Local property of a class of m subharmonic functions and applications

In the paper we introduce a new class of msubharmonic functions with finite weighted complex mHessian. We prove that this class has local property. As an application, we give a lower estimate for the log canonical threshold of plurisubharmonic functions in the class Em(Ω).

and applications

Le Mau Hai*, Nguyen Xuan Hong* and Vu Viet Hung**

*Department of Mathematics, Hanoi National University of Education,

Hanoi, Vietnam

**Department of Mathematics, Physics and Informatics,

Tay Bac University, Son La, Viet NamE-mail: mauhai@fpt.vn, xuanhongdhsp@yahoo.com

and viethungtbu@gmail.com

Abstract

In the paper we introduce a new class of m-subharmonic functions with finite weighted complex m-Hessian We prove that this class has local prop- erty As an application, we give a lower estimate for the log canonical thresh- old of plurisubharmonic functions in the class Em(Ω).

Ω = S

i∈I

Ωi then ϕ ∈ J (Ω) As well known that, the class E (Ω) introduced andinvestigated in [Ce2] is a local class Moreover, in [Ce2] Cegrell has proved the classE(Ω) is the biggest class on which the complex Monge-Amp`ere operator (ddc.)niswell defined as a Radon measure and it is continuous under decreasing sequences

An another weighted energy class Eχ(Ω) which extends the classes Ep(Ω) and

F (Ω) in [Ce1] and [Ce2] introduced and investigated recently by Benelkourchi,Guedj and Zeriahi in [BGZ] is as follows Let χ : R− −→ R+ be a decreasingfunction Then, as in [BGZ], we define

2010 Mathematics Subject Classification: 32U05, 32U15, 32U40, 32W20.

Key words and phrases: subharmonic functions, weighted energy classes of subharmonic functions, complex m-Hessian, local property, the log canonical threshold.

m-1

where E0(Ω) is the cone of bounded plurisubharmonic functions ϕ defined on Ωwith finite total Monge-Amp`ere mass and lim

z→ξϕ(z) = 0 for all ξ ∈ ∂Ω Notethat from Corollary 4.4 in [Be], it follows that if ϕ ∈ Eχ(Ω) then lim

z→ξϕ(z) = 0for all ξ ∈ ∂Ω Hence if ϕ ∈ Eχ(Ω) then ϕ /∈ Eχ(D) with D a relatively compacthyperconvex domain in Ω Thus the class Eχ(Ω) is not a ”local” one In this paper

by relying on ideas of the paper of Benelkourchi, Guedj and Zeriahi in [BGZ] and

on Cegrell’s classes of m-subharmonic functions introduced and studied recently

in [Ch] we introduce weighted energy classes of m-subharmonic functions Fm,χ(Ω)and Em,χ(Ω) However, under slight hypotheses for weights χ we achieve that theclass Fm,χ(Ω) is a convex cone (see Proposition 3.4 below) We also show that thecomplex Hessian operator Hm(u) = (ddcu)m∧ βn−m is well defined on the class

Em,χ(Ω) where β = ddckzk2denotes the canonical K¨ahler form of Cn Futhermore,

we prove that the class Em,χ(Ω) is a local class (see Theorem 4.6 in Section 4below) Next, based on a recent result of J.-P Demailly and Pham Hoang Hiep

on a lower bound of the log canonical threshold for the class eE(Ω) in [DH] weestablish a lower estimate for the log canonical threshold of plurisubharmonicfunctions in the class Em(Ω)

The paper is organized as follows Beside the introduction the paper has threesections In Section 2 we recall the definitions and results concerning to m-subharmonic functions which were introduced and investigated intensively in re-cent years by many authors, such as, in [Bl1], [DiKo], [SA] We also recall theCegrell’s classes of m-subharmonic functions Fm(Ω) and Em(Ω) introduced andstudied in [Ch] In Section 3 we introduce the new two weighted energy classes

of m-subharmonic functions Fm,χ(Ω) and Em,χ(Ω) Section 4 is devoted to provethe local property of the class Em,χ(Ω) under some extra assumptions on weights

χ To arrive this property of the class Em,χ(Ω) we need a result about sion for the class Fm,χ(Ω) (see Lemma 4.5 below) which is of interest Finally,

subexten-by relying on the local property of the class Em,χ(Ω), in section 5 we give a lowerestimate for the log canonical threshold of plurisubharmonic functions in the class

Em(Ω)

Some elements of pluripotential theory that will be used throughout the papercan be found in [BT1], [Kl], [Ko1], [Ko2] while elements of the theory of m-subharmonic functions and the complex Hessian operator can be found in [Bl1],[DiKo], [SA] Now we recall the definition of some Cegrell classes of plurisub-harmonic functions (see [Ce1] and [Ce2]), as well as, the class of m-subharmonicfunctions introduced by Blocki in [Bl1] and the classes Em0(Ω) and Fm(Ω) intro-duced and investigated by Lu Hoang Chinh in [Ch] recently Let Ω be an opensubset in Cn By β = ddckzk2 we denote the canonical K¨ahler form of Cn withthe form of volume dVn= n!1βnwhere d = ∂ + ∂ and dc= ∂−∂4i , hence, ddc= 2i∂∂.2.1 As in [Ce1] and [Ce2] we define the classes E0(Ω) and F (Ω) as follows Let Ω

be a bounded hyperconvex domain That means that Ω is a connected, boundedopen subset and there exists a negative plurisubharmonic function % such that

for all c < 0 the set Ωc= {z ∈ Ω : %(z) < c} b Ω Set

2.2 We recall the class of m-subharmonic functions introduced and investigated

in [Bl1] recently For 1 ≤ m ≤ n, we define

b

Γm= {η ∈ C(1,1) : η ∧ βn−1≥ 0, , ηm∧ βn−m≥ 0},

where C(1,1) denotes the space of (1, 1)-forms with constant coefficients

Definition 2.1 Let u be a subharmonic function on an open subset Ω ⊂ Cn u

is said to be a m-subharmonic function on Ω if for every η1, , ηm−1 in bΓm theinequality

ddcu ∧ η1∧ ∧ ηm−1∧ βn−m≥ 0,

holds in the sense of currents

By SHm(Ω) we denote the set of m-subharmonic functions on Ω while SHm−(Ω)denotes the set of negative m-subharmonic functions on Ω Before to formulatebasic properties of m-subharmonic functions, as in [Bl1], we recall the following.For λ = (λ1, , λn) ∈ Rn and 1 ≤ m ≤ n, define

Proposition 2.2 Let Ω be an open set in Cn Then we have

a) P SH(Ω) = SHn(Ω) ⊂ SHn−1(Ω) ⊂ · · · ⊂ SH1(Ω) = SH(Ω) Hence,

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

b) If u is C2 smooth then it is m-subharmonic if and only if the form ddcu ispointwise in bΓm

c) If u, v ∈ SHm(Ω) and α, β > 0 then αu + βv ∈ SHm(Ω)

h) Let u1, , up ∈ SHm(Ω) and χ : Rp → R be a convex function which

is non decreasing in each variable If χ is extended by continuity to a function[−∞, +∞)p → [−∞, ∞), then χ(u1, , up) ∈ SHm(Ω)

Example 2.3 Let u(z1, z2, z3) = 5|z1|2+4|z2|2−|z3|2 By using b) of Proposition2.2 it is easy to see that u ∈ SH2(C3) However, u is not a plurisubharmonicfunction in C3because the restriction of u on the line (0, 0, z3) is not subharmonic.Now as in [Bl1] and [DiKo] we define the complex Hessian operator of locallybounded m-subharmonic functions as follows

Definition 2.4 Assume that u1, , up ∈ SHm(Ω) ∩ L∞loc(Ω) Then the complexHessian operator Hm(u1, , up) is defined inductively by

ddcup∧ · · · ∧ ddcu1∧ βn−m= ddc(upddcup−1∧ · · · ∧ ddcu1∧ βn−m).From the definition of m-subharmonic functions and using arguments as in theproof of Theorem 2.1 in [BT1] we note that Hm(u1, , up) is a closed positivecurrent of bidegree (n − m + p, n − m + p) and this operator in continuous underdecreasing sequences of locally bounded m-subharmonic functions Hence, for

p = m, ddcu1∧ · · · ∧ ddcum∧ βn−mis a nonnegative Borel measure In particular,when u = u1 = · · · = um ∈ SHm(Ω) ∩ L∞loc(Ω) the Borel measure

Hm(u) = (ddcu)m∧ βn−m,

is well defined and is called the complex Hessian of u

2.3 Similar as in pluripotential theory now we recall a class of m-subharmonicfunctions introduced and investigated in [Bl1] recently

Definition 2.5 A m-subharmonic function u ∈ SHm(Ω) is called m-maximal ifevery v ∈ SHm(Ω), v ≤ u outside a compact subset of Ω implies that v ≤ u onΩ

By M SHm(Ω) we denote the set of m-maximal functions on Ω One of tial results in [Bl1] is Theorem 3.6 saying that a m-subharmonic function on

essen-a bounded domessen-ain Ω ⊂ Cn belongs to M SHm(Ω) if and only if it solves thehomogeneous Hessian equation Hm(u) = (ddcu)m∧ βn−m= 0

2.4 Next, we recall the classes E0

m(Ω) and Fm(Ω) introduced and investigated in[Ch] First we give the following

Let Ω be a bounded domain in Cn Ω is said to be m-hyperconvex if there exists

a continuous m-subharmonic function u : Ω −→ R− such that Ωc= {u < c} b Ωfor every c < 0 As above, every plurisubharmonic function is m-subharmonicwith m ≥ 1 then every hyperconvex domain in Cnis m-hyperconvex Let Ω ⊂ Cn

be a m-hyperconvex domain Put

where Hm(u) = (ddcu)m∧ βn−m denotes the Hessian measure of u ∈ SHm−(Ω)

∩L∞(Ω) From Theorem 3.14 in [Ch] it follows that if u ∈ Fm(Ω), the complexHessian Hm(u) = (ddcu)m∧ βn−m is well defined and is a Radon measure on Ω2.5 We recall the notion of m-capacity introduced in [Ch]

Definition 2.6 Let E ⊂ Ω be a Borel subset The m-capacity of E with respect

tmCm {ϕ < −2t}
≤ (ddcϕ) ∧ βn−m {ϕ < −t}

Proof Let v ∈ SHm(Ω), −1 < v < 0 For all t > 0 we have the following inclusion:

Em,χ(Ω) we note that in the case χ(t) ≡ 1 for all t < 0 we get the pluricomplexenergy classes Fm(Ω) and Em(Ω) introduced and investigated by Lu Hoang Chinh

in [Ch]

b) In the case m = n the class Fn,χ(Ω) coincides with the class of monic functions with weak singularities E−χ(Ω) early introduced and investigated

plurisubhar-by S.Benelkourchi, V.Guedj and A Zeriahi in [BGZ]

c) In the case m = n and χ(t) ≡ 1 for all t < 0 the classes Fn,χ(Ω) and En,χ(Ω)coincide with the classes F (Ω) and E (Ω) introduced and investigated in [Ce2]

We need the following lemma.

Lemma 3.3 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 followinghold:

j

(ddcu)m∧ βn−m

b) Since {λu + (1 − λ)v < −t} ⊂ {u < −t} ∪ {v < −t} so we have

Ω

χ(u) +1

j

(ddcu)m∧ βn−m+

Z

Ω

χ(v) +1

j

(ddcv)m∧ βn−m

It is clear that {αuj} ⊂ E0

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

Hence, the desired conclusion follows

Proposition 3.5 Let χ : R−−→ R+ be a decreasing function such that χ(2t) ≤aχ(t) for all t < 0 with some a > 1 Then for every u ∈ Fm,χ(Ω), there exists asequence {uj} ⊂ E0

m(Ω) ∩ C(Ω) such that uj & u and

uj = sup{ϕ ∈ SHm−(Ω) : ϕ ≤ j − 1

j wj on Ωj}.

It is easy to see that uj & u on Ω By Theorem 1.2.7 in [Bl3] and Proposition 3.2

in [Bl1] we get uj ∈ C(Ω) Moreover, since wj ≤ uj so uj ∈ E0

m(Ω) ∩ C(Ω) Now,since vj & u as j → ∞ and u ≤ wk so there exists j0 such that vj 0 ≤ k−1k wk on

Ωk Therefore, vj0 ≤ uk on Ω Lemma 3.3 implies that

Proof Without loss of generallity 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

Now we give the following definition which is similar as in [HHQ] for harmonic functions.

plurisub-Definition 4.1 A class J (Ω) ⊂ SHm−(Ω) is said to be a local class if ϕ ∈ J (Ω)then ϕ ∈ J (D) for all hyperconvex domain D b Ω and if ϕ ∈ SHm−(Ω), ϕ|Ω j ∈

In the sequel of the paper we will use the following notation We will write

”A B” if there exists a constant C such that A ≤ CB

Proposition 4.2 Put

K = {χ : R−−→ R+: χ is deareasing and − t2χ00(t) tχ0(t) χ(t), ∀t < 0}.Then the class K has the following properties

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 and, more general, |tk|χ(t) ∈ K for all k =

0, 1, 2,

Proof The proof is easy from direct calculations and we omit

The following result is necessary for the proof of the local property of the class

The next lemma is a crucial tool for the proof of the local property of the class

Em,χ and for lower bounds of the log canonical threshold in the class Em(Ω) insection 5

Lemma 4.4 Let Ω be a hyperconvex domain in Cn Assume that u ∈ SHm−(Ω) ∩C(Ω) and χ ∈ K with χ00(t) ≥ 0 for every t < 0 Then for every open ballB(x, r) b Ω and 1 ≤ m ≤ n the following holds:

Proof It suffices to prove lemma holds for the case u ∈ P SH−(Ω) ∩ C(Ω) For

ε > 0, put hε(z) := |z − x|2− r2− max(|z − x|2− r2, −ε) Using integration byparts we infer that

and we are done

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

Lemma 4.5 Assume that Ω b eΩ and u ∈ Fm,χ(Ω) Then there exists a eu ∈

Fm,χ(eΩ) such that u ≤ u on Ω.e

Proof We split the proof into three steps

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

w ≤ v on eΩ} ∈ Em0(eΩ) ∩ C(eΩ) and (ddcev)m∧ βn−m = 0 on {ev < v} ∩ eΩ Indeed,let ϕ ∈ Em0(eΩ) ∩ C(eΩ) such that ϕ ≤ inf

e Ω

v on eΩ\({v < v} ∩ eΩ)

Since ev is upper semicontinuous so ε = − sup

K

(ev − v) > 0 Choose δ ∈ (0, 1) suchthat −δ infΩev < ε We have (1 − δ)e ev ≤ev + ε ≤ v on K Hence, (1 − δ)ev + δw1 ≤ v

on eΩ, and, we get (1 − δ)ev + δw1 =ev Thus, w ≤ev on {v < v} ∩ ee Ω Hence, ev ism-maximal in {ev < v} ∩ eΩ By [Bl1] we get (ddc

ev)m∧ βn−m= 0 on {ev < v} ∩ eΩ.Step 2 Next, we prove that if u ∈ Em0(Ω) ∩ C(Ω) then there exists u ∈ Ee m0(eΩ),(ddcu)e m∧ βn−m= 0 on (eΩ\Ω) ∪ ({eu < u} ∩ Ω) and (ddceu)m∧ βn−m≤ (ddcu)m∧

m(eΩ)∩C(eΩ) and (ddcu)e m∧βn−m= 0 on {ev < v}∩ eΩ = (eΩ\Ω)∪({u < u}∩Ω) Lete

K be a compact set in {u = u} ∩ Ω Then for ε > 0 we have K b {ee u + ε > u} ∩ Ω

=Z

K

1{

e u+ε>u}(ddcmax(eu + ε, u))m∧ βn−m

≤Z

{ uej =u j }∩Ω

χ(uj)(ddcuj)m∧ βn−m

≤Z

j→∞uej ∈ Fm,χ(eΩ) andu ≤ u on Ω The proof is complete.e

The following result deal with the local property of the class Em,χ(Ω) Namely,

we have the following:

Theorem 4.6 Let Ω be a hyperconvex domain in Cn and m be an integer with

1 ≤ m ≤ n Assume that u ∈ SHm−(Ω) and χ ∈ K such that χ(2t) ≤ aχ(t) withsome a > 1 Then the following statements are equivalent

a) u ∈ Em,χ(Ω)

b) For every K b Ω0 b Ω and every {uj} ⊂ E0

m(Ω) ∩ C(Ω) such that uj & u

a)⇒b) Let r > 0 and x1, , xN ∈ K such that K b SN

j=1

B(xj, r) b Ω0 ByLemma 4.4 we have

for every p = 0, 1, , m Put euj = sup{ϕ ∈ SHm−(U ) : ϕ ≤ uj on Ω0} ∈ E0

m(U ).Let Ω0 b Ω1b b Ωm b K Since uj ≤uej on U and (ddc

vj ∈ Fm,χ(Ω) such thatevj ≤ vj Proposition 3.4 implies thatv :=e ev1+ · · · +evs∈

Fm,χ(Ω) and, hence, max(ev, u) ∈ Fm,χ(Ω) Moreover, since max(v, u) = u on Ωe 0

so u ∈ Em,χ(Ω) The proof is complete

From the above theorem we find out an interesting property of the class Em,χ(Ω).Corollary 4.7 Assume that Ω is a hyperconvex domain and χ ∈ K satisfies allhypotheses of Theorem 4.6 Then Em,χ(Ω) ⊂ Em−1,χ(Ω)

Proof Assume that u ∈ Em,χ(Ω) Let K b Ω Take a domain Ω0 with K b Ω0 b

Ω By Theorem 4.6 there exists a sequence {uj} ⊂ E0

m(Ω)∩C(Ω) such that uj & u

Let h ∈ Em−10 (Ω) be chosen For each j > 0 take mj > 0 such that uj ≥ mjh on

Ω0 Put vj = max(uj, mjh) ∈ E0

m−1(Ω) and vj = uj on Ω0 Note that vj & u on

Ω0 and (ddcvj)p∧ βq = (ddcuj)p∧ βq on Ω0 for 1 ≤ p ≤ m and 1 ≤ q ≤ m − 1.Futhermore, we may assume that vj|Ω0 ≤ −1 Hence, |vj|m ≥ |vj|m−1 on Ω0 forall j ≥ 1 Now we have

Therefore, again by Theorem 4.6 we get u ∈ Em−1,χ(Ω) The proof is complete

threshold in the class Em(Ω) ∩ P SH(Ω)

In this section by relying on the local property of the class Em(Ω) in the previoussection and a recent result about a lower bound for the log canonical threshold ofplurisubharmonic functions in the class eE(Ω) presented in [DH] we give a lowerbound for the log canonical threshold of plurisubharmonic functions in the class

Em(Ω) ∩ P SH(Ω) First we need the following lemma

Lemma 5.1 Assume that u ∈ Em(Ω) Then for every 1 ≤ p ≤ m − 1 and forevery K b Ω we have

B(0,r 2 )

(|z|2− r2

2)(ddcv)m∧ βn−m

≥ −r2 2