m \wedge \omega^{n-m} = e^{u_j} f_j dV_X.\] From the domination principle (Proposition 8), we deduce that \(u_j\) decreases to some functions \(u \in \mathcal{SH}_m(X,\omega)\). Passing to the limit, it follows from the continuity of Hessian operator along decreasing sequence ([29]) that \[(\omega+ dd^cu)^m \wedge \omega^{n-m} = e^{ u} fdV_X.\] Applying Theorem B, we obtain a uniform bound for \(u\). ◻
Monotonicity of the sequence \(u_j\) is not guaranteed when \(\lambda=0\). Our approach is to construct a supersolution and a subsolution and demonstrate that they coincide, as in [15] and [30].
Theorem 30 (Existence of bounded solution). Let \(f\), \(u\) and the constant \(c\) be as above. Then \(u \in \mathcal{SH}_m(X,\omega) \cap L^{\infty}(X)\) is a solution to \[(\omega + dd^cu)^m \wedge \omega^{n-m} = c f dV_X.\] Moreover, \(u\) is uniformly bounded.
Proof. We define \[\rho_j \mathrel{\vcenter{:}}= \left(\sup_{k \ge j} u_k \right)^*, \; v_j \mathrel{\vcenter{:}}= P_{\omega,m}\left(\inf_{k \ge j}u_k\right),\] where \(P_{\omega,m}\left(\inf_{k \ge j}u_k\right)\) denotes the largest \((\omega,m)\)-subharmonic function lying below \(\inf_{k \ge j}u_k\). Observe that \(\rho_j\) is a decreasing sequence that converges to \(u\), and \(v_j\) is an increasing sequence that converges to an \((\omega,m)\)-subharmonic function \(v\). In particular, by construction, we have \(u \ge v\). We are going to show that \(u=v\).
Set \(c_j^- f_j^- \mathrel{\vcenter{:}}= \inf_{k\ge j} c_k f_k\). Then it follows from Lemma 18 that \[(\omega + dd^c\rho_j)^m \wedge \omega^{n-m} \ge c_j^- f_j^- dV_X.\] Then we deduce from the continuity of Hessian operator along monotone sequences (see [29]) that \[\label{eqn:supersol} (\omega + dd^cu)^m \wedge \omega^{n-m} \ge c f dV_X.\tag{11}\] On the other hand, we write \((\omega+ dd^cv_j)^{m}\wedge \omega^{n-m} = e^{v_j - v_j}c_j f_j dV_X\). Define \(c_j^+ f_j^+ \mathrel{\vcenter{:}}= \sup_{k\ge j} c_k f_k\). It also follows from Lemma 18 that \[(\omega + dd^cv_j)^m \wedge\omega^{n-m} \le e^{v_j - \inf_{k \ge j}u_k} c_j^{+} f_j^+ dV_X.\] Let \(j \to +\infty\). Together with 11 we deduce that \[(\omega+ dd^cv)^m \wedge \omega^{n-m} \le e^{v- u}cf dV_X \le e^{v-u} (\omega+ dd^cu)^m \wedge \omega^{n-m} .\] By the domination principle (Proposition 8), we conclude that \(v \ge u\). Hence, \(v = u\) is a bounded solution to the complex Hessian equation: \[(\omega + dd^cu)^m \wedge \omega^{n-m} = c f dV_X.\] The uniform bound for \(u\) follows from Theorem B. ◻
Proof of Theorem C. Existence of continuous solutions. We now prove that the sequence \(\{ u_j \}\) converges uniformly to \(u\) as \(j \rightarrow +\infty\). We recall that when \(\lambda >0\), the \(L^{\infty}\)-estimate implies that \(e^{\lambda u }f \le f\), allowing us to reduce to the case where \(\lambda =0\). Fix \(j_0\) large enough. For each \(j>j_0\), we infer by Theorem 24 that\[(u_j - u)_+ \le C_1 \left\|(u_j - u)_+f^{\frac{n}{m} } \right\|_{\chi} ^{\frac{1}{n+1}}, \; (u - u_j )_+ \le C_1 \left\|(u - u_j)_+f_j^{\frac{n}{m} } \right\|_{\chi} ^{\frac{1}{n+1}}.\] We are going to show that \(\left\|(u_j - u)_+f ^{\frac{n}{m} } \right\|_{\chi} \rightarrow 0\) as \(j \rightarrow +\infty\). Fixing \(s >0\), we proceed to compute the corresponding integrability. \[\begin{align} &\int_X \chi \left( \frac{(u_j-u)_+f^{\frac{n}{m} }}{s} \right) dV_X \\= & \int_X \left( \frac{(u_j-u)_+f^{\frac{n}{m} }}{s} \right) \log^n \left( 1+\frac{(u_j-u)_+f^{\frac{n}{m} } }{s} \right) \left(h \circ \log \circ \log \left( 3+\frac{(u_j-u)_+f^{\frac{n}{m} }}{s} \right)\right)^ndV_X. \end{align}\] As \(u\) and \(u_j\)’s are uniformly bounded, it follows that \((u_j -u)+ \le C_0\) for some uniform constant \(C_0\), and since \(h\) is increasing, we deduce that \[\begin{gather} \int_X \chi \left( \frac{(u_j-u)_+f^{\frac{n}{m} }}{s} \right) dV_X \le \int_X \left( \frac{(u_j - u)_+f^{\frac{n}{m} }}{s} \right) \log^n \left( 1+\frac{C_0 f^{\frac{n}{m} }}{s} \right) \\ \cdot \left(h \circ \log \circ \log \left( 3+\frac{C_0f^{\frac{n}{m} }}{s} \right)\right)^n dV_X. \end{gather}\]
By the dominated convergence theorem, we conclude that \[\left\|\frac{(u_j - u)_+ f^{\frac{n}{m} }}{s} \right\|_{\chi} \rightarrow 0 \text{, for all } 0<s<+\infty,\] and consequently \[\|(u_j - u)_+f^{\frac{n}{m} } \|_{\chi} ^{\frac{1}{n+1}} \rightarrow 0.\] Applying the same argument, we obtain that \(\left\|(u - u_j)_+f_j^{\frac{n}{m} } \right\|_{\chi} ^{\frac{1}{n+1}} \le \left\|(u - u_j)_+f^{\frac{n}{m} } \right\|_{\chi} ^{\frac{1}{n+1}} \rightarrow 0\). Therefore, \(\{u_j\}_{j=1}^{\infty}\) is a Cauchy sequence in \(\mathcal{SH}_m(X,\omega)\cap \mathcal{C}^0(X)\). It follows that the bounded solution \(u\) to the complex Hessian equation \[(\omega + dd^cu )^m \wedge \omega^{n-m} = c e^{\lambda u} f \omega^n, \text{ where }f \in L^{\chi} ,\] is continuous.
Continuity of bounded solutions. Assume now that \(v \in \mathcal{SH}_m(X,\omega)\) is another bounded solution to the complex Hessian equation \[(\omega + dd^cv)^m \wedge \omega^{n-m} = c f dV_X.\] Then \(v\) is continuous. Indeed, the uniform bound for \(v\) ensures that \(\widetilde{f} = e^{-v}f \in L^{\chi}\). We deduce from the previous argument that there exist \(\widetilde{u} \in \mathcal{SH}_m(X,\omega)\cap \mathcal{C}^0(X)\) solving \[(\omega + dd^c\widetilde{u})^m \wedge \omega^{n-m} = e^{\widetilde{u}}\widetilde{f}dV_X.\] It follows that \[(\omega + dd^c\widetilde{u})^m \wedge \omega^{n-m} = e^{\widetilde{u}-v} f dV_X = e^{\widetilde{u}-v} (\omega + dd^cv)^m \wedge \omega^{n-m}.\] Applying the Domination Principle (Proposition 8) yields that \(v = \widetilde{u}\). Therefore, \(v \in \mathcal{C}^0(X)\), as claimed. ◻