Local property of maximal plurifinely plurisubharmonic functions

# Local property of maximal plurifinely plurisubharmonic functions

Nguyen Xuan Hong 111This work is finished during the first author’s post-doctoral fellowship of the Vietnam Institute for Advanced Study in Mathematics. He wishes to thank this institution for their kind hospitality and support. Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy Street, Caugiay District, Hanoi, Vietnam Hoang Viet Vietnam Education Publishing House, Hanoi, Vietnam
###### Abstract

In this paper, we prove that a continuous plurisubharmonic functions defined in an open set in is maximal if and only if it is locally maximal.

###### keywords:
plurifine pluripotential theory, plurisubharmonic functions, maximal plurisubharmonic functions
###### Msc:
[2010] 32U05, 32U15

## 1 Introduction

The plurifine topology on a Euclidean open set of is the smallest topology that makes all plurisubharmonic function on continuous. El Kadiri K03 () defined in 2003 the notion of finely plurisubharmonic function in a plurifine open subset of and studied properties of such functions. These functions are introduced as plurifinely upper semicontinuous functions, of which the restriction to complex lines are finely subharmonic, where a finely subharmonic function is defined on on a fine domain is a finely upper semi-continuous and satisfies an appropriate modification of the mean value inequality. In MW10 () El Marzguioui and Wiegerinck studied the continuity properties of the plurifinely plurisubharmonic functions. El Kadiri, Fuglede and Wiegerinck KFW11 () proved in 2011 the most important properties of the plurifinely plurisubharmonic functions. El Kadiri and Wiegerinck KW14 () defined in 2014 the Monge-Ampère operator on finite plurifinely plurisubharmonic functions in plurifinely open sets and show that it defines a positive measure. El Kadiri and M. Smit KS14 () introduced and studied in 2014 the notion of -maximal -plurisubharmonic functions, which extends the notion of maximal plurisubharmonic functions on a Euclidean domain to an -domain of in a natural way.

There is a natural questions that whether an locally maximal plurisubharmonic function on an open set of also maximal in (see question 4.17 in KS14 ()). El Kadiri and M. Smit KS14 () gave an example to show that this result is not valid when the function is not finite. The aim of this paper is to give a positive answer for this question when the function is continuous. Namely, we will prove the following theorem.

Main theorem. Let be an open set in . Assume that is a continuous plurisubharmonic function in . Then is maximal in if and only if it is locally maximal in .

Klimek Kl91 () proved that a locally bounded plurisubharmonic function defined in an Euclidean open set is maximal if and only if , and therefore, the bounded plurisubharmonic functions defined in an Euclidean open set is maximal if and only if it is locally maximal. Notice that for bounded plurisubharmonic functions defined in an open set , the complex Monge-Ampère operator is locally defined in (see KW14 ()), and therefore, in if and only if is locally maximal in (see KS14 ()). Hence, it need to find an another approach in studying the local property of maximal plurisubharmonic functions. Techniques used in the proof of the main theorem come from HHo131 () (also see HHo132 ()).

The paper is organized as follows. In section 2 we recall some notions of plurifine pluripotential theory. In Section 3 we prove main theorem.

## 2 Preliminaries

Some elements of pluripotential theory (plurifine pluripotential theory) that will be used throughout the paper can be found in Bl2009 ()-W12 ().

2.1. The plurifine topology

The plurifine topology on a Euclidean open set of is the smallest topology that makes all plurisubharmonic functions on continuous.

Notions pertaining to the plurifine topology are indicated with the prefix to distinguish them from notions pertaining to the Euclidean topology on . For a set we write for the closure of in the one point compactification of , for the -closure of and for the -boundary of .

A local basis is given by the sets where be Euclidean open balls of center , radius ; and with .

The plurifine topology is quasi-Lindelöf, that is, every arbitrary union of open sets is the union of a countable subunion and a pluripolar set.

2.2. plurisubharmonic functions

Let be an open set in . A function is said to be plurisubharmonic function if it is upper semicontinuous, and for every complete line in , the restriction of to any component of the finely open subset of is either finely subharmonic or .

The set of all plurisubharmonic functions in is denoted by .

Let . We say that is maximal in if for every bounded open set of with , and for every function that is bounded from above on and extends upper semicontinuously to with on implies on .

The set of all maximal plurisubharmonic functions in is denoted by .

The function is called locally (resp. locally) maximal in if for every there exists an Euclidean open (resp. open) neighbourhood of such that is maximal on .

## 3 Proof of main theorem

First we give the following.

###### Proposition 3.1.

Let be an open set in . Assume that is a bounded plurisubharmonic function in . Then, the following conditions are equivalent

(a) .

(b) , for every pluriharmonic functions in .

(c) For every and for every open set with we have

 supG(v−u)≤supΩ∖G(v−u).
###### Proof.

(a) (b) is obvious.

(a) (c). Let and let be an open set with . Put

 M:=supΩ∖G(v−u).

Without loss of generality we can assume that . Then in . In particular, on , and hence, in . Therefore,

 supG(v−u)≤M=supΩ∖G(v−u).

(c) (a). Let be an open set in with , and let such that is bounded from above on , extends upper semicontinuously to , and on . Put

 φ:={max(v,u) on G,u on Ω∖G.

Thanks to Proposition 2.3 in KS14 () we have . It follows that

 supG(v−u)≤supG(φ−u)≤supΩ∖G(φ−u)=0.

Hence, in . The proof is complete. ∎

###### Proposition 3.2.

Let be open sets in . Assume that is bounded, locally maximal, plurisubharmonic function in . Then is maximal in .

###### Proof.

Let be a plurisubharmonic function in and let is a bounded open set in such that . Choose such that . Let . Put , . Choose such that and

 limj→+∞[vε(pj)−u(pj)]=supG(vε−u).

Let such that and is maximal plurisubharmonic function in . Without loss of generality we can assume that . Put

 gε,j(z):=ε|z−pj|2−ε|z|2, z∈Cn.

It is clear that are pluriharmonic functions in . Following Proposition 3.1 we have are maximal plurisubharmonic functions in , and hence, again by Proposition 3.1, we get

 supG(vε−u) =limj→+∞[v(pj)−u(pj)−gε,j(pj)] ≤supG∩B(p,2r)(v−u−gε,j) ≤sup(Ω∩B(p,3r))∖(G∩B(p,2r))(v−u−gε,j) ≤max(sup(Ω∩B(p,3r))∖G(v−u−gε,j),supG∖B(p,2r)(v−u−gε,j)) ≤max(sup(Ω∩B(p,3r))∖G(vε−u),supG∖B(p,2r)(vε−u−εr2)).

Therefore,

 supΩ∩B(0,R)(vε−u) =max(supG∩B(0,R)(vε−u),sup(Ω∩B(0,R))∖G(vε−u)) ≤max(supΩ∩B(p,3r)(vε−u)−εr2,sup(Ω∩B(0,R))∖G(vε−u)) ≤max(supΩ∩B(0,R)(vε−u)−εr2,sup(Ω∩B(0,R))∖G(vε−u)).

It follows that

 supΩ∩B(0,R)(vε−u)=sup(Ω∩B(0,R))∖G(vε−u).

Hence,

 supG(v−u) ≤supG(vε−u)≤supΩ∩B(0,R)(vε−u)−εr2 =sup(Ω∩B(0,R))∖G(vε−u)−εr2

Let we obtain that

 supG(v−u)≤supΩ∖G(v−u).

Thanks to Proposition 3.1 this implies that is maximal plurisubharmonic function in . The proof is complete. ∎

We now give the proof of main theorem.

###### Proof of main theorem.

The proof of the necessity is obvious. We now give the proof of the sufficiency. By Proposition 3.2 it remains to prove that is locally maximal in , and hence, without loss of generality we can assume that . Let is a bounded open set in with , and let such that is bounded from above on , extends upper semicontinuously to and on . Let . Put

 vε(z):={max(v(z)+ε(|z|2−R2),u(z)) if z∈G,u(z) if z∈Ω∖G.

Thanks to Proposition 2.3 in KS14 () we have . Assume that

 supG(vε−u)>δ0>0.

Choose such that , for all and

 limj→+∞[vε(pj)−u(pj)]=supG(vε−u)=supΩ(vε−u).

First, we claim that

 limsupj→+∞vε(pj)≤vε(p).

Indeed, let . Since is continuous in so there exist a smooth functions defined in such that

 u≤f≤u+δ on ¯¯¯¯G.

Choose such that

 f=φ−ψ on B(0,R).

Put

 w:={max(vε+ψ,f+ψ) in G,f+ψ in B(0,R)∖G.

Since on , from Proposition 2.3 in KS14 () we have , and hence, by Proposition 2.14 in KFW11 () we have . Therefore, be upper semicontinuous function in . Since

 vε(pj)−f(pj)≥vε(pj)−u(pj)−δ>0,

we have . It follows that

 limsupj→+∞vε(pj) =limsupj→+∞[w(pj)−ψ(pj)] ≤w(p)−ψ(p)≤max(vε(p),f(p)) ≤max(vε(p),u(p)+δ).

Letting we obtain that

 limsupj→+∞vε(pj)≤max(vε(p),u(p))=vε(p).

This proves the claim.

Now, since is continuous function, we have

 vε(p)−u(p) ≤supΩ(vε−u)=limsupj→+∞[vε(pj)−u(pj)] =limsupj→+∞vε(pj)−u(p) ≤vε(p)−u(p).

Therefore,

 vε(p)−u(p)=supG(vε−u)=supΩ(vε−u)>0.

It follows that . Thanks to Theorem 3.1 in MW10 () this implies that is open neighbourhood of .

Let and let such that , and is maximal plurisubharmonic function in . Without loss of generality we can assume that is bounded on . Put

 g(z):=ε|z−p|2−ε(|z|2−R2), z∈Cn.

Since is a pluriharmonic functions in , by Proposition 3.1 we have is maximal plurisubharmonic function in . Let . Again by Proposition 3.1, we get

 supΩ(vε−u) =vε(p)−u(p) =v(p)+δϕ(p)−u(p)−g(p) ≤sup(B(p,2r)∩{ϕ>−2})∖(B(p,r)∩{ϕ>−1})(v+δϕ−u−g) ≤max(supB(p,2r)∩{−2<ϕ≤−1}(v+δϕ−u−g),sup(B(p,2r)∩{ϕ>−2})∖B(p,r)(v+δϕ−u−g)) ≤max(supB(p,2r)∩{−2<ϕ≤−1}(vε−δ−u),sup(B(p,2r)∩{ϕ>−2})∖B(p,r)(vε+δϕ−u−εr2)) ≤max(supΩ(vε−u)−δ,supB(p,2r)∩{ϕ>−2}(vε+δϕ−u−εr2))

It implies that

 supΩ(vε−u) ≤supB(p,2r)∩{ϕ>−2}(vε+δϕ−u−εr2) ≤supΩ(vε−u)+δsupB(p,2r)ϕ−εr2.

Let , we obtain

 supΩ(vε−u)≤supΩ(vε−u)−εr2.

This is impossible. Thus,

 supG(vε−u)≤0.

Therefore,

 supG(v−u)≤supG(vε−u)+εR2≤εR2.

Letting we obtain that

 supG(v−u)≤0.

It follows that in . Hence, is maximal plurisubharmonic function in . The proof is complete. ∎

Main Theorem together with Proposition 2.5 in KS14 () and Theorem 4.15 in KS14 () gives

###### Corollary 3.3.

Let be an open set in . Assume that is a continuous plurisubharmonic function in . Then is maximal in if and only if on .

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