The Hölder continuous subsolution theorem for complex Hessian equations

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The Hölder continuous subsolution theorem for complex Hessian equations Tome 7 (2020), p. 981-1007.

<http://jep.centre-mersenne.org/item/JEP_2020__7__981_0>

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THE HÖLDER CONTINUOUS SUBSOLUTION THEOREM FOR COMPLEX HESSIAN EQUATIONS

by Amel Benali & Ahmed Zeriahi

Abstract. — LetΩbCⁿbe a bounded stronglym-pseudoconvex domain (16m6n) andµ a positive Borel measure with finite mass onΩ. We solve the Hölder continuous subsolution problem for the complex Hessian equation(dd^cu)^m∧β^n−m=µonΩ. Namely, we show that this equation admits a unique Hölder continuous solution onΩwith given Hölder continuous boundary values if it admits a Hölder continuous subsolution onΩ. The main step in solving the problem is to establish a new capacity estimate showing that them-Hessian measure of a Hölder continuousm-subharmonic function on Ωwith zero boundary values is dominated by them-Hessian capacity with respect toΩwith an (explicit) exponentτ >1.

Résumé(Le théorème des sous-solutions Hölder continues pour les équations hessiennes com- plexes)

SoitΩbCⁿun domaine borné fortementm-pseudoconvexe (16m6n) etµune mesure de Borel positive de masse finie surΩ. Nous démontrons que l’équation hessienne complexe (dd^cu)^m∧β^n−m =µsurΩadmet une solution Hölder continue surΩ pour une donnée au bord Hölder continue si (et seulement si) elle admet une sous-solution Hölder continue surΩ.

L’étape principale dans la résolution du problème consiste à établir une nouvelle estimation capacitaire, qui montre que la mesurem-hessienne complexe d’une fonctionm-sous-harmonique Hölder continue surΩavec valeur au bord nulle est dominée par la capacitém-hessienne par rapport àΩavec un exposant expliciteτ >1.

Contents

1. Introduction. . . 982

2. Preliminary results. . . 985

3. Subharmonic envelopes and obstacle problems. . . 994

4. Hessian measures of Hölder continuous potentials. . . 998

5. Proofs of the main results. . . .1001

References. . . .1006

2020Mathematics Subject Classification. — 31C45, 32U15, 32U40, 32W20, 35J96.

Keywords. — Complex Monge-Ampère equations, complex Hessian equations, Dirichlet problem, obstacle problems, maximal subextension, capacity.

The second author was partially supported by the ANR project GRACK.

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1. Introduction

Complex Hessian equations are important examples of fully non-linear PDE’s of second order on complex manifolds. They interpolate between (linear) complex Poisson equations (m = 1) and (non linear) complex Monge-Ampère equations (m=n). They appear in many geometric problems, including theJ-flow [SW08] and quaternionic geometry [AV10]. They have attracted the attention of many researchers these last years as we will mention below.

1.1. Statement of the problem. — Let Ω b Cⁿ be a bounded domain and let m be a fixed integer such that1 6m6n. We consider the following general Dirichlet problem for the complexm-Hessian equation.

The Dirichlet problem. — Let g ∈ C⁰(∂Ω) be a continuous function (the boundary data) and µ be a positive Borel measure on Ω (the right hand side). The problem is to find a necessary and sufficient condition on µ such that the following problem admits a solution:

(1.1)











U ∈ SH_m(Ω)∩ C⁰(Ω)

(dd^cU)^m∧β^n−m=µ onΩ (†)

U_|∂Ω=g on∂Ω (††)

The equation (†) must be understood in the sense of currents on Ω as it will be explained in Section 2. The equality (††) means that lim_z→ζU(z) = g(ζ) for any pointζ∈∂Ω.

Recall that for a real functionu∈ C²(Ω) and each integer 16k6n, we denote byσ_k(u)the continuous function defined at each pointz∈Ωas thek-th symmetric polynomial of the eigenvaluesλ(z) := (λ1(z), . . . λn(z))of the complex Hessian matrix ∂²u

∂zj∂zk(z)

ofui.e.,

σk(u)(z) := X

16j1<···<jk6n

λj₁(z)· · ·λj_k(z), z∈Ω.

We say that a real functionu ∈ C²(Ω) is m-subharmonic on Ωif for any k such that16k6m, we haveσk(u)>0pointwise onΩ.

Form= 1,σ₁(u) = (1/4)∆uand for m=n, σ_n(u) = det

∂²u

∂zj∂zk(z)

. Therefore, 1-subharmonic means subharmonic andn-subharmonic means plurisubharmonic.

As observed by Z. Błocki ([Bło05]), it is possible to define a general notion of m- subharmonic functions using the theory ofm-positive currents (see Section 2). More- over, it is possible to define the k-Hessian measure (dd^cu)^k∧β^n−k when16k6m for any (locally) boundedm-subharmonic functionuonΩ(see Section 2).

Whenµ= 0, the Dirichlet problem (1.1) can be solved using the Perron method as for the complex Monge-Ampère equation (see [Bło05], [Cha16a]).

Wheng= 0andµis a positive Borel measure onΩ, the Dirichlet problem is much more difficult. A necessary condition for the existence of a solution to (1.1) is the existence of a subsolution.

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Therefore, a particular case of the Dirichlet problem (1.1) we are interested in can be formulated as follows.

The Hölder continuous subsolution problem. — Letµbe a positive Borel measure onΩ.

Assume that there exists a function ϕ ∈ SHm(Ω)∩ C^α(Ω) satisfying the following condition:

(1.2) µ6(dd^cϕ)^m∧β^n−monΩ, and ϕ_|∂Ω= 0.

(1) Does the Dirichlet problem (1.1) admit a Hölder continuous solution U_µ,g for any boundary datagwhich is Hölder continuous on∂Ω?

(2) In this case, is it possible to estimate precisely the Hölder exponent of the solutionUµ,g in terms of the Hölder exponents ofϕandg?

Our goal in this paper is to answer the first question on the existence of a Hölder continuous solution and give an explicit lower bound of the Hölder exponent of the solution in terms of the Hölder exponent of the subsolution when the measureµhas finite total mass.

1.2. Known results. — There have been many articles on the subject. We will only mention those that are relevant to our study and closely related to our work. The terminology used below will be defined in the next section.

Assume thatΩis a smooth stronglym-pseudoconvex domain. When the boundary datagis smooth and the right hand sideµ=f λ_2nis a measure with a smooth positive densityf >0, S. Y. Li proved in [Li04] that the problem has a unique smooth solution. Later, Z. Błocki introduced the notion of weak solution and solved the Dirichlet problem for the homogeneous Hessian equation in the unit ball inCⁿ([Bło05]). When the density06f ∈L^p(Ω) withp > n/m, Dinew and Kołodziej proved the existence of a continuous solution ([DK14]). Assuming moreover that g is Hölder continuous onΩ, Ngoc Cuong Nguyen proved the Hölder continuity of the solution under an additional assumption on the densityf ([Ngu14]) and M. Charabati proved the Hölder continuity of the solution for general densities ([Cha16b]).

On the other hand, S. Kołodziej [Koł05] proved that the Dirichlet problem has a bounded plurisubharmonic solution if (and only if) it has a bounded subsolution with zero boundary values. This is known as the bounded subsolution theorem for plurisubharmonic functions. The same result was proved for the Hessian equation by Ngoc Cuong Nguyen in [Ngu12].

The Hölder continuous subsolution problem stated above has attracted a lot of attention these last years and was formulated in [DGZ16] for the complex Monge- Ampère equation.

It has been solved for the complex Monge-Ampère by Ngoc Cuong Nguyen in [Ngu18, Ngu20]. Recently S. Kołodziej and Ngoc Cuong Nguyen solved the Hölder subsolution problem for the Hessian equation under the restrictive assumption that the measureµis compactly supported onΩ(see [KN20b], [KN20a]).

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1.3. Main new results. — In this paper we will solve the Hölder continuous subsolution problem for Hessian equations whenµis any positive Borel measure with finite mass on Ω.

Our first main result gives a new comparison inequality which will be applied to positive Borel measures without restriction on their support.

TheoremA. — Let ΩbCⁿ be a bounded strongly m-pseudoconvex domain. Let ϕ∈ SHm(Ω)∩ C^α(Ω) with 0< α 61 such that ϕ= 0 in ∂Ω. Then for any r such that 0< r < m/(n−m), there exists a constantA >0such that for every compactK⊂Ω,

Z

K

(dd^cϕ)^m∧β^n−m6A

[Cap_m(K,Ω)]^1+ε+ [Cap_m(K,Ω)]^1+mε ,

whereε:=αr/((2−α)m+α)>0.

The capacity Cap_m(K,Ω) will be defined in the next section. The constantA in the theorem is explicit (see Formula (5.13)). Observe that the most relevant case in the application of this inequality will be whenCap_m(K,Ω)61. In this case the right exponent isτ = 1 +αr/((2−α)m+α).

Theorem A substantially improves a recent result of [KN20a] who proved an estimate of this kind when the compact set K ⊂ Ω⁰ is contained in a fixed open set Ω⁰bΩ, i.e.,K stays away from the boundary ofΩ.

When m = n a better estimate was obtained in [Ngu18] using the exponential integrability of plurisubharmonic functions which fails whenm < n.

As a consequence of Theorem A, we will deduce the following result which solves the Hölder continuous subsolution problem.

TheoremB. — Let ΩbCⁿ be a bounded strongly m-pseudoconvex domain andµ a positive Borel measure on Ωwith finite mass. Assume that there exists ϕ∈ E_m⁰(Ω)∩ C^α(Ω) with 0< α <1 such that

(1.3) µ6(dd^cϕ)^m∧β^n−m weakly onΩ, and ϕ≡0on ∂Ω.

Then for any continuous function g ∈ C^2α(∂Ω), there exists a unique function U = U_g,µ∈ SH_m(Ω)∩ C⁰(Ω) such that

(dd^cU)^m∧β^n−m=µ, and U =g on ∂Ω.

Moreover,U ∈ C^α⁰(Ω) for anyα⁰ such that 0< α⁰<(α^m/2^m−1)·γ(m, n, α), where

γ(m, n, α) := mα

m(m+ 1)α+ 2(n−m)·

Recall that by definition when α = 1/2, g ∈ C¹(∂Ω) means that g is Lipschitz and when 1/2 < α <1 and 2α = 1 +θ with 0 < θ <1, g ∈ C^2α(∂Ω) means that g∈ C¹(∂Ω)and and∇g is Hölder continuous of exponentθon∂Ω.

Let us give a rough idea of the proofs of these results.

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Idea of the proof of TheoremA. — The general idea of the proof is inspired by [KN20a].

However, since our measure is not compactly supported nor of finite mass, we need to control the behaviour of them-Hessian measure ofϕclose to the boundary. This will be done in several steps in Section 3 and Section 4.

– The first step is to estimate the mass of the m-Hessian measure σm(ϕ) of a Hölder continuousm-subharmonic functionϕin terms of its regularizationϕ_δ on any compact set inΩδ. This requires to consider them-subharmonic envelope ofϕδ onΩ and provide a precise control on itsm-Hessian measure (see Theorem 3.3).

– The second step is to estimate the mass ofσ_m(ϕ)on a compact set close to the boundary in terms of its Hausdorff distance to the boundary (see Lemma 4.1).

Idea of the proof of TheoremB. — The proof will be in two steps.

– The first step relies on a standard method which goes back to [EGZ09] (see also [GKZ08]) in the case of the complex Monge-Ampère equation. This method consists in proving a semi-stability inequality estimating sup_Ω(v−u)+ in terms of k(v−u)₊kL¹(Ω,µ), where uis the bounded m-subharmonic solution to the Dirichlet problem (1.1) andvis any boundedm-subharmonic function with the same boundary values asu, under the assumption that the measureµis dominated by them-Hessian capacity with an exponentτ >1 (see Definition 2.18).

– The second step uses an idea which goes back to [DDG⁺14] in the setting of compact Kähler manifolds (see also [GZ17]). It has been also used in the local setting in [Ngu18] and [KN20a]. It consists in estimating theL¹(µ)-norm ofv−uin terms of theL¹(λ_2n)-norm of(v−u)whereuis the bounded solution to the Dirichlet problem andvis a boundedm-subharmonic function onΩclose to the regularizationu_δ ofu.

This step requires that the measure µis well dominated by the m-Hessian capacity, which is precisely the content of our Theorem A. Then using the Poisson-Jensen formula as in [GKZ08], we see that theL¹-norm of(u_δ−u)isO(δ²)(see Lemma 2.3) and Lemma 2.5 allows us to finish the proof.

Acknowledgements. — The authors are indebted to Hoang Chinh Lu for his very careful reading of the first version of this paper and for valuable comments that helped to correct some errors and to improve the presentation of the paper. They also would like to thank Vincent Guedj for interesting discussions and Ngoc Cuong Nguyen for useful exchanges about his earlier work on this subject.

This project started when the first author was visiting the Institute of Mathematics of Toulouse (IMT) during the springs 2017 and 2018. She would like to thank IMT for the invitation and for providing excellent research conditions.

The authors thank the referees for their useful comments.

2. Preliminary results

In this section, we recall the basic properties ofm-subharmonic functions and some results we will use throughout the paper.

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2.1. Hessian potentials. — For a hermitiann×nmatrix a= (a_j,k) with complex coefficients, we denote by λ1, . . . λn the eigenvalues of the matrix a. For anyk such that16k6nwe define thek-th trace ofaby the formula

sk(a) := X

16j₁<···<jk6n

λj₁· · ·λj_k,

which is the k-th elementary symmetric polynomial of the eigenvalues (λ1, . . . , λn) ofa.

Let Cⁿ(1,1) be the space of real (1,1)-forms on Cⁿ with constant coefficients, and define the cone ofm-positive(1,1)-forms onCⁿ by

(2.1) Θm:={θ∈Cⁿ(1,1);θ∧βⁿ⁻¹>0, . . . , θ^m∧β^n−m>0}.

Definition2.1

(1) A smooth (1,1)-form θ on Ω is said to be m-positive on Ωif for any z ∈ Ω, θ(z)∈Θ_m.

(2) A function u : Ω → R∪ {−∞} is said to be m-subharmonic on Ω if it is subharmonic onΩ(not identically−∞on any component) and for any collection of smoothm-positive(1,1)−formsθ₁, . . . , θ_m−1 onΩ, the following inequality

dd^cu∧θ1∧ · · · ∧θm−1∧β^n−m>0, holds in the sense of currents onΩ.

We denote bySHm(Ω)the positive convex cone ofm-subharmonic functions onΩ.

We give below the most basic properties ofm-subharmonic functions that will be used in the sequel.

Proposition2.2

(1) Ifu∈ C²(Ω), thenuism-subharmonic onΩif and only if (dd^cu)^k∧β^n−k >0 pointwise on Ωfork= 1, . . . , m.

(2) PSH(Ω) =SH_n(Ω)(SH_n−1(Ω)(· · ·(SH₁(Ω) =SH(Ω).

(3) SHm(Ω)⊂L¹_loc(Ω) is a positive convex cone.

(4) Ifuism-subharmonic onΩandf :I→Ris a convex, increasing function on some interval containing the image ofu, then f◦uism-subharmonic onΩ.

(5) The limit of a decreasing sequence of functions inSHm(Ω) ism-subharmonic onΩ when it is not identically−∞on any component.

(6) Let u∈ SHm(Ω) and v ∈ SHm(Ω⁰), where Ω⁰ ⊂Cⁿ is an open set such that Ω∩Ω⁰ 6=∅. If u>v on Ω∩∂Ω⁰, then the function

z7−→w(z) :=

(max(u(z), v(z)) ifz∈Ω∩Ω⁰ u(z) ifz∈ΩrΩ⁰ ism-subharmonic on Ω.

Another ingredient which will be important is the regularization process. Letχbe a fixed smooth positive radial function with compact support in the unit ballB⊂Cⁿ

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andR

Cⁿχ(ζ)dλ2n(ζ) = 1. For anyδ such that0< δ < δ0:= diam(Ω), we set χδ(ζ) = 1

δ²ⁿχ(ζ/δ) and Ωδ ={z∈Ω ; dist(z, ∂Ω)> δ}.

Letu∈ SHm(Ω)⊂L¹_loc(Ω)and define its standardδ-regularization by the formula

(2.2) u_δ(z) :=

Z

Ω

u(z−ζ)χ_δ(ζ)dλ_2n(ζ), z∈Ω_δ.

Then it is easy to see thatuδ ism-subharmonic and smooth onΩδ and decreases tou onΩasδdecreases to0.

The following lemma was proved in [GKZ08].

Lemma2.3. — Let u ∈ SHm(Ω)∩L¹(Ω). Then for δ ∈ ]0, δ0[, its δ-regularization extends toCⁿ by the formula

(2.3) uδ(z) :=

Z

Ω

u(ζ)χδ(z−ζ)dλ2n(ζ), z∈Cⁿ,

and have the following properties:

(1) u_δ is a smooth function onCⁿ which ism-subharmonic on Ω_δ; (2) (uδ) decreases touonΩasδ decreases to0 and

Z

Ωδ

(uδ(z)−u(z))dλ2n(z)6anδ² Z

Ωδ

dd^cu∧βⁿ⁻¹,

wherea_n>0 is a uniform constant independent ofuandδ.

Let us introduce the notion of strong m-pseudoconvexity that will be used in the sequel.

Definition2.4. — We say that the open setΩis stronglym-pseudoconvex ifΩadmits a defining functionρwhich is smooth strictlym-subharmonic in a neighbourhood ofΩ and|∇ρ|>0on∂Ω ={ρ= 0}. In this case we can choose ρso that

(2.4) (dd^cρ)^k∧β^n−k>βⁿ forksuch that16k6m, pointwise onΩ.

The following lemma is analogous to a lemma proved in [GKZ08] using mean values rather than convolution.

Lemma2.5. — Let ΩbCⁿ be a bounded domain andu∈ SH(Ω)∩L^∞(Ω). Assume that u is Hölder continuous near ∂Ω with exponent α ∈ ]0,1[. Then the following properties are equivalent:

(i) ∃c1>0,uδ :=u ? χδ6u+c1δ^α in Ωδ, (ii) ∃c₂>0,sup_B(z,δ)u6u+c₂δ^α in Ω_δ.

A similar lemma has been recently proved in the compact Hermitian manifold setting in [LPT20]. A slight modification of the proof of [GKZ08] with an observation from [LPT20] works also in our context as it is explained in [Zer20].

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Remark2.6. — Recall thatuis Hölder continuous near ∂Ωwith exponent α∈]0,1]

if there exists δ1 >0 small enough and a constant κ >0 such that for any ζ ∈∂Ω and anyδ∈]0, δ₁[,

sup

z∈Ω(ζ,δ)

|u(z)−u(ζ)|6κδ^α, where Ω(ζ, δ) := Ω∩B(ζ, δ).

Assume that there exists two functions v, w defined and Hölder continuous with exponent α on a neighbourhood U of ∂Ω in Ω such that v 6 u 6 w on U and v=u=won∂Ω. Thenuis Hölder continuous with exponentαnear∂Ω.

2.2. Complex Hessian operators. — Following [Bło05], we can define the Hessian operators acting on (locally) bounded m-subharmonic functions as follows. Given u₁, . . . , u_k ∈ SHm(Ω)∩L^∞(Ω) (16k6m), one can define inductively the following positive(m−k, m−k)-current onΩ

dd^cu₁∧ · · · ∧dd^cu_k∧β^n−m:=dd^c(u₁dd^cu₂∧ · · · ∧dd^cu_k∧β^n−m).

In particular, ifu∈ SHm(Ω)∩L^∞_loc(Ω), the positive current(dd^cu)^m∧β^n−mcan be identified to a positive Borel measure on Ω, the so called m-Hessian measure of u denoted by:

σm(u) := (dd^cu)^m∧β^n−m.

Observe that when m= 1, σ1(u) =dd^cu∧βⁿ⁻¹ is the Riesz measure of u (up to a positive constant), whileσn(u) = (dd^cu)ⁿis the complex Monge-Ampère measure ofu.

It is then possible to extend Bedford-Taylor theory to this context. In particular, Chern-Levine Nirenberg inequalities holds and the Hessian operators are continuous under local uniform convergence and pointwise a.e. monotone convergence on Ω of sequences of functions inSH(Ω)∩L^∞_loc(Ω)(see [Bło05], [Lu12]).

We defineE_m⁰(Ω)to be the positive convex cone of negative functionsφ∈ SH⁻_m(Ω)∩

L^∞(Ω)with zero boundary values such that Z

Ω

(dd^cφ)^m∧β^n−m<+∞.

These are the “test functions” inm-Hessian potential theory. The formula of integra- tion by parts is valid for these functions.

More generally it follows from [Lu12, Lu15] that the following property holds: ifφ∈ E_m⁰(Ω) andu, v∈ SHm(Ω)∩L^∞(Ω)withu60, then forksuch that06k6m−1, (2.5)

Z

Ω

(−φ)dd^cu∧(dd^cv)^k∧β^n−k−16 Z

Ω

(−u)dd^cφ∧(dd^cv)^k∧β^n−k−1.

An important tool in the corresponding potential theory is the comparison principle.

Proposition 2.7. — Assume that u, v ∈ SH_m(Ω)∩L^∞(Ω) and for any ζ ∈ ∂Ω, lim inf_z→ζ(u(z)−v(z))>0. Then

Z

{u<v}

(dd^cv)^m∧β^n−m6 Z

{u<v}

(dd^cu)^m∧β^n−m.

Consequently, if (dd^cu)^m∧β^n−m6(dd^cv)^m∧β^n−mweakly on Ω, thenu>v on Ω.

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It follows from the comparison principle that if the Dirichlet problem (1.1) admits a solution, then it is unique.

Let us recall the following estimates due to Cegrell ([Ceg04]) for the complex Monge-Ampère operators and extended by Charabati to complex Hessian operators ([Cha16b]).

Lemma2.8. — Let u, v, w∈ E_m⁰(Ω). Then for any16k6m−1 Z

Ω

dd^cu∧(dd^cv)^k∧(dd^cw)^m−k−1∧β^n−m6Im(u)^1/mIm(v)^k/mIm(w)^{(m−k−1)/m}, whereI_m(u) :=R

Ω(dd^cu)^m∧β^n−m.

In particular, ifΩis stronglym-pseudoconvex, then Z

Ω

dd^cu∧(dd^cw)^k∧β^n−k−16c_m,n(I_m(u))^1/m(I_m(w))^k/m,

and Z

Ω

dd^cu∧βⁿ⁻¹6cm,n(Im(u))^1/m, wherec_m,n>0 is a uniform constant.

The following consequence will be useful in the sequel. This result is usually stated for plurisubharmonic functions on a bounded domain with boundary values0. Let us give a more general version using Cegrell inequalities.

Corollary2.9. — Let Ωb Cⁿ be a bounded strongly m-pseudoconvex domain. As- sume that u, v ∈ SHm(Ω) ∩L^∞(Ω) satisfy u 6 v on Ω and for any ζ ∈ ∂Ω, lim_z→ζ(u(z)−v(z)) = 0. Then

Z

Ω

Proof. — The proof is standard but let us repeat it here for the convenience of the reader. We can assume thatIm(u) :=R

Ω(dd^cu)^m∧β^n−m<+∞.

Letρ: Ω→]− ∞,0[be a defining m-subharmonic function on a neighbourhood ofΩ. In particularR

Ω(dd^cρ)^m∧β^n−m<+∞. For fixedε >0, the functionuε:=u+ερ is a boundedm-subharmonic function such thatuε=v on∂Ωand{uε< v}= Ω.

Applying Proposition 2.7, we obtain Z

Ω

(dd^cuε)^m∧β^n−m.

Observe that

(dd^cuε)^m∧β^n−m= (dd^cu)^m∧β^n−m+

m

X

j=1

C_m^jε^j(dd^cu)^m−j∧(dd^cρ)^j∧β^n−m.

By Lemma 2.8, there exists a constant C > 0 depending on Im(u) and Im(ρ) such that for any16j6m, we have

Z

Ω

(dd^cu)^m−j∧(dd^cρ)^j∧β^n−m6C.

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Therefore, for anyε∈]0,1[, Z

Ω

(dd^cu)^m∧β^n−m+C2^mε.

Lettingε→0, we obtain the required inequality.

2.3. The bounded subsolution theorem. — Let Ω b Cⁿ be a bounded strongly m-pseudoconvex domain. Assume there exists v∈ SHm(Ω)∩L^∞(Ω)such that (2.6) µ6(dd^cv)^m∧β^n−monΩ and v|∂Ω≡0.

Ngoc Cuong Nguyen proved that under this condition, the Dirichlet problem (1.1) admits a unique boundedm-subharmonic solution (see [Ngu12]).

Theorem2.10 ([Ngu12]). — Let ΩbCⁿ be a bounded strongly m-pseudoconvex do- main and µ a positive Borel measure on Ω satisfying the condition (2.6). Then for any g ∈ C⁰(∂Ω), there exists a unique U = Ug,µ ∈ SHm(Ω)∩L^∞(Ω) such that (dd^cU)^m∧β^n−m=µ onΩandU|∂Ω≡g.

2.4. The viscosity comparison principle. — In order to prove Theorem A, we will need to prove an important result (Theorem 3.3). The proof of this result uses the viscosity comparison principle which was established for complex Hessian equations by H. C. Lu ([Lu13]) in the spirit of the earlier work by P. Eyssidieux, V. Guedj and the second author on complex Monge-Ampère equations ([EGZ11]).

To state this comparison principle we need some definitions. Let Ω b Cⁿ be a bounded domain andF : Ω×R→Ra continuous functionnon-decreasingin the last variable.

Definition2.11. — Letu: Ω→R∪ {−∞}be a function and qbe aC² function in a neighborhood ofz0∈Ω. We say thatqtouchesufrom above (resp. below) atz0 if q(z₀) =u(z₀)andq(z)>u(z)(resp.q(z)6u(z)) for everyzin a neighborhood ofz₀. Definition 2.12. — An upper semicontinuous function u : Ω → R is a viscosity subsolution to the equation

(2.7) (dd^cu)^m∧β^n−m=F(z, u)βⁿ,

if for anyz₀∈Ωand anyC²functionqwhich touchesufrom above atz₀then σ_m(q)>F(·, q(z0))βⁿ, atz=z₀.

We will also say thatσ_m(u)>F(·, u)βⁿin the viscosity sense atz₀andqis an upper test function foruatz₀.

Definition2.13. — A lower semicontinuous functionv: Ω→Ris a viscosity supersolution to (2.7) if for anyz0∈X and anyC² functionqwhich touchesvfrom below atz0,

[(dd^cq)^m∧β^n−m]+6F(z, q)βⁿ, at z=z0.

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Here [α^m∧β^n−m]+ is defined to beα^m∧β^n−m ifα ism-positive and 0 otherwise.

We will also say thatσm(v)+6F(·, v)βⁿ in the viscosity sense atz0andqis a lower test function forvat z₀.

Remark2.14. — If v∈ C²(Ω) thenσ_m(v)>F(z, v)βⁿ (resp.[σ_m(v)]₊ 6F(z, v)βⁿ) holds onΩin the viscosity sense iff it holds in the usual sense.

Definition2.15. — A continuous functionu: Ω→Ris a viscosity solution to (2.7) if it is both a subsolution and a supersolution.

The first important result in this theory compares the viscosity and potential sub- solutions.

Proposition 2.16 ([Lu13]). — Let u be a bounded upper semi-continuous function inΩ. Then the inequality

(2.8) σm(u)>F(·, u)βⁿ

holds in the viscosity sense on Ω if and only if uis m-subharmonic and (2.8) holds in the potential sense onΩ.

Now we can state the viscosity comparison principle.

Theorem 2.17 ([Lu13]). — Let u : Ω → R be a bounded viscosity subsolution and v: Ω→Rbe a viscosity supersolution of the equation

σm(u) =F(·, u)βⁿ,

onΩ. If u6v on ∂Ωthenu6v onΩ.

For more details on this theory we refer to [Lu13] and [EGZ11] in the complex case and to [CIL92] for the real case.

2.5. Weak stability estimates. — An important tool in dealing with our problems is the notion of capacity. This was introduced by Bedford and Taylor in their pio- neering work for the complex Monge-Ampère operator (see [BT82]). Let us recall the corresponding notion of capacity we will use here (see [Lu12], [SA13]). Let ΩbCⁿ be a stronglym-pseudoconvex domain. Them-Hessian capacity is defined as follows.

For any compact setK⊂Ω, Cap_m(K,Ω) := supR

K(dd^cu)^m∧β^n−m;u∈ SHm(Ω),−16u60 . We can extend this capacity as an outer capacity on Ω. Given a set S ⊂ Ω, we define the inner capacity ofS by the formula

Cap_m(S,Ω) := sup{Cap_m(K,Ω) ;K compact, K⊂S}.

The outer capacity ofS is defined by the formula

Cap^∗_m(S,Ω) := inf{Cap_m(U,Ω) ;U is open, U ⊃S},

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It is possible to show thatCap^∗_m(·,Ω)is a Choquet capacity and then any Borel set B⊂Ωis capacitable and for any compact setK⊂Ω,

(2.9) Cap_m(K,Ω) =

Z

Ω

(dd^cu^∗_K)^m∧β^n−m,

whereuK is the relative equilibrium potential of (K,Ω)defined by the formula:

u_K:= sup{u∈ SHm(Ω) ;u 6 −1K onΩ}, andu^∗_K is its upper semi-continuous regularization onΩ(see [Lu12]).

It is well known thatu^∗_K ism-subharmonic onΩ,−16u^∗_K 60, u^∗_K =−1 almost everywhere (with respect toCap_m) onΩandu^∗_K →0as z→∂Ω(see [Lu12]).

We will use the following definition.

Definition2.18. — Letµbe a positive Borel measure onΩand letA, τ >0be positive numbers. We say thatµis dominated by them-Hessian capacity with parameters (A, τ)if for any compact subsetK⊂ΩwithCap_m(K,Ω)61,

(2.10) µ(K)6ACap_m(K,Ω)^τ.

Observe that by capacitability, this inequality is then satisfied for any Borel set K⊂Ω.

Let us mention that S. Kołodziej was the first to relate the domination of the measureµby the Monge-Ampère capacity to the regularity of the solution to complex Monge-Ampère equations (see [Koł96]).

Using his idea, Eyssidieux, Guedj and the second author were able to establish in [EGZ09] a weak stability L¹-L^∞ estimate for bounded solutions to the Dirichlet problem for the complex Monge-Ampère equation. This result is the main tool in deriving estimates on the modulus of continuity of solutions to the complex Monge- Ampère and Hessian equations.

The following examples are due to Dinew and Kołodziej (see [DK14]).

Example2.19

(1) Dinew and Kołodziej proved in [DK14] that the volume measure λ2n is dominated by capacity. Namely for any r such that 1 < r < m/(n−m), there exists a constantN(r)>0such that for any compact subsetK⊂Ω,

(2.11) λ_2n(K)6N(r) Cap_m(K,Ω)^1+r.

Observe that this estimate is sharp in terms of the exponent when m < n. This can be seen by taking Ω = B the unit ball and K :=Bs ⊂B the closed ball of radius s∈]0,1[, sinceCap_m(Bs,B)≈s^2(n−m)ass→0(see [Lu12]). When m=nwe know that the domination is much more precise (see [ÅCK⁺09]).

(2) Let 0 6f ∈L^p(Ω) with p > n/m. Then n(p−1)/p(n−m) >1. By Hölder inequality and inequality (2.11) we obtain: for any τ ∈ ]1, n(p−1)/p(n−m)[ there exists a constantM(τ)>0such that for any compact set K⊂Ω,

(2.12)

Z

K

f dλ2n6M(τ)kfkpCap_m(K,Ω)^τ.

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Theorem A will provide us with many new examples. Condition (2.10) plays an important role in the following stability result which will be a crucial point in the proof of our theorems (see [EGZ09, GKZ08, Cha16b]).

Proposition 2.20. — Let µ be a positive Borel measure on Ω dominated by the m-Hessian capacity with parameters (A, τ)such thatτ >1.

Then for any u, v ∈ SHm(Ω)∩L^∞(Ω) such that (dd^cu)^m∧β^n−m 6µon Ω and lim inf∂Ω(u−v)>0, we have

(2.13) sup

Ω

(v−u)+62k(v−u)+k^1/(m+1)_1,µ +Ck(v−u)+k^γ_1,µ, wherek(v−u)₊k_1,µ:=R

Ω(v−u)₊dµ and (2.14) C:= 1 + 2^τA^1/m

1−2^1−τ, γ=γ(τ, m) := τ−1 τ(m+ 1)−m·

Observe that the most relevant case in applications is whenk(v−u)+k1,µis small.

So the right exponent isγ <1/(m+ 1).

Proof. — The proof uses an idea which goes back to Kołodziej ([Koł96]) with some simplifications due to Guedj, Eyssidieux and the second author (see [EGZ09, GKZ08]).

It relies on the following estimates: for anyt >0,s >0 (2.15) t^mCap_m({u < v−s−t},Ω)6

Z

{u<v−s}

Indeed lett > 0, s >0 be fixed andw ∈ SHm(Ω) be given such that −16w60.

Then

{u−v <−s−t} ⊂ {u−v < tw−s} ⊂ {u−v <−s}bΩ.

It follows that t^m

Z

{u−v<−s−t}

(dd^cw)^m∧β^n−m6 Z

{u<v−s−t}

(dd^c(v+tw))^m∧β^n−m

6 Z

{u<v+tw−s}

(dd^c(v+tw))^m∧β^n−m.

On the other hand the comparison principle yields Z

{u<v+tw−s}

(dd^c(v+tw))^m∧β^n−m6 Z

{u<v+tw−s}

(dd^cu)^m∧β^n−m

6 Z

{u<v−s}

The last two inequalities imply (2.15).

Applying inequality (2.15) with the parameter(s/2, s/2)instead of(t, s)and taking into account thatuis a supersolution, we obtain

Cap_m({u < v−s},Ω)62^ms^−m Z

{u<v−s/2}

(dd^cu)^m∧β^n−m

62^m+1s^−m−1 Z

Ω

(v−u)+dµ.

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Sets0:= 2k(v−u)+k^1/(m+1)_1,µ . Then for anys>s0, (2.16) Cap_m({u < v−s},Ω)61.

Fixε >0and s>0. Then applying inequality (2.15) withs0+s+εinstead ofs and taking into account the fact that(dd^cu)^m∧β^n−m6µweakly onΩ, we get (2.17) t^mCap_m({u < v−s₀−ε−s−t},Ω)6

Z

{u<v−s₀−ε−s}

dµ.

Setf(s) =fε(s) := Cap_m({u−v <−s−s0−ε},Ω)^1/m. By (2.16), we havef(s)61.

Hence sinceµis dominated by capacity, it follows that for anyt >0 ands >0, tf(s+t)6A^1/mf(t)^1+a, wherea:=τ−1>0.

It follows from [EGZ09, Lem. 2.4]) thatf(s) = 0for anys>S_∞where S_∞:= 2A^1/m

1−2^−a[f(0)]^a,

Thus v−u 6 s0+ε+S_∞ almost everywhere on Ω and then the inequality holds everywhere on Ω, i.e.,

max(v−u)+ 6s0+ε+ 2A^1/m

1−2^−a Cap_m({v−u > ε},Ω)^a Applying (2.15) witht=εands= 0we obtain

Cap_m({v−u > ε},Ω)62ε^−m−1k(v−u)₊k_1,µ. As a consequence of the previous estimate, we obtain

sup

Ω

(v−u)62k(v−u)+k^1/(m+1)_1,µ +ε+C⁰ε^−a(m+1)k(v−u)+k^a_1,µ,

whereC⁰:= 2^a+1A^1/m/(1−2^−a). Setε:=k(v−u)+k^γ_1,µ, with

γ:= a

1 +a(m+ 1) = τ−1 (τ−1)(m+ 1) + 1. Then

sup

Ω

(v−u)₊62k(v−u)₊k^1/(m+1)_1,µ +Ck(v−u)₊k^γ_1,µ,

whereC:=C⁰+ 1 = 1 + 2^a+1A^1/m/(1−2^−a) = 1 + 2^τA^1/m/(1−2^1−τ).

3. Subharmonic envelopes and obstacle problems

Here we prove some results that will be used in the proof of Theorem A. Since they are of independent interest, we will state them in the most general form and give complete proofs.

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3.1. Subharmonic envelopes. — Let Ω b Cⁿ and h : Ω → R is a non positive bounded Borel function and define the corresponding projection:

(3.1) eh=P_m,Ω(h) := (sup{v∈ SHm(Ω) ;v6hin Ω})^∗.

Observe that we do not need to take the upper semi-continuous regularization ifhis upper semi-continuous on Ω. On the other hand, we can easily see that

Pm,Ω(h) := sup{v∈ SHm(Ω) ; v6hquasi everywhere onΩ},

wherev 6hquasi everywhere onΩmeans that the exceptional set wherev>hhas zeroCap_m-capacity.

This is a classical construction in potential theory and has been considered in Complex Analysis first by H. Bremermann in [Bre59], J. B. Walsh in [Wal69] and also by J. Siciak in [Sic81]. Later it has been studied by Bedford and Taylor when solving the Dirichlet problem for the the complex Monge-Ampère equation ([BT76], [BT82].

In the setting of compact Kähler manifolds it has bee considered R. Berman and J.-P. Demailly in [BD12] and later in [Ber19]. It has been also considered recently in [GLZ19] in connexion with the supersolution problem for complex Monge-Ampère equations, where a precise estimate of its complex Monge-Ampère measure was given.

We will extend these last results to Hessian equations.

Lemma 3.1. — Let Ω b Cⁿ be a bounded strongly m-pseudoconvex domain and h a bounded lower semi-continuous function on Ω. Then the function eh := Pm,Ω(h) satisfies the following properties:

(i) eh∈ SH_m(Ω)∩L^∞(Ω), andeh6ha.e. on Ω;

(ii) if his continuous onΩ, theneh is continuous onΩand satisfies the following properties

(3.2) lim

Ω3z→ζeh(z) =h(ζ), ζ∈∂Ω,

(iii) R

Ω(eh−h)(dd^ceh)^m∧β^n−m= 0.

Proof. — Observe thatmin_Ωh6eh6max_ΩhonΩ.

(1) Property (i) follows from the general theory (see [Lu12]).

(2) Property (ii) can be proved using the perturbation method due to J. B. Walsh (see [Wal69]). Let us recall the argument for completeness.

We first prove thatehsatisfies (3.2) meaning that it has boundary values equal toh and then it extends as a function onΩwhich is continuous on∂Ω. Indeed fixε >0 and leth⁰be aC²approximating function onΩsuch thath−ε6h⁰6honΩ. Letρ be the stronglym-subharmonic defining function forΩ. Then there exists a constant A >0 such thatu:=Aρ+h⁰ ism-subharmonic on Ωand u6h⁰ 6honΩ. Hence

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by definition of the envelope, we haveu6eh6honΩ. Therefore, for any ζ∈∂Ω, h(ζ)−ε6h⁰(ζ) = lim

Ω3z→ζu(z) 6lim inf

Ω3z→ζeh(z)6lim sup

Ω3z→ζ

eh(z)6h(ζ).

Sinceε >0 is arbitrary, we obtain the identity (3.2). We can then extendehtoΩby settingeh(ζ) =h(ζ)forζ∈∂Ω. To prove continuity ofehonΩ, we use the perturbation argument of J. B. Walsh. Fix δ > 0 small enough, a∈Cⁿ such that|a| 6δ and set Ωa := (−a) + Ω. We define the modulus of continuity of eh near the boundary as follows:

eκ

eh(δ) := sup{|eh(z)−eh(ζ)|;z∈Ω, ζ∈∂Ω,|z−ζ|6δ.

Then, since eh = h is uniformly continuous on ∂Ω, we see that lim_δ→0+eκ

eh(δ) = 0.

By definition ofeκ

eh, for any z∈Ω∩∂Ω_a, we have eh(z+a)6eh(z) +κe

eh(δ)6eh(z) +eκ

eh(δ) +κ_h(δ),

whereκh(δ)is the modulus of continuity ofhonΩ. Therefore, by the gluing principle, the function defined by

v(z) :=

(max{eh(z),eh(z+a)−eκ

eh(δ)−κ_h(δ)} ifz∈Ω∩Ω_a

eh(z) ifz∈ΩrΩa

ism-subharmonic onΩand satisfiesv6honΩ. Hencev6ehonΩand then eh(z+a)−eκ

eh(δ)−κh(δ)6eh(z),

for anyz∈Ω∩Ωa with|a|6δ. This proves thatehis uniformly continuous onΩ.

(3) Property (iii) follows by a standard balayage argument in potential theory which goes back to Bedford and Taylor for the complex Monge-Ampère equation

([BT76], [BT82], see also [GLZ19]).

Remark 3.2. — The proof above does not give any information on the modulus of continuity ofehin terms of the modulus of continuity ofh. In other words we do not know ifeκ

eh is comparable toκh.

However ifhisC²-smooth onΩ, the functionu:=Aρ+h, considered in the proof above with h⁰ =h, is m-subharmonic onΩ, Lipschitz onΩ and satisfiesu6eh6h on Ω. Then this implies that eκ

eh(δ) 6 κ_h(δ) +κ_u(δ) 6 Cκ_h(δ), where C > 0 is a uniform constant. Therefore, the modulus of continuity of eh satisfies the inequality κ

eh(δ)6C⁰κ_h(δ), where C⁰ >0is an absolute constant.

This information is not needed here, but it is worth mentioning that this an interesting open problem related to the regularity of solutions to obstacle problems.

We will come back to this in a subsequent work.

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3.2. An obstacle problem

Theorem 3.3. — Let h ∈ C²(Ω). Then eh := Pm,Ωh ∈ SHm(Ω) ∩C⁰(Ω) and its m-Hessian measure satisfies the following inequality:

(3.3) (dd^ceh)^m∧β^n−m61_{e_h=h}σ⁺_m(h), in the sense of currents onΩ.

Here for a functionh∈ C²(Ω), we set

σ_m⁺(h) :=1Gσm(h),

pointwise on Ω, whereG is the set of pointsz ∈Ωsuch that dd^ch(z)∈Θm i.e., the (1,1)-form dd^ch(z)ism-positive (see Definition 2.1).

Proof. — To prove (3.3), we proceed as in [GLZ19], using an idea which goes back to R. Berman [Ber19]. Thanks to the property (ii) of Lemma 3.1, it is enough to prove that

(3.4) (dd^ceh)^m∧β^n−m6σ_m⁺(h), in the sense of currents onΩ. We proceed in two steps.

(1) Assume first that Ω is smooth strongly m-pseudoconvex and h ∈ C²(Ω) and consider the following Dirichlet problem for the complexm-Hessian equation depending on the parameterj∈N,

(3.5) (dd^cu)^m∧β^n−m=e^j(u−h)σ_m⁺(h), u=hin ∂Ω.

By [Lu13], for each j∈N, there exists a unique continuous solution uj∈ SHm(Ω)∩ C⁰(Ω)to this problem (see also [Cha16b]).

Our goal is to prove that the sequence(uj)_j∈Nincreases toehuniformly onΩ. We argue as in [GLZ19] with obvious modifications. RecallhisC²in Ω. Then by defini- tionhis a viscosity supersolution to the Dirichlet problem (3.5). Moreover, by Propo- sition 2.16,ujis a viscosity subsolution to the Dirichlet problem (3.5). By the viscosity comparison principle Theorem 2.17, we conclude thatuj6hinΩsinceuj=hon∂Ω.

Therefore, the pluripotential comparison principle (Proposition 2.7) implies that (uj) is an increasing sequence. On the other hand, by Theorem 2.10 there exists a boundedm-subharmonic functionψonΩwhich is a solution to the complex Hessian equation

σm(ψ) =e^ψ−hσ⁺_m(h)

with ψ=hon∂Ω. Moreover, for anyj ∈N, one can easily check that the function defined by the formula

ψj:= (1−1/j)eh+ (1/j)(ψ−mlogj)

is a (pluripotential) subsolution to the equation (3.5), sinceeh6h on Ω. Hence by Proposition 2.7 we haveψj6uj onΩ.

Summarizing, we have proved that for anyj∈N,ψ_j 6u_j6ehonΩ. Therefore, 06eh−uj6eh−ψj= (1/j)(eh−ψ+mlogj) onΩfor anyj∈N^∗.

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This proves that uj converges toehuniformly on Ω. Then sinceuj 6honΩ, taking the limit as j → +∞ in (3.5) we obtain inequality (3.4) by the continuity of the Hessian operators for uniform convergence (see [Lu12]).

(2) For the general case of a boundedm-hyperconvex domain, we approximateΩby an increasing sequence(Ωj)_j∈Nof smooth stronglym-pseudoconvex domains such that for any j∈N, Ωj+1⊂Ωj andΩ =∪j∈NΩj. Then it is easy to see that the sequence (P_m,Ωh_j) decreases to P_m,Ωhon Ω (see [GLZ19]). Thus the result follows from the previous case by the continuity of the Hessian operator for monotone sequences.

It is worth mentioning that these envelopes have been considered by several authors in the context of compact Kähler manifolds. WhenhisC²it was proved recently that P(h) isC^1,1 (see [CZ19], [Tos18], [Ber19]) and equality holds in (3.3), which means thatP(h)is a solution to an obstacle problem (see [BD12]).

We can address a similar question.

Question. — Is it true that eh is C^1,1 locally on Ω when h is C² on Ω? Is there equality in (3.3)?

Corollary 3.4. — Let Ω b Cⁿ be a strongly m-pseudoconvex domain. Let u ∈ SHm(Ω) a negativem-subharmonic function. Then there exists a decreasing sequence (u_j)of continuousm-subharmonic functions onΩwith boundary values0 which con- verges pointwise to uonΩ.

Proof. — We can assume thatuis bounded onΩand extend it as a semi-continuous function on Ω. Let (h_j)_j∈_Nbe a decreasing sequence of smooth functions in a neighbourhood ofΩwhich converges touinΩ. For eachj∈N, consider them-subharmonic envelopevj :=PΩhj onΩ and setuj := max{vj, jρ} on Ω, whereρis a continuous m-subharmonic defining function forΩ. Then by Lemma 3.1, (u_j)is a decreasing sequence of continuousm-subharmonic functions on Ωwhich converges touonΩ.

Applying the smoothing method of Richberg it is possible to construct a decreasing sequence of smooth m-subharmonic functions on Ω which converges to uin Ω (see [Pli14]).

4. Hessian measures of Hölder continuous potentials

In this section we will prove two important results which will be used in the proof of the main theorems stated in the introduction.

4.1. Hessian mass estimates near the boundary. — Here we prove a comparison inequality which seems to be new even in the case of a complex Monge-Ampère measure.

Lemma4.1. — Let ΩbCⁿ be a bounded strongly m-pseudoconvex domain and ϕ∈ SHm(Ω)∩ C^α(Ω)(0< α61) withϕ≡0 on∂Ω. Then for any Borel setK⊂Ω, we

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have

Z

K

(dd^cϕ)^m∧β^n−m6L^m[δK(∂Ω)]^mα Cap_m(K,Ω),

where

δK(∂Ω) := sup

z∈K

dist(z, ∂Ω)

andL >0 is the Hölder norm ofϕ.

The constantδK(∂Ω)is the Hausdorff distance ofKto the boundary in the sense thatδ_K(∂Ω)6εmeans thatK is contained in theε-neighbourhood of ∂Ω.

The relevant point here is that the estimate takes care of the behaviour at the boundary. It shows in particular that if the volume of the compact set is fixed, the capacity tends to +∞ when the compact set approaches the boundary at a rate controlled by the Hausdorff distance of the compact to the boundary.

Proof. — By inner regularity, we can assume that K ⊂ Ω is compact. Since ϕ is Hölder continuous on Ω, we have ϕ(ζ)−ϕ(z)6 L|ζ−z|^α for any ζ ∈∂Ω and any z∈Ω.

Fix a compact setK⊂Ω. Sinceϕ= 0in ∂Ω, it follows that for anyz∈K,

−ϕ(z)6κ[dist(z, ∂Ω)]^α6L[δK(∂Ω)]^α=:a.

Therefore, the function v:=a⁻¹ϕ∈ SHm(Ω) andv60onΩandv>−1 inK. Fix ε >0 and let uK be the relative extremalm-subharmonic function of(K,Ω). Then K ⊂ {(1 +ε)u^∗_K < v} ∪ {u_K < u^∗_K}. Since the set{u_K < u^∗_K} has zero m-capacity (see [Lu12]), it follows from the comparison principle that for anyε >0,

Z

K

{(1+ε)u^∗_K<v}

(dd^cv)^m∧β^n−m

6(1 +ε)^m Z

{(1+ε)u^∗_K<v}

(dd^cu^∗_K)^m∧β^n−m

6(1 +ε)^mCap_m(K,Ω).

The last inequality follows from (2.9). The estimate of the lemma follows by letting

ε→0.

4.2. Hölder continuity of Hessian measures. — In order to prove the Hölder continuous subsolution theorem we need an additional argument following an idea which goes back to [DDG⁺14] and used in a systematic way in [Ngu18] (see also [KN20a]).

Given a continuous functiong ∈ C⁰(∂Ω)and a real numberR >0, we denote by E_m^g(Ω, R)the convex set of boundedm-subharmonic functionsvonΩsuch thatv=g on∂Ωnormalized by the mass condition R

Ω(dd^cv)^m∧β^n−m6R.

In order to prove Theorem B, we will need the following Lemma:

Lemma4.2. — Let ϕ ∈ E_m⁰(Ω)∩ C^α(Ω) (0 < α 6 1) and g ∈ C⁰(∂Ω) and R > 0.

Then for any 1 6 k 6m, there exists Ck = C(k, m,Ω, R) >0 such that for every

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u, v∈ E_m^g(Ω, R)

(4.1)

Z

Ω

|u−v|(dd^cϕ)^k∧β^n−k6Ck[ku−vk1]^α^k, provided that ku−vk₁:=R

Ω|u−v|dλ_2n61, whereα_k :=α^k2^−k.

Proof. — Recall the following notation for the complex Hessian measure ofϕ:

σ_k(ϕ) := (dd^cϕ)^k∧β^n−k 16k6m.

Observe that for any ε > 0, uε := max{u−ε, v} ∈ E_m^g(Ω) and uε = v near the boundary ∂Ω. By the the comparison principle, this implies that uε ∈ E_m^g(Ω, R).

Therefore, replacing u byuε, we can assume that u >v on Ω and u= v near the boundary ∂Ω. Then the inequality (4.1) will follow from this case since |u−v| = (max{u, v} −u) + (max{u, v} −v).

On the other hand by approximation on the supportS ofu−vwhich is compact, we can assume thatuandv are smooth on a neighbourhood ofS.

We will argue by induction on k. For k = 0, the inequality is obviously satisfied withC0= 1. Assume that the inequality holds for some integerksuch that06k < m i.e.,

(4.2)

Z

Ω

(u−v)σ_k(ϕ)6C_k[ku−vk1]^α^k. We will show that there existsCk+1>0such that

Z

Ω

(u−v)σk+1(ϕ)6Ck+1[ku−vk1]^α^k+1.

We will approximateϕby smooth functions. We first extendϕas a Hölder continuous function onCⁿ. Indeed recall that for anyz, ζ∈Ω, we have ϕ(z)6ϕ(ζ) +κ|z−ζ|^α. Then it is easy to see that the following function

(4.3) ϕ(z) := sup{ϕ(ζ)−κ|z−ζ|^α; ζ∈Ω}, z∈Cⁿ.

is Hölder continuous of orderαonCⁿ andϕ=ϕonΩ. For simplicity, we will denote this extension by ϕ. Then we denote by ϕδ the smooth approximants of ϕ on Cⁿ, obtained by Formula (2.3). By Lemma 2.3 forδ∈]0, δ0[,ϕδ ∈ SHm(Ωδ)∩ C^∞(Cⁿ).

To prove the required estimate, we write Z

Ω

(u−v)(dd^cϕ)^k+1∧β^n−k−1=A+B,

where

A:=

Z

Ω

(u−v)dd^cϕδ∧(dd^cϕ)^k∧β^n−k−1,

and

B :=

Z

Ω

(u−v)dd^c(ϕδ−ϕ)∧(dd^cϕ)^k∧β^n−k−1.

(22)

We estimate each term separately. Fix δ ∈]0, δ0[. Since ϕ is Hölder continuous, we have|ϕδ(z)−ϕ(z)|6κδ^α for anyz∈Ω. Moreover, differentiating the formula (2.2), we obtain forz∈Ω,

(4.4) dd^cϕ_δ(z)6M₁κδ^α

δ² β,

whereM₁>0 depends only onΩandχ. By (4.4) and (4.2), we have (4.5) |A|6M₁κδ^α

δ² Z

Ω

(u−v)σ_k(ϕ)6M₁C_kκδ^α−2C_k[ku−vk₁]^α^k.

To estimateB, observe that, sinceu−v= 0near the boundary, we can integrate by parts to get the following formula

B= Z

Ω

(ϕ−ϕ_δ)dd^c(u−v)∧(dd^cϕ)^k∧β^n−k−1,

and then

|B|6 Z

Ω

|ϕδ−ϕ|dd^c(u+v)∧(dd^cϕ)^k∧β^n−k−1.

Therefore, since|ϕδ−ϕ|6κδ^α onΩ, it follows that (4.6) |B|6(Ik(u, ϕ) +Ik(v, ϕ))κδ^α, where I_k(u, ϕ) := R

Ωdd^cu∧(dd^cϕ)^k ∧β^n−k−1. Observe that by Lemma 2.8 and the normalization mass condition, there exists a constantd(m, n)>0 such that for any k such that1 6k 6m, Ik(u, ϕ) +Ik(v, ϕ)6d(m, n). Combining this with the inequalities (4.5) and (4.6), we obtain forδ∈]0, δ0[,

Z

Ω

(u−v)σ_k+1(ϕ)6M₁C_kκδ^α−2[ku−vk₁]^α^k+d(m, n)κδ^α. Since[ku−vk161., we can takeδ=δ₀p

[ku−vk1]^α^k < δ₀ in the last inequality to obtain

Z

Ω

(u−v)σk+1(ϕ)6(M1Ck+d(m, n))κp

[ku−vk1]^α^kα

=C_k+1[ku−vk₁]^α^k+1,

whereαk+1:=αk(α/2).

It is an open problem to know the precise modulus of continuity of the Hes- sian measure σm(ϕ) acting on the space of normalized m-subharmonic potentials E_m^g(Ω, R)⊂L¹(Ω, λ2n). We do not know either if the lemma is true when the total mass of the Hessian measureσ_m(ϕ)onΩis infinite.

5. Proofs of the main results

In this section we will give the proofs of Theorem A and Theorem B stated in the introduction using the previous results.