Quasi-plurisubharmonic envelopes 1: Uniform estimates on K\"ahler manifolds

We develop a new approach to $L^{\infty}$-a priori estimates for degenerate complex Monge-Amp\`ere equations on complex manifolds. It only relies on compactness and envelopes properties of quasi-plurisubharmonic functions. Our method allows one to obtain new and efficient proofs of several fundamental results in K\"ahler geometry as we explain in this article. In a sequel we shall explain how this approach also applies to the hermitian setting producing new relative a priori bounds, as well as existence results.


Introduction
Complex Monge-Ampère equations have been one of the most powerful tools in Kähler geometry since Yau's solution to the Calabi conjecture [Yau78].A notable application is the construction of Kähler-Einstein metrics: given (X, ω) a compact Kähler manifold of complex dimension n and µ an appropriate volume form normalized by µ(X) = X ω n , one seeks for a solution ϕ : X → R to (ω + dd c ϕ) n = e −λϕ µ, where d = ∂ +∂, d c = i(∂ −∂) and λ ∈ R is a constant whose sign depends on that of c 1 (X).The metric ω ϕ := ω + dd c ϕ is then Kähler-Einstein as Ric(ω ϕ ) = λω ϕ .
When λ ≤ 0, Yau [Yau78] (see also [Aub78] when λ < 0) showed the existence of a unique (normalized) solution ϕ by establishing a priori estimates along a continuity method, the most delicate one being the uniform a priori estimate that he established by using Moser iteration process.
In recent years degenerate complex Monge-Ampère equations have been intensively studied by many authors.In relation to the Minimal Model Program, they led to the construction of singular Kähler-Einstein metrics (see [EGZ09,GZ,BBEGZ] and the references therein).The main analytical input came here from pluripotential theory which allowed Kolodziej [Kol98] to establish uniform a priori estimates when µ = f dV X has density in L p for some p > 1.
Using different methods (Gromov-Hausdorff techniques), the case λ > 0 (Yau-Tian-Donaldson conjecture) has been settled by Chen-Donaldson-Sun [CDS15,Don18].Again establishing a uniform a priori estimate in this context turned out to be the most delicate issue, a key step being obtained by Donaldson-Sun [DS14] through a refinement of Hörmander L 2 -techniques.An alternative pluripotential variational approach has been developed by Berman-Boucksom-Jonsson in [BBJ21], based on finite energy classes studied in [GZ07] and variational tools obtained in [BBGZ13].This approach has been pushed one step further by Li-Tian-Wang who have settled the case of singular Fano varieties [LTW20].
The main goal of this article is to provide yet another approach for establishing such uniform a priori estimates.While the pluripotential approach consists in measuring the Monge-Ampère capacity of sublevel sets (ϕ < −t), we directly measure the volume of the latter, avoiding delicate integration by parts.Our approach thus extends with minor modifications to the hermitian (non Kähler) setting, providing several new results that will be discussed in a companion paper [GL21]: the hermitian setting introduces several technicalities and new challenges that might affect the clarity of exposition and could scare the Kähler reader away.
In the whole article we let thus X denote a compact Kähler manifold of complex dimension n.We fix ω a closed semi-positive (1, 1)-form which is big, i.e.
We let PSH(X, ω) denote the set of ω-plurisubharmonic functions: these are functions u : X → R ∪ {−∞} which are locally given as the sum of a smooth and a plurisubharmonic function, and such that ω + dd c u ≥ 0 is a positive current.
Our first main result is a brand new proof of the following a priori estimate : Theorem A. Let ω be semi-positive and big.Let µ be a probability measure such that PSH(X, ω) ⊂ L m (µ) for some m > n.
for some uniform constant T µ = T (A m (µ)) which depends on an upper bound on Hölder inequality shows that this result covers the case when µ = f dV X is absolutely continuous with respect to Lebesgue measure, with density f belonging to L p , p > 1, or to an appropriate Orlicz class, as we explain in Section 2.2.
A crucial particular case of this estimate is due to Kolodziej [Kol98].Other important special cases have been previously obtained in [EGZ09,EGZ08,DP10].
Our new method covers all these settings at once, it also permits to recover the main estimates of [BEGZ10] (big cohomology classes) and [DnGG20] (collapsing families) as we explain in Sections 3.1 and 3.2.A slight refinement of our technique allows one to establish an important stability estimate (see Theorem 2.4).
There are several geometric situations when one can not expect the Monge-Ampère potential ϕ to be globally bounded.We next consider the equation where the density f ∈ L 1 (X) does not belong to any good Orlicz class.Since the measure µ = f dV X is non pluripolar, there exists a unique finite energy solution ϕ (see [GZ], [Din09]).It is crucial to understand its locally bounded locus.
As ω is a semi-positive and big (1, 1) form, we can find ρ an ω-psh function with analytic singularities such that ω + dd c ρ ≥ δω X is a Kähler current (see [DP04, Theorem 0.5]).For ψ quasi-psh and c > 0, we set where ν(ψ, x) denotes the Lelong number of ψ at x.A celebrated theorem of Siu ensures that for any c > 0, the set E c (ψ) is a closed analytic subset of X.
Our second main result provides the following a priori estimate, which extends a result of DiNezza-Lu [DnL17]: and ψ is a quasi-psh function.Then there exists a unique ϕ ∈ E(X, ω) such that where α, β > 0 depend on an upper bound for ||g|| L p and 1 p + 1 q = 1.Again the proof we provide is direct, and can be extended to the hermitian setting (see [GL21]).We finally show in Section 4 how the same arguments can be applied to efficiently solve the Dirichlet problem in pseudoconvex domains.
Comparison with other works.Yau's proof of his famous L ∞ -a priori estimate [Yau78] goes through a Moser iteration process.Although Yau could deal with some singularities, the method does not apply when the right hand side is too degenerate (see however [Cao85,Tos10] for further applications of Yau's method).
An important generalization of Yau's estimate has been provided by Kolodziej [Kol98] using pluripotential techniques.These have been further generalized in [EGZ09, EGZ08, DP10, BEGZ10] in order to deal with less positive or collapsing families of cohomology classes on Kähler manifolds.As this approach relies on delicate integration by parts, it is difficult to extend to the hermitian setting.
Blocki has provided a different approach in [Blo05] based on the Alexandroff-Bakelman-Pucci maximum principle and a local stability estimate due to Cheng-Yau (L 2 -case) and Kolodziej (L p -case).This has been pushed further by Szekelehydi in [Szek18].It requires the reference form ω to be strictly positive.
A PDE proof of the L ∞ -estimate has been very recently provided by Guo-Phong-Tong [GPT21] using an auxiliary Monge-Ampère equation, inspired by the recent breakthrough by Chen-Cheng on constant scalar curvature metrics [CC21].
Our approach consists in showing that the sublevel set (ϕ < −t) becomes the empty set in finite time by directly measuring its µ-size.It only uses weak compactness of normalized ω-plurisubharmonic functions and basic properties of quasi-psh envelopes, allowing us to deal with semi-positive forms.
Acknowledgements.This work has benefited from State aid managed by the ANR projects PARAPLUI and ANR-11-LABX-0040.We thank Ahmed Zeriahi for carefully reading the paper and giving numerous useful comments.

Quasi-plurisubharmonic envelopes
In the whole article we let X denote a compact Kähler manifold of complex dimension n ≥ 1.We fix ω a smooth closed real (1, 1)-form on X.
Constant functions are ω-psh functions if (and only if) ω is semi-positive.A C 2 -smooth function u has bounded Hessian, hence εu is ω-psh if 0 < ε is small enough and ω is positive.It is useful to consider as well the case when ω is not necessarily positive, in order to study big cohomology classes (see section 3.1).
The set PSH(X, ω) is a closed subset of L 1 (X), for the L 1 -topology.Subsets of ω-psh functions enjoy strong compactness and integrability properties, we mention notably the following: for any fixed r ≥ 1, • PSH(X, ω) ⊂ L r (X); the induced L r -topologies are equivalent; • PSH A (X, ω) := {u ∈ PSH(X, ω), −A ≤ sup X u ≤ 0} is compact in L r .We refer the reader to [Dem, GZ] for further basic properties of ω-psh functions.
1.1.2.Monge-Ampère measure.The complex Monge-Ampère measure (ω + dd c u) n = ω n u is well-defined for any ω-psh function u which is bounded, as follows from Bedford-Taylor theory (see [BT82] for the local theory, and [GZ] for the compact Kähler context).It also makes sense in the ample locus of a big cohomology class [BEGZ10], as we shall briefly discuss in section 3.1.
The mixed Monge-Ampère measures (ω + dd c u) j ∧ (ω + dd c v) n−j are also well defined for any 0 ≤ j ≤ n, and any bounded ω-psh functions u, v.We note for later use the following classical inequality: Lemma 1.3.Let ϕ, ψ be bounded ω-psh functions such that ϕ ≤ ψ, then Proof.To simplify notations we just treat the case j = n.It follows from Bedford-Taylor theory [BT82] that for any bounded ω-psh functions ϕ, ψ, We shall also need the following (see [GZ,Proposition 10.11]): Proposition 1.4.[Domination principle] If u, v are bounded ω-psh functions such that u ≥ v a.e. with respect to ω n u .Then u ≥ v. 1.2.Envelopes.Upper envelopes of (pluri)subharmonic functions are classical objects in Potential Theory.They were considered by Bedford and Taylor to solve the Dirichlet problem for the complex Monge-Ampère equation in strictly pseudo-convex domains [BT76].We consider here envelopes of ω-psh functions.
Definition 1.5.Given a Lebesgue measurable function h : X → R, we define the ω-psh envelope of h by where the star means that we take the upper semi-continuous regularization.
The following is a combination of [GLZ19, Propositions 2.2 and 2.5, Lemma 2.3]: Proposition 1.6.If h is bounded from below and quasi-continuous, then • P ω (h) is a bounded ω-plurisubharmonic function; • P ω (h) ≤ h in X \ P , where P is pluripolar; Recall that a function h is quasi-continuous if for any ε > 0, there exists an open set G of capacity smaller than ε such that h is continuous in X \ G. Quasi-psh functions are quasi-continuous (see [BT82]), as well as differences of the latter: we shall use this fact during the proof of Theorem 3.3. When ] and one can further has

1.2.2.
A key lemma.The following is a key technical tool to our new approach: Lemma 1.7.Fix χ : R − → R − a concave increasing function such that χ ′ (0) ≥ 1.
When these functions are less regular we take a different route.We set

Global L ∞ bounds
In this section we prove Theorem A, as well as a stability estimate.
Theorem 2.1.Let ω be semi-positive and big.Let µ be a probability measure such that PSH(X, ω) ⊂ L m (µ) for some m > n.
for some uniform constant T µ = T (A m (µ)) which depends on an upper bound on Let us stress that this result is not new: it can be derived from the celebrated a priori estimate of Kolodziej [Kol98], together with its extensions [EGZ09, EGZ08, DP10].We provide here an elementary proof that does not use the theory of Monge-Ampère capacities, and merely relies on the compactness properties of sup-normalized ω-psh functions and Lemma 1.7.
Proof.Shifting by an additive constant, we normalize ϕ by sup X ϕ = 0. Set Our goal is to establish a precise bound on T max .By definition, −T max ≤ ϕ almost everywhere with respect to µ, hence everywhere by the domination principle (Proposition 1.4), providing the desired a priori bound Osc X (ϕ) ≤ T max .
Controlling the norms ||u|| L m .We are going to choose below the weight χ in such a way that X (χ ′ • ϕ) n+2ε dµ = B ≤ 2 is a finite constant under control.This provides a uniform lower bound on sup X u as we now explain: indeed 0 We infer that u belongs to a compact set of ω-psh functions, hence its norm ||u|| L m (µ) is under control with Choice of χ.Lebesgue's formula ensures that if g : R + → R + is an increasing function such that g(0) = 1, then we define χ by imposing χ(0) = 0, χ ′ (0) = 1, and This choice guarantees that χ : R − → R − is concave increasing with χ ′ ≥ 1, and
2.2.Absolutely continuous measures.Assume µ = f dV X is absolutely continuous with respect to a volume form dV X , with density 0 ≤ f ∈ L p (dV X ) for some p > 1.Since PSH(X, ω) ⊂ L r (dV X ) for any 1 ≤ r < +∞, we obtain , for all u ∈ PSH(X, ω), where 1/p + 1/q = 1, so that PSH(X, ω) ⊂ L m (dµ) for all m ≥ 1.Thus Theorem 2.1 applies to this type of measures, providing a new proof of the celebrated a priori estimate of Kolodziej [Kol98] (see also [EGZ09]).
As in [Kol98] our technique also covers the case of more general densities as we briefly indicate.Let w : R + → R + be a convex increasing weight.A measurable function f belongs to the Orlicz class L w (dV X ) if there exists α > 0 such that The Luxembourg norm of f is defined as it turns L w (dV X ) into a Banach space.
If w * denotes the conjugate convex weight of w (its Legendre transform), Hölder-Young inequality ensures that for all measurable functions f, g, We refer the reader to [RR] for more information on Orlicz classes.
Theorem 2.1 thus allows to reprove [Kol98, Theorem 2.5.2]: Corollary 2.2.Let µ = f dV X be a probability measure.Let w : R + → R + be a convex increasing weight that grows at infinity at least like t(log t) m with m > n.
If f belongs to the Orlicz class L w then any solution Proof.While this was not required for the case of L p densities, we need here to invoke Skoda's uniform integrability result (see [GZ,Theorem 8.11]): there exists α > 0 and The reader will check that, as s → +∞, the conjugate weight w * (s) grows like It follows therefore from Young inequality that any ω-psh function u satisfies Thus PSH(X, ω) ⊂ L m (µ) and the conclusion follows from Theorem 2.1.
One can slightly improve the assumption on the density as in [Kol98, Theorem 2.5.2],we leave the technical details to the interested reader.

Stability estimate.
We now establish the following stability estimate, which can be seen as a refinement of [GZ12, Proposition 5.2].
Controlling the norms ||u|| L m .We choose χ below s.t.X (χ ′ • (ϕ − φ)) n+2c dµ ≤ B is under control.This provides a uniform lower bound on sup X u.Indeed our normalizations yield Thus u belongs to a compact set of ω-psh functions: its norm ||u|| L q (µ) is under control for any q ≤ m.
It follows therefore from Chebyshev inequality that Choice of χ.Fix T 0 ∈ (0, T max ).We set g(t) = [χ ′ (−t)] n+2c and define χ by imposing χ(0) = 0, χ ′ (0) = 1, and This choice guarantees that It follows from Theorem 2.1 that T max ≤ T µ is uniformly bounded from above, hence B := 1 + T µ is under control.Together with (2.3) and (2.4) we thus obtain , Multiplying by h ′ , integrating between 0 and t, we infer that for all t ∈ [0, T 0 ], Recall that we have set m = n + ε so that Raising both sides of (2.6) to power 1/β we obtain We integrate between 0 and T 0 and make the change of variables we obtain the desired estimate.

Refinements and extensions
We explain now how minor modifications of the proof of Theorem 2.1 provide other important uniform estimates in various contexts of Kähler geometry.
3.1.Big cohomology classes.Let θ be a smooth closed (1, 1)-form that represents a big cohomology class α.We set The latter is a θ-psh function with minimal singularities, i.e. any other θ-psh function ϕ satisfies ϕ ≤ V θ + C for some constant C. It is locally bounded in the ample locus Amp(α), a Zariski open subset of X where the cohomology class α behaves like a Kähler class.
The Monge-Ampère measure (θ + dd c ϕ) n of a θ-psh function with minimal singularities is well defined in Amp(α), and one can show that it has finite mass independent from ϕ and equal to the volume of the class α.
We refer the reader to [BEGZ10] for more details on these notions and focus here on slightly extending [BEGZ10, Theorem B] by our new approach: Theorem 3.1.Let µ be a probability measure on X.If PSH(X, θ) ⊂ L m (µ) for some m > n, then there exists a unique ϕ ∈ PSH(X, θ) with minimal singularities such that for some uniform constant T µ .
Proof.It follows from [BEGZ10, Theorem A] that there exists a unique finite energy solution ϕ.The key point for us here is to establish the a priori estimate.Note that ϕ ≤ V θ since sup X ϕ = 0 Our goal is to show that V θ − T max ≤ ϕ, obtaining a uniform upper bound on T max .A difficulty lies in the fact that θ is not a positive form.We consider the positive current ω = θ + dd c V θ and set φ = ϕ − V θ ≤ 0. Observe that Our plan is thus to show that the "ω-psh" function φ is bounded.
As in the proof of Theorem 2.1 we let χ : R − → R − denote a concave increasing function such that χ(0) = 0 and χ ′ (0) = 1.We set ψ = V θ + χ • φ and consider The envelopes in the context of big cohomology classes enjoy similar properties as those reviewed in Section 1.2.In particular the complex Monge-Ampère measure (θ + dd c P θ (ψ)) n is concentrated on the contact set {P θ (ψ) = ψ} (see [GLZ19, Theorem 2.7]) and the big-version of Lemma 1.7 holds, showing that The rest of the proof is identical to that of Theorem 2.1.

Degenerating families.
Families of Kähler-Einstein varieties have been intensively studied in the past decade, requiring one to analyze the associated family of complex Monge-Ampère equations.We refer the reader to [Tos09, Tos10, ST12, GTZ13, DnGG20, Li20] for detailed examples and geometrical motivations.
The most delicate situation is when the volume of the fiber collapses.Theorem 2.1 yields a uniform bound in this case, providing an alternative proof and an extension of the main results of [EGZ08, DP10]: Corollary 3.2.Let ω t be a family of semi-positive and big forms on X, and assume there is a fixed form Θ such that 0 ≤ ω t ≤ Θ.We let V t := X ω n t > 0 denote the volume of (X, ω t ).Let µ be a probability measure.If PSH(X, Θ) ⊂ L m (µ) for some m > n, then any solution ϕ The point here is that the estimate is uniform in t although the volumes V t may degenerate to zero (volume collapsing).
This uniform estimate shows in particular that in many geometrical contexts, a uniform control on the L n+ε -norm of the Monge-Ampère potentials ϕ t suffices to obtain a L ∞ -control of the latter.
One can obtain similarly uniform estimates when the underlying complex structure is also changing: let X be an irreducible and reduced complex Kähler space, and let π : X → D denote a proper, surjective holomorphic map such that each fiber X t = π −1 (t) is an n-dimensional, reduced, irreducible, compact Kähler space, for any t ∈ D. Given ω a Kähler form on X and ω t := ω |Xt , one can consider the complex Monge-Ampère equations 1 V (ω t + dd c ϕ t ) n = µ t , where • the volume V = Xt ω n t turns out to be independent of t, and • µ t is a family of probability measures on each fiber X t (e.g. the normalized Calabi-Yau measures of a degenerating family of Calabi-Yau manifolds).In many concrete geometrical situations (see e.g.[GTZ13,DnGG20,Li20]), one can check that A m (ω t , µ t ) ≤ A is uniformly bounded from above for some m > n (often any m > 1).If one can further uniformly compare sup Xt ϕ t and Xt ϕ t ω n t V , then Theorem 2.1 then applies and provides a uniform L ∞ -estimate.It is thus sometimes not necessary to establish a uniform Skoda integrability theorem in families (compare with [DnGG20,Li20b]).

3.3.
Relative a priori L ∞ -bounds.Fix ω a semi-positive and big (1, 1) form, and ρ an ω-psh function with analytic singularities such that ω + dd c ρ ≥ δω X is a Kähler current which is smooth in the ample locus Amp(ω).We normalize ρ so that sup X ρ = 0 and set V = X ω n > 0.
We consider in this section the degenerate complex Monge-Ampère equation where µ is a probability measure whose density f ∈ L 1 (X) does not belong to any good Orlicz class (see section 2.2).Since µ does not charge pluripolar sets, there exists a unique "finite energy solution" ϕ ∈ E(X, ω) (see [GZ]), but one cannot expect anylonger that ϕ is globally bounded on X.
Given ψ a quasi-plurisubharmonic function on X and c > 0, we set the Legendre transform of w.Thus χ 1 : R − → R − is a convex increasing weight such that χ 1 (−∞) = −∞ and as follows from the additive version of Hölder-Young inequality and the compactness of sup-normalized ω-psh functions.
It follows that ϕ belongs to a compact subset of the finite energy class E χ 1 (X, ω), hence for all λ ∈ R, for some C λ independent of ϕ (see [GZ07,GZ] for more information).
The envelope construction.Let u = P (2ϕ − ψ) denote the greatest ω-psh function that lies below 2ϕ − ψ.Since h = 2ϕ − ψ is bounded from below and quasi-continuous, it follows from Proposition 1.6 that the measure (ω + dd c u) n is supported on the contact set C = {u = 2ϕ − ψ}.Thus Since v ≤ w on X, it follows from Lemma 1.3 that where • v = u + ψ is 2ω-psh and u + ψ ≤ 2ϕ = w on X; • {u + ψ = 2ϕ} coincides with the contact set C.
Therefore, it follows from (3.3) that using that sup X u ≤ c 2 is uniformly bounded from above, as we explain below.It follows from Hölder inequality and (3.2) that the measure ge −2ϕ/a dV X satisfies the assumption of Theorem 2.1.We infer that u ≥ −M is uniformly bounded below, hence The desired a priori estimate follows with β = M/2 and α = max(1, δ/2K).
Bounding sup X u from above.We can assume without loss of generality that sup X ψ = 0. Consider G = { ψ > −1}; this is a non empty plurifine open set.
It follows from [GZ, Theorem 9.17

The local context
4.1.Cegrell classes.We fix Ω ⊂ C n a bounded hyperconvex domain, i.e. there exists a continuous plurisubharmonic function ρ : Ω → [−1, 0) whose sublevel sets {ρ < −c} ⋐ Ω are relatively compact for all c > 0. Let T (Ω) denote the set of bounded plurisubharmonic functions u in Ω such that lim z→ζ u(z) = 0, for every ζ ∈ ∂Ω, and Ω (dd c u) n < +∞.Cegrell [Ceg98,Ceg04] has studied the complex Monge-Ampère operator (dd c •) n and introduced different classes of plurisubharmonic functions on which the latter is well defined: • DMA(Ω) is the set of psh functions u such that for all z 0 ∈ Ω, there exists a neighborhood V z 0 of z 0 and u j ∈ T (Ω) a decreasing sequence which converges to u in V z 0 and satisfies sup j Ω (dd c u j ) n < +∞.• a function u belongs to F(Ω) iff there exists u j ∈ T (Ω) a sequence decreasing towards u in all of Ω, which satisfies sup j Ω (dd c u j ) n < +∞; • a function u belongs to E p (Ω) if there exists a sequence of functions u j ∈ T (Ω) decreasing towards u in Ω with sup j Ω (−u j ) p (dd c u j ) n < +∞.• a function u belongs to F p (Ω) if there exists a sequence of functions u j ∈ T (Ω) decreasing towards u in Ω with sup j Ω [1 + (−u j ) p ](dd c u j ) n < +∞.
Given u ∈ E p (Ω) we define the weighted energy of u by The operator (dd c •) n is well defined on these sets, and continuous under decreasing limits.If u ∈ E p (Ω) for some p > 0 then (dd c u) n vanishes on all pluripolar sets [BGZ09, Theorem 2.1].If u ∈ E p (Ω) and Ω (dd c u) n < +∞ then u ∈ F p (Ω).Also, note that Cegrell has characterized the range of the complex Monge-Ampère operator acting on the classes E p (Ω): Theorem 4.1.[Ceg98, Theorem 5.1] Let µ be a probability measure in Ω.There exists a function u ∈ F p (Ω) such that (dd c u) n = µ if and only if F p (Ω) ⊂ L p (Ω).
A simplified variational proof of this result has been provided in [ACC12].Proof.We first explain why the integrability condition F(Ω) ⊂ L m (Ω, µ) is equivalent to the finiteness of A m .Indeed, if A m is not finite then there exists a sequence (u j ) in T (Ω) such that Ω (dd c u j ) n ≤ 1 but Ω |u j | m dµ ≥ 4 jm .Let u := +∞ j=1 2 −j u j .Then, by [Ceg04, Corollary 5.6], we have u ∈ F(Ω), but It follows from Theorem 4.1 that there exists ϕ ∈ F(Ω) such that (dd c ϕ) n = µ.We assume for the moment that u ∈ T is bounded and we establish a uniform bound for ϕ.Set T max := sup{t > 0 : µ(ϕ < −t) > 0}.Our goal is to establish a precise bound on T max .By definition, −T max ≤ ϕ almost everywhere with respect to µ, hence (dd c max(ϕ, −T max )) n ≥ (dd c ϕ) n and the domination principle, [GZ, Corollary 3.31], gives ϕ ≥ −T max , providing the desired a priori bound |ϕ| ≤ T max .
Although the function ψ is not psh, this provides a bound from above on the positivity of dd c ψ which allows to control the Monge-Ampère of its envelope, see [DnGL20, Lemma 4.1 and Lemma 4.2], (dd c u) n ≤ 1 {u=ψ} (dd c ψ) n ≤ (χ ′ • ϕ) n µ.
The above inequalities hold for smooth functions and the general case of bounded psh functions can be obtained as in the proof of Lemma 1.7.
We thus get a uniform control on the Monge-Ampère mass of u: We are going to choose below the weight χ in such a way that Ω (χ ′ • ϕ) n dµ = B ≤ 2 is a finite constant under control.This provides a uniform upper bound on ||u|| L m (µ) .Using Chebyshev inequality we thus obtain Setting g(t) = [χ ′ (−t)] n we define χ by imposing χ(0) = 0, χ ′ (0) = 1, and This choice guarantees that

Conclusion.
We set h(t) = −χ(−t) and work with the positive counterpart of χ.Note that h(0) = 0 and h ′ (t) = [g(t)] 1 n is positive increasing, hence h is convex increasing (so χ is concave increasing and negative).
We integrate this inequality as in the proof of Theorem 2.1 and obtain for some uniform constant C ′ depending on n, m, A m .