Local property of a class of msubharmonic functions

Let Ω be a hyperconvex domain in C n . By PSH(Ω) (resp. PSH−(Ω)) we denote the cone of plurisubharmonic functions (resp. negative plurisubharmonic functions) on Ω. In 17, the authors introduced and investigated the notion of local class as follows. A class J (Ω) ⊂ PSH−(Ω) is said to be a local class if ϕ ∈ J (Ω) then ϕ ∈ J (D) for all hyperconvex domains D b Ω and if ϕ ∈ PSH−(Ω),ϕ|Ωi ∈ J (Ωi),∀i ∈ I with Ω = S i∈I Ωi then ϕ ∈ J (Ω). As is well known, Błocki (see 9) proved the class E (Ω) introduced and investigated by Cegrell in 11, is a local class. Moreover, in 11 Cegrell has proved this class is the biggest on which the complex MongeAmpere operator ` (ddc .) n is well defined as a Radon measure and it is continuous under decreasing sequences. On the other hand, another weighted energy class Eχ (Ω) which extends the classes Ep(Ω) and F(Ω) in 10 and 11 introduced and investigated recently by Benelkourchi, Guedj and Zeriahi in 5 is

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Local property of a class of m-subharmonic functions

Vu Viet Hung

Received: date / Accepted: date

Abstract In the paper we introduce a new class of m-subharmonic functions with finiteweighted complex m-Hessian We prove that this class has local property

Keywords m-subharmonic functions, weighted energy classes of m-subharmonic functions,complex m-Hessian, local property

Mathematics Subject Classification (2010) 32U05, 32U15, 32U40, 32W20

1 Introduction

Let Ω be a hyperconvex domain in Cn By PSH(Ω ) (resp PSH−(Ω )) we denote thecone of plurisubharmonic functions (resp negative plurisubharmonic functions) on Ω In[17], the authors introduced and investigated the notion of local class as follows A class

J (Ω) ⊂ PSH−(Ω ) is said to be a local class if ϕ ∈J (Ω) then ϕ ∈ J (D) for all perconvex domains Db Ω and if ϕ ∈ PSH−(Ω ), ϕ|Ωi ∈J (Ωi), ∀i ∈ I with Ω = S

hy-i∈IΩithen ϕ ∈J (Ω) As is well known, Błocki (see [9]) proved the class E (Ω) introduced andinvestigated by Cegrell in [11], is a local class Moreover, in [11] Cegrell has proved thisclass is the biggest on which the complex Monge-Amp`ere operator (ddc.)nis well defined

as a Radon measure and it is continuous under decreasing sequences On the other hand,another weighted energy classEχ(Ω ) which extends the classesEp(Ω ) andF (Ω) in [10]and [11] introduced and investigated recently by Benelkourchi, Guedj and Zeriahi in [5] is

Vu Viet Hung

Department of Mathematics, Physics and Informatics, Tay Bac University, Son La, Viet Nam

E-mail: viethungtbu@gmail.com

as follows Let χ : R−−→ R+be a decreasing function Then, as in [5], we define

z→ξϕ (z) = 0 for all ξ ∈ ∂ Ω Hence if ϕ ∈Eχ(Ω ) then

ϕ /∈Eχ(D) with D a relatively compact hyperconvex domain in Ω Thus the classEχ(Ω ) isnot a ”local” one In this paper by relying on ideas from the paper of Benelkourchi, Guedjand Zeriahi in [5] and on Cegrell classes of m-subharmonic functions introduced and studiedrecently in [13] we introduce weighted energy classes of m-subharmonic functionsFm,χ(Ω )andEm,χ(Ω ) Under slight hypotheses for weights χ we achieve that the classFm,χ(Ω ) is

a convex cone (see Proposition 2 below) We also show that the complex Hessian operator

Hm(u) = (ddcu)m∧ βn−mis well defined on the classEm,χ(Ω ) where β = ddckzk2denotesthe canonical K¨ahler form of Cn Futhermore, we prove that the classEm,χ(Ω ) is a localclass (see Theorem 2 in Section 4 below) In this article, we prove the following main result.Theorem 4.6 Let Ω be a hyperconvex domain in Cn and m be an integer with1 ≤ m ≤

n Assume that u∈ SHm−(Ω ) and χ ∈K such that χ00(t) ≥ 0, ∀t < 0 Then the followingstatements are equivalent

a) u∈Em,χ(Ω )

b) For all Kb Ω , there exists a sequence {uj} ⊂E0

m(Ω ) ∩C (Ω),uj& u on K such that

supj

Finally, using the main results above, we prove an interesting corollary Namely, we haveCorollary 4.7 Assume that Ω is a bounded hyperconvex domain and χ ∈K satisfies allhypotheses of Theorem 2 ThenEm,χ(Ω ) ⊂Em−1,χ(Ω )

The paper is organized as follows Beside the introduction the paper has three sections InSection 2 we recall the definitions and results concerning to m-subharmonic functions whichwere introduced and investigated intensively in recent years by many authors, see [6], [14],[24] We also recall the Cegrell classes of m-subharmonic functionsF (Ω ) and E (Ω )

introduced and studied in [13] In Section 3 we introduce two new weighted energy classes

of m-subharmonic functionsFm,χ(Ω ) andEm,χ(Ω ) Section 4 is devoted to the proof ofthe local property of the classEm,χ(Ω ) under some extra assumptions on weights χ Toshow this property of the classEm,χ(Ω ) we need a result about subextension for the class

Fm,χ(Ω ) (see Lemma 5 below) which is of independent interest Finally, by relying on thelocal property of the classEm,χ(Ω ), we prove a corollary for this class

2 Preliminairies

Some elements of pluripotential theory that will be used throughout the paper can be found

in [1], [19], [20], [23], while elements of the theory of m-subharmonic functions and thecomplex Hessian operator can be found in [6], [14], [24] Now we recall the definition ofsome Cegrell classes of plurisubharmonic functions (see [10] and [11]), as well as, the class

of m-subharmonic functions introduced by Błocki in [6] and the classesE0

m(Ω ) andFm(Ω )introduced and investigated by Lu Hoang Chinh in [13] recently Let Ω be an open subset in

Cn By β = ddckzk2

we denote the canonical K¨ahler form of Cnwith the volume element

dVn=n!1βnwhere d = ∂ + ∂ and dc=∂ −∂

4i , hence, ddc=2i∂ ∂ 2.1 As in [10] and [11] we define the classesE0(Ω ) andF (Ω) as follows Let Ω be abounded hyperconvex domain That means that Ω is a connected, bounded open subset andthere exists a negative plurisubharmonic function ρ such that for all c < 0 the set Ωc= {z ∈

2.2 We recall the class of m-subharmonic functions introduced and investigated in [6] cently For 1 ≤ m ≤ n, we define

re-b

Γm= {η ∈ C(1,1): η ∧ βn−1≥ 0, , ηm∧ βn−m≥ 0},where C denotes the space of (1, 1)-forms with constant coefficients

Definition 1 Let u be a subharmonic function on an open subset Ω ⊂ Cn u is said to be am-subharmonic function on Ω if for every η1, , ηm−1in bΓmthe inequality

ddcu∧ η1∧ ∧ ηm−1∧ βn−m≥ 0,holds in the sense of currents

By SHm(Ω ) (resp SHm−(Ω )) we denote the cone of m−subharmonic functions (resp tive m−subharmonic functions) on Ω Before formulating basic properties of m-subharmonic,

nega-we recall the following (see [6])

For λ = (λ1, , λn) ∈ Rnand 1 ≤ m ≤ n, define

Sm(λ ) = ∑

1≤ j1<···< j m ≤n

λj1· · · λj m.Set

1 ≤ m ≤ n, then u ∈ SHr(Ω ), for every 1 ≤ r ≤ m

b) If u is C2smooth then it is m-subharmonic if and only if the form ddcu is pointwise inb

f) If{uj}∞

j=1is a decreasing sequence of m-subharmonic functions then so is u= lim uj

g) Let ρ ≥ 0 be a smooth radial function in Cn vanishing outside the unit ball andsatisfying R

where ρε(z) := 1

ε2nρ (z/ε ) and Ωε= {z ∈ Ω : d(z, ∂ Ω ) > ε} Then uε∈ SHm(Ωε) ∩C∞(Ωε)and uε↓ u as ε ↓ 0

h) Let u1, , up∈ SHm(Ω ) and χ : Rp→ R be a convex function which is non ing in each variable If χ is extended by continuity to a function [−∞, +∞)p→ [−∞, ∞),then χ(u1, , up) ∈ SHm(Ω )

decreas-Example 1 Let u(z1, z2, z3) = 5|z1|2+ 4|z2|2− |z3|2 By using b) of Proposition 1 it is easy

to see that u ∈ SH2(C3) However, u is not a plurisubharmonic function in C3because therestriction of u on the line (0, 0, z3) is not subharmonic

Now as in [6] and [14], we define the complex Hessian operator of locally bounded subharmonic functions as follows

m-Definition 2 Assume that u1, , up∈ SHm(Ω ) ∩ L∞

loc(Ω ) Then the complex Hessian ator Hm(u1, , up) is defined inductively by

oper-ddcup∧ · · · ∧ ddcu1∧ βn−m= ddc(upddcup−1∧ · · · ∧ ddcu1∧ βn−m)

From the definition of m-subharmonic functions and using arguments as in the proof ofTheorem 2.1 in [1] we note that Hm(u1, , up) is a closed positive current of bidegree(n − m + p, n − m + p) and this operator in continuous under decreasing sequences of lo-cally bounded m-subharmonic functions Hence, for p = m, ddcu1∧ · · · ∧ ddcum∧ βn−mis anonnegative Borel measure In particular, when u = u1= · · · = um∈ SHm(Ω ) ∩ L∞

loc(Ω ) theBorel measure

Hm(u) = (ddcu)m∧ βn−m,

is well defined and is called the complex Hessian of u

2.3 Similarly to in pluripotential theory now we recall a class of m-subharmonic functionsintroduced and investigated in [6] recently

Definition 3 A m-subharmonic function u ∈ SHm(Ω ) is called m-maximal if every v ∈

SH (Ω ), v ≤ u outside a compact subset of Ω implies that v ≤ u on Ω

By MSHm(Ω ) we denote the set of m-maximal functions on Ω Theorem 3.6 in [6] claimsthat a locally bounded m-subharmonic function u on a bounded domain Ω ⊂ Cnbelongs toMSHm(Ω ) if and only if it solves the homogeneous Hessian equation Hm(u) = (ddcu)m∧

βn−m= 0

2.4 Next, we recall the classesE0

m(Ω ) andFm(Ω ) introduced and investigated in [13] First

we give the following

Let Ω be a bounded domain in Cn Ω is said to be m-hyperconvex if there exists a uous m-subharmonic function u : Ω −→ R−such that Ωc= {u < c} b Ω for every c < 0

contin-As above, every plurisubharmonic function is m-subharmonic with m ≥ 1 then every convex domain in Cnis m-hyperconvex Let Ω ⊂ Cnbe a m-hyperconvex domain Set

Em=Em(Ω ) =u ∈ SH−

m(Ω ) : ∀ U b Ω , ∃ v ∈ Fm(Ω ), v = u on U 2.5 We recall the notion of m-capacity introduced in [13]

Definition 4 Let E ⊂ Ω be a Borel subset The m-capacity of E with respect to Ω is definedby

Cm(E) = Cm(E, Ω ) = sup

nZ

(ddcu)m∧ βn−m: u ∈ SHm(Ω ), −1 ≤ u ≤ 0

o

Proposition 2.10 in [13] gives some elementary properties of the m-capacity similar as thecapacity presented in [1] Namely, we have:

3 The classesFm,χ(Ω ),Em,χ(Ω )

From now to the end of the paper we assume that Ω is a bounded hyperconvex domain

in Cn Now we introduce two the weighted pluricomplex energy classes of m-subharmonicfunctions defined as follows

Definition 5 Let χ : R−−→ R+be a decreasing function and 1 ≤ m ≤ n We define

Fm,χ(Ω ) = {u ∈ SHm−(Ω ) : ∃ {uj} ⊂E0

m(Ω ), uj& u on Ωsup

Em(Ω ) introduced and investigated in [13]

b) In the case m = n the classFn,χ(Ω ) coincides with the class of plurisubharmonic tions with weak singularitiesE−χ(Ω ) erase early introduced and investigated in [5].c) In the case m = n and χ(t) ≡ 1 for all t < 0 the classesFn,χ(Ω ) andEn,χ(Ω ) coincidewith the classesF (Ω) and E (Ω) in [11]

func-We need the following lemma

Lemma 2 Let χ : R−−→ R+be a decreasing function such that χ(2t) ≤ aχ(t) with some

a> 1 Assume that 1 ≤ m ≤ n and u, v ∈E0

m(Ω ) Then the following hold:

Then χjis a strictly decreasing function, χ < χj< χ +1j and χj(2t) ≤ max(a, 2).χj(t) forevery t < 0 Moreover, since {v < −t} ⊂ {u < −t} for every t > 0 so by Lemma 1 we have

(ddcu)m∧ βn−m

In general case we set Φj(t) = min(χ(t); − jt) Then Φjare decreasing functions such that

Φj(0) = 0 and Φj% χ on (−∞, 0) By first case, we have

Proof a) It suffices to prove that the conclusion holds for the classFm,χ(Ω ) Assume that

u∈Fm,χ(Ω ) and u ≤ v, v ∈ SHm−(Ω ) From the Definition 5, there exists a sequence {uj} ⊂

E0

m(Ω ), uj& u on Ω with

supj

b) First, we prove that if u ∈Fm,χ(Ω ) then αu ∈Fm,χ(Ω ) Indeed, let k ∈ N∗ with

2k> α and let {uj} ⊂E0

m(Ω ), uj& u on Ω withsup

j

Z

Ω

χ (uj)(ddcuj)m∧ βn−m< ∞

It is clear that {αuj} ⊂E0

m(Ω ), αuj& αu on Ω Moreover, since χ(αuj) ≤ χ(2kuj) ≤

Hence, the desired conclusion follows

Proposition 3 Let χ : R−−→ R+ be a decreasing function such that χ(2t) ≤ aχ(t) forall t< 0 with some a > 1 Then for every u ∈Fm,χ(Ω ), there exists a sequence {uj} ⊂

E0

m(Ω ) ∩C (Ω) such that uj& u and

supj

j→ ∞ and u ≤ wkso there exists j0such that vj0≤k−1

k wkon Ωk Therefore, vj0≤ ukon Ω Lemma 2 implies that

Hm(u) = (ddcu)m∧ βn−mis well defined as a positive Radon measure on Ω

Proof Without loss of generality we can assume that χ(t) > 0 for every t < 0 Let u ∈

Em,χ(Ω ) and z0∈ Ω Take a neighbourhood ω b Ω of z0 and a sequence {uj} ⊂E0

m(Ω )such that sup

ω

u1< 0, uj& u on ω and

supj

supj

supj

Z

Ω

Hm(uej) < ∞,

and it follows thatue∈Fm(Ω ) It is easy to see thateu= u on ω and this yields that u ∈

Em(Ω ) Theorem 3.14 in [13] implies that Hm(u) is a positive Radon measure on Ω Theproof is complete

Now we prove our main result about the local property of the classEm,χ(Ω )

4 The local property of the classEm,χ(Ω )

First we give the following definition which is similar as in [17] for plurisubharmonic tions

func-Definition 6 A classJ (Ω) ⊂ SH−

m(Ω ) is said to be a local class if ϕ ∈J (Ω) then ϕ ∈

J (D) for all hyperconvex domains D b Ω and if ϕ ∈ SH−

a) If χ1, χ2∈K and a1, a2≥ 0 then a1χ1+ a2χ2∈K

b) If χ1, χ2∈K then χ1.χ2∈K

c) If χ ∈K then χp∈K for all p > 0

d) If χ ∈K then (−t)χ(t) ∈ K More generally |tk|χ(t) ∈K for all k = 0,1,2, Proof The proof is standard hence we omit it

Remark 2 If χ ∈K then χ(2t) ≤ aχ(t),∀t < 0 with some a > 1 Indeed, by hypothesis

t χ0(t) ≤ Cχ(t),C = constant > 0 We set s(t) =(−t)χ (t)C Then s0(t) ≥ 0, ∀t < 0, hence s(t) is

an increasing function This implies that s(2t) ≤ s(t) and we have χ(2t) ≤ 2C

χ (t)

The following result is necessary for the proof of the local property of the classEm,χ(Ω ).

Lemma 3 Let u, v ∈ SH−m(Ω ) ∩ L∞(Ω ) with u ≤ v on Ω , χ ∈K and T = ddc

On the other hand

ddc(Φ χ(u)) ∧ T ≥ −Aχ(u)ddckzk2∧ T + Φ χ0(u)ddcu∧ T + Φ χ00(u)du ∧ dcu∧ T

− χ0(u)(udΦ ∧ dcΦ +1

In the case χ00(u) ≤ 0, we have the following

vddc(Φ χ(u)) ∧ T ≤ −Auχ(u)ddckzk2∧ T + uχ0(u)ddcu∧ T

+ u min{χ00(u), 0}du ∧ dcu∧ T − u2

χ0(u)dΦ ∧ dcΦ ∧ T

− χ0(u)du ∧ dcu∧ T

In the case χ00(u) ≥ 0, from (1) and we note that Φvχ00(u)du ∧ dcu∧ T ≤ 0 and it is easy toobtain above estimates Now, we have the following estimates

On other hand, by hypothesis about the classK we have uχ0(u) ≤ c1χ (u) and (−u2)χ0(u) ≤

c1(−u)χ(u), uχ00(u) ≤ c2(−χ0(u)) Therefore

0 , ψ|Ω000= 1, supp ψ b Ω00, then we have

The next lemma is a crucial tool for the proof of the local property of the classEm,χ(Ω ).

Lemma 4 Let Ω be a hyperconvex domain in Cnand1 ≤ m ≤ n Assume that u ∈E0

m(Ω )and χ ∈K such that χ00(t) ≥ 0, ∀t < 0 Then for Ω0b Ω there exists a constant C = C(Ω0)such that the following holds:

χk−1(x)dx From the hypothesis

χ ∈K then χ(2t) ≤ aχ(t) and it is easy to check that χk∈K and χ(t)(−t)k χk(t)

χ (t)(−t)k

Now, choose R > 0 large enough such that kzk2≤ R2 on Ω Let ϕ ∈E0

m(Ω ) and A > 0such that kzk2− R2≥ Aϕ on Ω0 Set h = max(kzk2− R2; Aϕ) then h ∈E0

By integration by parts we have

supj≥1

Z

χ (uj)(ddcuj)m∧ βn−m< +∞

By dominated convergence theorem and (ddcuj)m−p∧ (ddckzk2)n−m+p is weakly gent to (ddcu)m−p∧ (ddckzk2)n−m+pin the sense of currents

We also need a following result on subextension for the classFm,χ(Ω )

Lemma 5 Assume that Ωb eΩ and u ∈Fm,χ(Ω ) Then there exists aue∈Fm,χ( eΩ ) suchthateu≤ u on Ω

Proof We split the proof into three steps

Step 1.We prove that if v ∈C (Ω ), v ≤ 0, suppve b eΩ thenev:= sup{w ∈ SHm−( eΩ ) : w ≤

von suppv Since ϕ ≤evsove∈E0

m( eΩ ) Moreover, by Proposition3.2 in [6] we haveve∈C (Ω ) Let w ∈ SHe m({ev< v}) such that w ≤veoutside a compactsubset K of {ve< v} Set

w1=

(max(w,v)e on {ev< v}

Step 2.Next, we prove that if u ∈E0

m(Ω ) ∩C (Ω) then there existsue∈E0

m( eΩ ), (ddcu)em∧

βn−m= 0 on ( eΩ \Ω ) ∪ ({ue< u} ∩ Ω ) and (ddc

eu)m∧ βn−m≤ (ddcu)m∧ βn−m on {ue=u} ∩ Ω Indeed, set

set in {ue= u} ∩ Ω Then for ε > 0 we have K b {eu+ ε > u} ∩ Ω so we have

=

Z

K

1{e u+ε>u}(ddcmax(ue+ ε, u))m∧ βn−m

≤

Z

K(ddcmax(eu+ ε, u))m∧ βn−m,where the equality in the second line follows by using the same arguments as in [2]( alsosee the proof of Theorem 3.23 in [13]) However, max(ue+ ε, u) & u on Ω as ε → 0 so by[24] it follows that (ddcmax(ue+ ε, u))m∧ βn−mis weakly convergent to (ddcu)m∧ βn−mas

ε → 0 On the other hand, 1Kis upper semicontinuous on Ω so we can approximate 1Kwith

a decreasing sequence of continuous functions ϕj Hence, we infer that

as j → ∞ This yields that (ddc

eu)m∧ βn−m≤ (ddcu)m∧ βn−mon {eu= u} ∩ Ω Step 3.Now, let uj∈E0

m(Ω ) ∩C (Ω) such that uj& u andsup