Mohamed ZAWAY

Abstract The aim of this paper is to study the operator ()q ∧T on some classes of plurisubharmonic (psh)functions,which are not necessary bounded,where T is a positive closed current of bidimension (q,q)on an open set Ω of Cn.The author introduces two classes FTp (Ω)and ETp (Ω)and shows first that they belong to the domain of definition of the operator ()q ∧T.Then the author proves that all functions that belong to these classes are CT-quasi-continuous and that the comparison principle is valid for them.

Keywords Positive closed current,Plurisubharmonic function,Capacity,Monge-Ampère Operator

1 Introduction

Let Ω be a bounded open set of Cnand denote by PSH(Ω)the set of plurisubharmonic(psh)functions on Ω.The definition of the complex Monge-Ampère operator ()non the set of psh functions was studied by Bedford and Taylor in [1],they proved that this operator is well defined on the set of locally bounded psh functions and they established the comparison principle to study the Dirichlet problem on PSH(Ω)∩L∞(Ω).The problem of extending its domain of definition has been treated by many other authors,in particular Cegrell introduced,between 1998 and 2004 (see [2–4]),a general class E(Ω): The class of psh functions which are locally equal to decreasing limits of bounded psh functions vanishing on ∂Ω with bounded Monge-Ampère mass on Ω.He proved that the Monge-Ampère operator is well defined on E(Ω)and that it is the largest domain of definition ofif it is required to be continuous under decreasing sequences.The study of this class leads to many results such as the comparison principle,the convergence in capacity and the solvability of the Dirichlet problem.This paper continues the studies of plurisubharmonic functions and the complex Monge-Ampère operator associated to a positive closed current T.

Throughout this paper,we denote by T a positive closed current of bidimension (q,q)on Ωwhere 1 ≤q ≤n.The operatorT was studied by Dabbek and Elkhadhra [5]in the case of bounded psh functions.We will extend here the domain of definition of this operator to some classes of unbounded psh functions,and its different properties will be studied.

In this paper we recall the classes FT(Ω)and ET(Ω)introduced by Hai and Dung in [7]where they proved that Monge-Ampère operatoris well defnied.For such classes one of the results in [7]was cited with incomplete proof,so we state it here and give a completed proof (see Lemma 2.2).

In Section 2 we introduce the class ETp(Ω)and show that the Monge-Ampère operatoris well defined on this class.Then we give some properties of the classes ETp(Ω)and FT(Ω).

In Section 3 we prove that every function in ETp(Ω)or in FT(Ω)is CT-quasi-continuous; it means that it is continuous outside a subset of small CT-capacity.The main tool will be an estimate of the growth of CT({u<−s}).Indeed we prove that

for every u ∈ETp(Ω)(resp.u ∈FT(Ω)).

In Section 4,we give the main result of this article (see Theorem 4.1).

2 The Classes ETp(Ω)and FTp(Ω)

2.1 Preliminary results

Throughout this paper,Ω will be a hyperconvex domain of Cn,therefore it is open,bounded,connected and there exists h ∈PSH−(Ω)such that for all c < 0,the set {z ∈Ω,h(z)< c} is relatively compact in Ω where PSH−(Ω)is the set of negative psh functions.We introduce the class ET0(Ω)associated to T,slightly different from the class ET0(Ω)introduced in[7],as follows:

Using the same proof as in [7],one can easily prove that this class is a convex cone and that for all ψ ∈PSH−(Ω)and ϕ ∈ET0(Ω)the function max(ϕ,ψ)∈ET0(Ω).

In this section we will introduce new energy classes ETp(Ω)and FTp(Ω)similar to Cegrell’s ones and we will prove that the Monge-Ampère operator is well defined on them.

Definition 2.1For every real p ≥1 we define ETp(Ω)as the set

When the sequence (ϕj)jassociated to ϕ can be chosen such that

we say that ϕ ∈FTp(Ω).

It’s easy to check that ET0(Ω)⊂FTp(Ω)⊂ETp(Ω)and that using Hölder’s inequality,one has(Ω)⊂(Ω)for all p2≤p1.

We recall the following result which will be useful to prove some properties of our classes.

Theorem 2.1(see [5])Suppose that u,v ∈ET0(Ω).If p ≥1,then for every 0 ≤s ≤q one has

We prove firstly that these two classes inherit some properties of the energy class ET0(Ω).

Theorem 2.2The classes ETp(Ω)and FTp(Ω)are convex cones.

ProofIt suffices to prove that u+v ∈ETp(Ω)for every u,v ∈ETp(Ω).Let(uj)jand(vj)jbe two sequences that decrease to u and v respectively as in Definition 2.1.We want to estimate

Thanks to Minkowsky inequality,it is enough to estimate the following terms:

for all 0

As these sequences are uniformly bounded by the definition of ETp(Ω),the result follows.

Proposition 2.1Let u ∈ETp(Ω)(resp.FTp(Ω))and v ∈PSH−(Ω).Then the function w:=max(u,v)is in ETp(Ω)(resp.in FTp(Ω)).

ProofLet (uj)jbe a sequence that decreases to u as in Definition 2.1 and take wj:=max(uj,v).The sequence (wj)decreases to w.So it is enough to prove that

Thanks to Theorem 2.1,one has

Therefore

The right-hand side is uniformly bounded because u ∈ETp(Ω)and the result follows.

The following theorem proves that the Monge-Ampère operatoris well defined on the new classes.

Theorem 2.3Let u ∈ETp(Ω)and let (uj)jbe a sequence of psh functions that decreases to u as in Definition 2.1.Then the sequence((ddcuj)q∧T)jconverges weakly to a positive measureµ and this limit is independent of the choice of the sequence (uj)j.We set (ddcu)q∧T :=µ.

ProofLet 0 ≤χ ∈D(Ω),δ=sup{u1(z); z ∈Suppχ} and ε >0.There exists a sequence(rj)jsuch that 0

Let

where dV is the normalized Lebesgue measure on the unit ball B.Then one has

The function urjis continuous,psh on{uj<}and uj≤urjon Ω.Let=max(urj+δ,2uj).Then the sequencedecreases to a psh functionand∈ET0(Ω)by Proposition 2.1.Furthermore,using the same technic of the previous proof,we obtain

The proof of the theorem will be complete if we show that

exists.

Let h be an exhaustion function in ET0(Ω).Then

Thanks to Dabbek-Elkhadhra [5],the sequence of measures (ddcmax(,−k))q∧T converges weakly for every k.So it is enough to control

This completes the proof of the theorem.

Theorem 2.4If u ∈ET1(Ω),then

Moreover,if vj∈PSH−(Ω)such that (vj)jdecreases to u,then

ProofSince u ∈ET1(Ω),there exists a sequence (uj)j⊂ET0such that

We then prove that

For every k ≥j and ε>0,one has

and

This goes to 0 when ε →0.By Theorem 2.3 we obtain

Now since −ujis lower semi-continuous,

Hence for all j,

It follows that

Thus

As (vk)kdecreases to u,we have vk∈ET1(Ω).It follows that

Moreover,(max(uj,vk))j∈N⊂ET0(Ω)and decreases to vkso thanks to (2.1),

By tending j →+∞,(2.1)–(2.3)give

Thus

With the same reason,as (max(uj,vk))k∈Ndecreases to uj,we have

Hence

The result follows from (2.4)–(2.5).

Remark 2.1We notice that if u ∈ET1(Ω)and (uj)jis a decreasing sequence to u as in Definition 2.1,then the sequencedecreases to

2.2 Comparison theorems

We recall two classes ET(Ω)and FT(Ω)introduced in[7]where the authors proved that the Monge-Ampère operator ()q∧T is well defined on them.

Definition 2.2We say that u ∈FT(Ω)if there exists a sequence (uj)j⊂ET0(Ω)which decreases to u such that

We said that u ∈ET(Ω)if for all z ∈Ω there exists a neighborhood ω of z and a function v ∈FT(Ω)such that u=v on ω.

As a consequence,for every p ≥1 one has FTp(Ω)⊂FT(Ω)⊂ET(Ω),but there is no relationship between ETp(Ω)and ET(Ω).

Lemma 2.1Let u,v ∈PSH(Ω)∩L∞(Ω)and let U be an open subset of Ω such that u=v near ∂U.Then

ProofLet uεand vεbe the usual regularizations of u and v respectively.Choose U′⊂⊂U such that u=v near ∂U′.If ε > 0 is small enough,one has uε=vεnear ∂U′,and if we take χ ∈D(U′)with χ=1 near {uεvε},then ddcχ=0 on {uεvε}.So

Hence

The result follows.

Corollary 2.1Let u,v ∈FT(Ω).Assume that there exists an open subset U of Ω such that u=v near ∂U.Then

ProofLet u,v ∈FT(Ω)and w ∈ET0(Ω)such that w(z)0 for all z.Then uj:=max(u,jw)and vj=max(v,jw)belong to ET0(Ω)and they are equal on ∂U.The result follows from the previous lemma.

Now we recall a result due to [7]and give a different proof.

Proposition 2.2(see [7])For u,v ∈FT(Ω)such that u ≤v on Ω,one has

ProofLet(uj)jand(vj)jbe the corresponding decreasing sequences to u and v respectively as in Definition 2.2.Replace vjby max(uj,vj),we can assume that uj≤vjfor all j ∈N.For h ∈ET0(Ω)and ε>0 we have

By tending ε to 0 we obtain

The result follows by choosing h decreasing to −1.

Lemma 2.2For all u ∈FT(Ω),there exists a sequence(uj)j⊂ET0(Ω)∩C()that decreases to u.

We claim that this lemma was cited in [7,Theorem 5.1]with incomplete proof.In fact the authors used a comparison theorem proved by Dabbek-Elkhadhra[5]only for bounded psh functions in FT(Ω)where functions are not in general bounded.

ProofWe refer to Cegrell [3,Theorem 2.1]for the construction of the sequence (uj)j.It remains to show that

As uj≥u then by Proposition 2.2 one has

3 CT-Quasi-continuity

Now we establish the quasi-continuity of psh functions belonging to FT(Ω)and ETp(Ω).We need to recall some notions given in [5](see also [9])about the capacity associated to T which is defined as

for any compact subset K of Ω.If E is a subset of Ω,we define

We refer to [5,9]for the properties of this capacity.

Definition 3.1(1)A subset A of Ω is said to be T-pluripolar if CT(A,Ω)=0.

(2)A psh function u is said to be quasi-continuous with respect to CT,if for every ε > 0,there exists an open subset Oεsuch that CT(Oε,Ω)<ε and u is continuous on ΩOε.

Proposition 3.1Let u ∈FT(Ω).Then for every s>0 one has

In particular,the set {u=−∞} is T-pluripolar.

ProofLet (uj)j⊂ET0(Ω)be a decreasing sequence to u on Ω as in Definition 2.2.Take s > 0,v ∈PSH(Ω,[−1,0])and let K be a compact subset in {uj≤−s}.Thanks to the comparison principle (for bounded psh functions),we have

It follows that

By tending j to infinity,we obtain

Corollary 3.1Every u ∈FT(Ω)is CT-quasi-continuous.

ProofLet u ∈FT(Ω)and ε > 0.Denote Bu(t):={z ∈Ω; u(z)< t},t ≤0.By Proposition 3.1,there exists sε≥1 such that CT(Bu(−sε),Ω)<.The function uε:=max(u,−sε)is bounded on Ω so thanks to Dabbek-Elkhadhra [5],there exists an open subset O in Ω such that CT(O,Ω)

To study the CT-quasi-continuity on ETp(Ω),we will proceed as in the previous case.

Proposition 3.2Let u ∈ETp(Ω)and (uj)j⊂ET0(Ω)decreases to u on Ω as in Definition 2.1.Then for every s>0 one has

In particular,the set {u=−∞} is T-pluripolar.

ProofLet s > 0,v ∈PSH(Ω,[−1,0]).Thanks to comparison principle (for bounded psh functions),we have

It follows that

By tending j to infinity,we obtain

By the same argument as in Corollary 3.1 we can easily deduce the following result.

Corollary 3.2Every function in ETp(Ω)is CT-quasi-continuous.

Now we need a first version of the comparison principle where one of the functions will be unbounded.This result was proved in [5]for bounded functions.

Theorem 3.1Let u ∈FT(Ω)and v ∈PSH(Ω)∩L∞(Ω)such that

Then

ProofFirstly we assume that u and v are continuous on a neighborhood W of SuppT.Without loss of generality,we can assume that u < v on W and u=v on ∂W.Let vε:=max(u,v −ε).Then one has vε=u on ∂W and

Since the family of measures (ddcvε)q∧T converges weakly to (ddcu)q∧T as ε →0,then we obtain

Let us now treat the general case.Replace u by u + δ if necessary,we can assume that lim inf(u −v)≥2δ.So there exists an open subset O ⊂⊂Ω such that u(z)≥v(z)+δ for all z ∈ΩO.Let (uk)kand (vj)jbe two smooth sequences of psh functions which decrease respectively to u and v on a neighborhood ofsuch that uk≥vjon∩SuppT for j ≥k.Using the previous argument we obtain

For ε>0,there exists an open subset G of Ω such that CT(G,Ω)<ε and u,v are continuous on ΩG.We can write v=ϕ+ψ where ϕ is continuous on Ω and ψ=0 on ΩG.Take U :={uk<ϕ} then

Since U ∪G={uk

Now as {uk

The continuity of u and v on ΩG gives that {u ≤v}G is a closed subset of Ω.It follows that

Thus

So

By tending ε to 0,we obtain

As{u+ρ

Recall that the Lelong-Demailly number of T with respect to a psh function ϕ is defined as the limit ν(T,ϕ):=where

The following result was proved in [6]but the author used Stokes formula where a regularity condition on ϕ was required.

Theorem 3.2Let ϕ ∈FT(Ω)such that eϕis continuous on Ω.Then for every s,t > 0 one has

In particular,

ProofLet t,s>0 and v ∈PSH(Ω,[−1,0]).For ε>0,we set vε=Thanks to Theorem 3.1 we have

By passing to the supremum over all v ∈PSH(Ω,[−1,0]),we obtain the following estimate:

By passing to the limit when ε →0,the left inequality in (3.1)is obtained.However,for the right inequality,we remark that the function ψ=is psh and satisfies −1 ≤ψ ≤0 on Ω.So by Corollary 2.1 and using the fact that ψ >−1 near ∂Bϕ(−t)we obtain

Thus the right inequality in (3.1)follows.

By the right inequality in (3.1),we have

If we take α>1 and s=αt in the left inequality in (3.1),we obtain

The result follows by letting α →+∞.

Remark 3.1By combining Proposition 3.2 and Theorem 3.2,it is easy to check that if ϕ ∈FTp(Ω)such that eϕis continuous on Ω then ν(T,ϕ)=0.

4 Main Result

The aim of this part is to prove the following main result.

Theorem 4.1(Comparison Principle)Let u ∈FT(Ω)and v ∈ET(Ω).Then

Before presenting the proof,we give some corollaries.

4.1 Consequences of Theorem 4.1

Corollary 4.1Let u,v ∈FTp(Ω)such that euis continuous on Ω.Then

ProofThanks to the comparison principle,we have

The result follows by the fact that ν(T,u)=0 because u ∈FTp(Ω).

Corollary 4.2Let u ∈FT(Ω)and v ∈FTp(Ω)such that evis continuous on Ω.We assume that

Then CT({u

ProofAssume CT({u0.Then there exists ψ ∈PSH(Ω,[0,1])such that

For ε>0 small enough,one has v+εψ ∈FT(Ω)so thanks to the comparison principle,

Hence

which is absurd.

4.2 Proof of Theorem 4.1

To prove this main result,we shall use similar Xing’s inequalities (see [10–11]for more details),generalized to ET(Ω).We start by recalling the following lemma.

Lemma 4.1(see [7])Let S be a positive closed current of bidimension (1,1)on Ω and u,v ∈PSH(Ω)∩L∞(Ω).Assume that u ≤v on Ω and

Then one has

for all k ≥1 and w ∈PSH(Ω,[0,1]).

Lemma 4.2Let u,v ∈PSH(Ω)∩L∞(Ω)such that u ≤v on Ω and

Then one has

for every r ≥1 and w1,··· ,wq∈PSH(Ω,[0,1]).

ProofLet K ⊂⊂Ω and assume that u=v on ΩK.Using Lemma 4.1 we obtain

In the general case,for every ε>0 we set vǫ=max(u,v −ε).Then vǫv on Ω and satisfies vǫ=u on ΩK for some K ⊂⊂Ω.Hence

Since vε−uv −u,the family of measures (ddcvε)q∧T converges weakly to (ddcv)q∧T as ε0 and the function r −w1is lower semicontinuous.Then by letting ε0,we obtain the desired inequality.

Proposition 4.1Let r ≥1 and w ∈PSH(Ω,[0,1]).

(1)For every u,v ∈FT(Ω)such that u ≤v on Ω one has

(2)Furthermore,(4.1)holds for u,v ∈ET(Ω)such that u ≤v on Ω and u=v on ΩK for some K ⊂⊂Ω.

Proof(1)Let u,v ∈FT(Ω)and um,vj∈ET0(Ω)which decrease to u and v respectively as in Definition 2.2.Replacing vjby max(uj,vj)we may assume that uj≤vjfor j ≥1.By Lemma 4.2 we have for m ≥j ≥1,

By approximating w by a sequence of continuous psh functions vanishing on ∂Ω (see [3])and using Proposition 2.2,we obtain that when m →+∞,

Since r −w is lower semi-continuous,we have

Hence by tending j →+∞,we obtain the result.

(2)Let G and W be open subsets of Ω such that K ⊂⊂G ⊂⊂W ⊂⊂Ω.There exists∈FT(Ω)such that≥v on Ω and=v on W.Letbe such that=u on G and=either.Since u=v=on W K,we have∈PSH−(Ω).It follows that∈FT(Ω),≤and=u on W.

Using (1)we obtain

Remark 4.1If we take w=0 and r=1 in Proposition 4.1,we obtain another proof of Proposition 2.2.

The following inequality is a generalization of Theorem 4.1 in [8]since we shall prove it for an arbitrary positive closed current T.

Theorem 4.2Let u,w1,··· ,wq−1∈FT(Ω)and v ∈PSH−(Ω).If we set S=ddcw1∧···∧ddcwq−1,then

ProofWe prove the theorem in two steps.First we assume that v ≡a < 0.Thanks to Lemma 2.2,there exist uj,wk,j∈ET0(Ω)∩C()such that (uj)jdecreases to u and (wk,j)jdecreases to wkfor each 1 ≤k ≤q −1.Since {uj>a} is open,one has

where Sj=ddcw1,j∧···∧ddcwq−1,j.As {u>a}⊂{uj>a} we obtain

It follows from [7]that

Hence

So

Now assume v ∈PSH−(Ω).Since {u>v}=it suffices to show that

for all a ∈Q−.As max(u,v)∈FT(Ω),by the first step,we have

The fact that max(u,v,a)=max(u,a)on the open set {a>v} gives

As {u>a>v} is contained in {u>a},in {max(u,v)>v} and in {a>v},then by combining the last equalities we obtain

We can now prove an inequality analogous to Demailly’s one.Such inequality was proved in [8]for the extremal case q=n.

Proposition 4.2(1)Let u,v ∈FT(Ω)such that (ddcu)q∧T({u=v=−∞})=0.Then

(2)Let µ be a positive measure vanishing on all pluripolar sets of Ω and u,v ∈ET(Ω)such that (ddcu)qT ≥µ,(ddcv)q∧T ≥µ.Then (ddcmax(u,v))q∧T ≥µ.

Proof(1)For each ǫ > 0,put Aǫ={u=v −ǫ}{u=v=−∞}.Since Aǫ∩Aδ=∅for ǫδ,there exists ǫj0 such that (ddcu)q∧T(Aǫj)=0 for j ≥1.On the other hand,since (ddcu)q∧T({u=v=−∞})=0 we have (ddcu)q∧T({u=v −ǫj})=0 for j ≥1.Using Theorem 4.2 it follows that

Letting j →+∞and by Theorem 2.3,we get

because max(u,v −ǫj)max(u,v)and 1l{u

(2)Argument is similar to that of (1).

Proposition 4.3Let u ∈FT(Ω),v ∈ET(Ω).Then

for w ∈PSH(Ω,[0,1])and all r ≥1.

ProofLet ε>0 and set=max(u,v −ε).By (4.1)we have

As {u ≤v −ε}⊂{u

Letting ε →0 we obtain

To conclude the proof of Theorem 4.1,it suffices to take w=0 and r=1 in the previous proposition.