A MONGE-AMPÈRE NORM FOR DELTA-PLURISUBHARMONIC

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A MONGE-AMPÈRE NORM FOR DELTA-PLURISUBHARMONIC

FUNCTIONS

URBAN CEGRELL^∗and JONAS WIKLUND

Abstract

We consider differences of plurisubharmonic functions in the energy classFas a linear space, and equip this space with a norm, depending on the generalized complex Monge-Ampère operator, turning the linear space into a Banach spaceδF. Fundamental topological questions for this space is studied, and we prove thatδFis not separable. Moreover we investigate the dual space.

The study is concluded with comparison betweenδFand the space of delta-plurisubharmonic functions, with norm depending on the total variation of the Laplace mass, studied by the first author in an earlier paper [7].

1. Introduction and notations

Convex-, subharmonic-, and plurisubharmonic functions are all convex cones in some larger linear space. Given any such cone,Ksay, we can investigate the space of differences from this coneδK. Such studies are often motivated by algebraic completion of the cone, and differences of convex functions were considered by F. Riesz in as early as 1911.

δ-convex functions, ord.c.functions as they sometimes are denoted, were studied by Kiselman [15], and Cegrell [8], and have been given attention in many areas ranging from nonsmooth optimization to super-reflexive Banach spaces [13].

δ-subharmonic where first given a systematic treatise in [3].δ-plurisubharmonic functions were studied by Cegrell [7], and Kiselman [15], where the topology was defined by neighbourhood basis of the form(U∩PSH)−(U∩ PSH),U a neighbourhood of the origin inL¹loc.

In this paper we study a subset ofδ-plurisubharmonic functions. Letbe a hyperconvex domain inCⁿ, thenF()is a convex cones in the linear space L¹loc(). LetδF()denote the set of functionsu∈L¹loc()that can be written asu=u¹−u², whereui ∈F().

∗Partially supported by the Swedish Research Council, contract number 621-2002-5308.

Received September 9, 2004.

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We will define a norm, depending on the Monge-Ampère operator, for functions in this class and discuss some of the topological questions that this norm raises.

For convenience we will denote the class of negative plurisubharmonic functions on a domainbyPSH⁻(), and as in [9] we will denote the class of bounded plurisubharmonic functions with boundary value zero and finite total Monge-Ampère mass byE⁰().

For the notation of the so called energy class F() on a hyperconvex domainwe refer to the paper [10]. As for now we remind the reader that the generalized complex Monge-Ampère operator is well defined inF(), and functions fromF()has finite total Monge-Ampère mass, but that the so called “comparison principle” do not hold in general, even if it is true that if u≥vonthen

(dd^cu)ⁿ ≤

(dd^cv)ⁿ.

In almost all results in this paper the domaindoes not matter much, except for the results in Section 5, and therefore we will often suppress the reference tofrom the notation.

This paper is an expanded version of a manuscript that can be found in the second author’s doctoral thesis [19]. The authors would like to thank Alexan- der Rashkovskii and Yang Xing for valuable comments and suggestions.

2. Definition of the norm

Deﬁnition2.1. Letbe a hyperconvex set inCⁿ. Assume thatu∈δF(), then we define the norm ofuto be:

u = inf

u¹−u²=u u1,u2∈F

(dd^c(u1+u2))ⁿ ¹_n

. Note that for functions u ∈ F we have uⁿ =

(dd^cu)ⁿ. To see this chooseu2 =0 in the infimum of the definition and henceuⁿ ≤

(dd^cu)ⁿ. For an inequality in the other direction letu¹, u² ∈F be any representation ofu= u1−u2. Sinceu2 ≤ 0 we haveu ≥u1−u2+2u2 = u1+u2, and thus

(dd^cu)ⁿ≤

(dd^c(u1+u2))ⁿthus

(dd^cu)ⁿ ≤ uⁿ. The following Lemma will be used repeatedly.

Lemma2.2. Supposeu, v∈F,h∈E⁰, and thatp, q are positive natural numbers such thatp+q=n. Then

−h(dd^cu)^p∧(dd^cv)^q≤

−h(dd^cu)ⁿ ^p_n

−h(dd^cv)ⁿ ^q_n

Proof. Cf. [10].

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The following inequality is very useful when working with the Monge- Ampère operator, and will be essential for our work.

Theorem2.3 (Błocki’s inequality, [4]). Let be an open subset of Cⁿ, and leth, u, v1, . . . , v2∈ PSH ∩C(). Furthermore, supposeu≤h, and u=hclose to∂, and that−1≤vj ≤0for1≤j ≤n. Then

(h−u)ⁿdd^cv1∧ · · · ∧dd^cvn≤n!

(−vn)(dd^cu)ⁿ. Lemma2.4. If λ∈Rthenλu = |λ|u.

Proof. Letλ≥0. From the definition, we have uⁿ= inf

u1−u2=u

(dd^c(u1+u2))ⁿ

= inf

u1−u2=u

dd^c

λ

λ(u¹+u²) _n

= inf

u1−u2=u

λ⁻ⁿ(dd^c(λu1+λu2))ⁿ

=λ⁻ⁿ inf

˜ u1− ˜u2=λu

(dd^c(u¹+u²))ⁿ =λ⁻ⁿλuⁿ. Henceλu = λu.

Ifλ <0 we haveλu= −λ(−u), and the same line of reasoning as above applies.

Lemma2.5. Suppose is a hyperconvex domain inCⁿ and that u, v ∈ F(), then

(dd^c(u+v))ⁿ ≤

(dd^cu)ⁿ ¹_n

+

(dd^cv)ⁿ ¹_n_n

Proof. Takeh∈E0and let us consider the left hand side in the inequality above.

−h(dd^c(u+v))ⁿ = n j=0

n

j −h(dd^cu)^n−j ∧(dd^cv)^j

≤ n j=0

n j

−h(dd^cu)ⁿ ^n−j_n

−h(dd^cv)ⁿ ^j_n

=

−h(dd^cu)ⁿ ¹_n

+

−h(dd^cv)ⁿ ¹_n_n

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where the inequality comes from the “Hölder-inequality” in Lemma 2.2. Fix w ∈, and takeh = max(k·g,−1), whereg(z, w)is the pluricomplex Green function with pole atw, thenh∈E0andh −1 onand the Lemma follows.

Now we are in a position to prove the triangle-inequality forδF.

Corollary2.6. Supposeis a hyperconvex domain inCⁿand thatu, v∈ δF(), then

(1) u+v ≤ u + v.

Proof. Take >0, then there isui, vi ∈F such that

(dd^c(u1+u2))ⁿ 1/n

<u +

and

(dd^c(v¹+v²))ⁿ 1/n

<v +.

According to Lemma 2.5 we have u + v −2 >

(dd^c(u1+u2))ⁿ 1/n

+

(dd^c(v1+v2))ⁿ 1/n

≥

(dd^c(u1+u2+v1+v2))ⁿ 1/n

,

and furthermore, sinceu1+v1−(u2+v2)=u−v,u1+v1andu2+v2are two of the functions in the set we take infimum over we have

(dd^c(u1+u2+v1+v2))ⁿ 1/n

≥ u+v.

Henceu+v ≤ u + v.

Lemma2.7. Ifu =0, thenu=0.

Proof. Take >0. Since u = inf

u¹−u²=u u1,u2∈F

(dd^c(u¹+u²))ⁿ ¹_n

. there isu˜i ∈F such that

(dd^c(u˜1+ ˜u2))ⁿ< .

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Take a sequence{vj} ⊂ E0∩C()¯ , such thatvj ˜u1+ ˜u2asj → ∞. Let t > 0 and defineht = max{vj,−t}. According to Błocki’s inequality (Theorem 2.3) we have

n! > n!

(dd^cvj)ⁿ>

(ht −vj)ⁿdV, hence

n! >ht−vjLⁿvol().

Lettingt 0 we get

n!

vol() >vjLⁿ,

independent ofj. Thusu1+u2Lⁿ < C , and letting→0 we getuLⁿ = 0, sou=0, except for a set of measure zero, but sinceu∈δFwe haveu≡0.

A remark on other energy classes

Since other type of energy-classes, for instanceEp()also are convex cones we can form the linear spacesδEp. It is natural to try to generalize our norm to a norm for these spaces. Consider a hyperconvex domain⊂ C², and the energy classE¹(). Since

(dd^cu)²is not finite in general we have to replace it with

−u(dd^cu)². Thus the natural generalization of the norm would be to takeu∈δE1, and set

q(u)= inf

u1−u2=u u1,u2∈E1

−(u1+u2) dd^c(u1+u2)2¹₃ .

Unfortunatelyqis not a norm, since it does not satisfy the triangle inequality.

Using the energy estimate in [11], and repeating the calculations in Lemma 2.5 we only getq(u+v)≤e²^/³(q(u)+q(v)).

3. On the Topology ofδF

Theorem3.1.(δF, · )is a Banach space.

Proof. Lemmata 2.4 and 2.7, and Corollary 2.6 shows that(δF, · )is a normed vector space. It remains to show completeness.

Suppose(un)is a Cauchy sequence inδF. For each integerk there is an integernksuch thatun−um<2^−kforn, m > nk. We choose thenk’s such thatnk+1> nk.

We haveunk = un1+(un2−un1)+ · · · +(unk−un(k−1)). Sinceunj ∈δF forj =1, . . . , kwe can writeunj −unj−1 =φ_j¹−φ_j², forφ_j¹, φ_j²∈F, where

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theφ_j¹andφ_j²are chosen such that unj−unj−1 =inf

(dd^c(ϕ¹+ϕ²))ⁿ 1/n

≥

(dd^c(φ_j¹+φ_j²))ⁿ 1/n

−2^−j⁻¹. Then we have

unk =un1+(φ2¹−φ2²)+ · · · +(φ_k¹−φ_k²)

=un1+(φ2¹+ · · · +φ_k¹)−(φ2²+ · · · +φ²_k) and since_k

j=2φ_j¹∈PSH⁻()is a decreasing sequence and dd^c

k j=2

φ_j¹

_n1/n

≤ dd^c k

j=2

φ_j¹+φ_j²

_n1/n

≤ k j=2

dd^c

φ_j¹+φ_j²

_n1/n

≤ k j=2

unj −unj−1 +2^−j−¹1/n= k j=2

2^−j +2^−j−¹1/n< 1

√n

2−1. Thus_k

j=2φ_j¹is an decreasing sequence of plurisubharmonic functions with bounded total mass, and in the same way_k

j=2φ_j²is. Thereforeunkis conver- gent to someu∈δF, and since(un)is a Cauchy sequenceun→u.

Lemma3.2. F is closed in the topology ofδF.

Proof. Take any Cauchy-sequence(um)in F. Choose a suitable sparse subsequence(u_m), thenup = u0+u1−u0+ · · · +up−up−1, and by the exact same reasoning as in the proof of completeness for δF, we get that up →u∈F.

Proposition3.3.The continuous functions are not dense inδF.Further- moreδF isnotseparable.

Proof. Let us denote the Lelong number ofuatxwithν(u, x). The Lelong number at the origin is of course a linear functional on all ofδF, furthermore ν(·,0)is a continuos linear functional onδF, by Theorem 4.3 or directly by the estimate:

(2πν(u, x))ⁿ≤(dd^cu)ⁿ({x}), for functionsu∈F (see e.g. [10]).

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For all functionsu∈PSH ∩C we haveν(u,0)=0, thus log|z|can not be approximated by continuos functions in our topology.

For the second statement of the proposition, let us assume thatδF is indeed separable. Let{ui}be a dense subset ofδF. It is well known that the set where the Lelong number is positive for a given functionu, is of Lebesgue measure zero. Thus the union of the sets where the Lelong number is positive for functions from{ui}is also of Lebesgue-measure zero. Take any pointxnot in this union, i.e.ν(ui, x)=0 for allui’s, and then we see thatv(z)=

log|z|δx

cannot be approximated from functions in{ui}.

A vector spaceLoverRwith an order structure defined by a binary relation

“≤” being reflexive, transitive and anti-symmetric is called anordered vector spaceoverRif the relation satisfies:

(1) translation-invariance,x≤y ⇒x+z≤y+zfor allx, y, z∈L (2) x ≤y⇒λx ≤λyfor allx, y∈Landλ >0.

Clearly every vector space of real-valued functionsf on a parameter setXis an ordered vector space under the natural orderf ≤g iff (x)≤g(x)for all x∈X.

IfLis a topological vector space, and an ordered vector space, we say that if is anordered topological vector spaceif the positive coneC = {x|x ≥0} is closed onL. In particularδF is an ordered topological vector space since {u∈δF |u≥0}is closed on the topology ofδF.

A comprehensive treatise of ordered topological vector spaces is found in the book of Schaefer and Wolff [18].

It is natural to ask wetherδF has even more ordered structure.

Remember that avector latticeis an ordered vector spaceLoverRsuch that sup(x, y)and inf(x, y)exist for every pair(x, y)∈L×L. For a vector lattice Lset|x| = sup(x,−x). Of courseδF is a vector lattice since sup(u, v) = max(u, v)exist and the same for infimum.

Given a topological vector spaceLoverR, with a vector lattice structure, a setX⊂Lis called solid ifx ∈Xand|x| ≤ |y|imply thaty ∈X.

We callLlocally solid if it has a 0-neighbourhood base of solid sets, i.e.

the norm is compatible with the lattice structure.

UnfortunatelyδF is not locally solid. It suffices to show that the the unit ballB⊂Lis notsolid, (see e.g. [18] or [17]), and this is showed in the example below.

Example3.4. Consider the functionf (ζ )=max(log|ζ|,−1)−

max(log|ζ|,−1/2) in the unit-discD in C¹. We have log|ζ| = π, and

|f| ≤ |log|ζ||.

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Since max(log|ζ|,−1)=pµ, and max(log|ζ|,−1/2)=pν, whereµand ν are the Lebesgue measure on the circles{|ζ| = e⁻¹} and {|ζ| = e⁻¹^/²}, and therefore have disjunct support we can calculate thatf =π+π. Thus δF(D)is not locally solid. In particular:δF is not a so-called Banach lattice.

(A Banach lattice is a locally solid Banach space.) 4. The dual space

Let us denote the topological dual ofδF by(δF).

It is natural to ask which elements of the dual can be given by Borel measures.

Theorem4.1.Takeψ ∈F. Suppose.∈(δF)is given by .(u)=

dd^cu∧(dd^cψ)ⁿ⁻¹,

then. = ψⁿ⁻¹, and ifψ =0there is no Borel measure onsuch that .(u)=

u dµ.

Proof. Letu∈F. According to Lemma 2.2 we have .(u)=

dd^cu∧(dd^cψ)ⁿ⁻¹≤

(dd^cu)ⁿ ¹_n

(dd^cψ)ⁿ ⁿ⁻¹_n

. Thus.(u)≤ u · ψⁿ⁻¹. Takef ∈δF and choose anyu, v∈F such that f =u−v, then

|.(f )| = |.(u−v)| ≤ |.(u)| + |.(v)| =.(u)+.(v)

=.(u+v)≤

(dd^c(u+v))ⁿ _n¹

· ψⁿ⁻¹,

and we get that

|.(f )| ≤ inf

u−v=f;u,v∈F

(dd^c(u+v))ⁿ 1/n

· ψⁿ⁻¹= f · ψⁿ⁻¹

On the other hand, takeu= ψ⁻¹ψ. Thenu =1 and .(u)=

dd^c(ψ⁻¹ψ)∧(dd^cψ)ⁿ⁻¹= ψⁿ⁻¹.

Thus . = sup

f=1

|.(f )| = ψⁿ⁻¹.

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To see that. is not given by a Borel measure, takeu, v ∈ F such that u=vnear∂. Then

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dd^cu∧(dd^cψ)ⁿ⁻¹=

dd^cv∧(dd^cψ)ⁿ⁻¹, by “Stokes’ theorem”, and if.(u)=

u dµthen

(v−u) dµ=0. Since C0^∞ ⊂ δE0 (see Lemma 3.1, [10]) it follows thatdµ has its support on the boundary of. But then.(u)=0 for allu∈E0. Take a sequence{uj} ⊂E0

such thatuj ψ, and by continuity we get

(dd^cψ)ⁿ =0, thusψ =0.

Example4.2. Supposeq >1. Letg∈L^q(). For anyu∈F(), define T (u)=

ug dV, thenT ∈(δF).

Proof. From [12] we have for everyu∈F with

(dd^cu)ⁿ ≤1 there is a constantA, depending only onsuch that

e^−udV ≤A. Thusu∈L^p,∀p. Theorem4.3. IfT is a linear functional onδF such thatT (x)≥ 0, for allx∈F, thenT is continuous.

Proof. Take a bounded sequence{fk} ⊂ δF, such that fk < M. By construction there isxk, yk ∈Fsuch thatfk=xk−yk, andxk+yk< M+1.

We havexk = fk+yk ≤ fk + yk ≤M+ yk ≤M+ xk+yk ≤ 2M+1, where the second to last inequality follows from thatyk ≥xk+yk, thus

(dd^cyk)ⁿ≤

(dd^c(xk+yk))ⁿ,

If T is bounded on all bounded sequences {xk} ⊂ F then |T (fk)| =

|T (xk)−T (yk)| ≤ |T (xk)| + |T (yk)|, andT (fk)would be bounded as well.

Suppose T is not continuous. Then there has to be a bounded sequence {fk} ⊂δF such that{T (fk)}is not bounded. Thus there has to be a bounded sequence{xk}inF such thatT (xk) > k >0.

Now defineφ=_∞

k=1k⁻²xk. SinceFis a convex cone and{xk}is bounded φ ∈ F. Note that T (φ) = T ^p1xk

+T ^∞_p+1xk

≥ T ^p1xk , since T ≥ 0 on F. But then T (φ) ≥ _p

1k⁻¹, for all positive numbers p, i.e.

T = +∞,and we have a contradiction.

Let us recall the notion of dual cones.

Deﬁnition4.4. IfCis a cone in the topological vector spaceL, thedual coneCofCis defined to be the set

C= {T ∈L|T (u)≥0 if u∈C}.

Theorem4.5.(δF)=F−F=δF.

Proof. This follows more or less immediately from [16], (see also Lemma 1 p. 218 [18]), since one can show thatF is a so called normal cone, but to avoid

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giving the rather abstract definitions of normal cones, we give a self contained proof.

TakeT ∈(δF)and definep:F →R₊byp(u):=sup{T (v)|u≤v≤ 0}. By the linearity ofT,p(λu)= λp(u), forλ≥0, and since{φ |u+v≤ φ≤0} ⊃ {φ |u≤φ ≤0} + {φ|v≤φ≤0},p(u+v)≥p(u)+p(v)also.

Thus the setV = {(t, u)|0≤t ≤p(u)} ⊂R×δF is a convex cone.

ClearlyR×δF is a normable space. Take a sequence{uk} ⊂F such that uk →0, ask → ∞. Ifϕ∈Fanduk ≤ϕ≤0 then

(dd^cϕ)ⁿ≤

(dd^cuk)ⁿ, and henceϕ ≤ uk, thusp(uk)→0, ask → ∞by the continuity ofT. We conclude that(1,0)∈ ¯V.

SinceδF is locally convex there is a closed real hyperplaneH = {t, u)| h(t, x)= −1}, separatingV¯ and(1,0)where we can choosehsuch thath≥0 onV andh(1,0) = −1. Since(R×δF) is algebraically isomorphic with (R⊕δF), (see Theorem 4.3 p. 137, [18]) we haveh(t, u)=αt+g(u). Now h(1,0)=α= −1.

Since(0, u)∈V, for allu∈F, andg ∈(δF), we haveg(u)≥0 onF according to our choice ofH.V was chosen such that(p(u), u)∈ V, hence h(p(u), u)= −p(u)+g(u)≥ 0, and we getT (u) ≤p(u)≤ g(u). To sum up:T = g−(g−T ), whereg−T ≥ 0. Note that by Theorem 4.3, linear operators that are positive onF are continuous.

We can extend the definition of the Monge-Ampère operator to the whole of δF. Suppose u ∈ δF, then u = u1 −u2, for some u1, u2 ∈ F, and we can define(dd^cu)ⁿ = _n

j=0(−1)^j ⁿ_j

(dd^cu1)^n−j ∧(dd^cu2)^j. To see that this definition is independent of the choice of the functions fromF, suppose u=u1−u2=v1−v2, and thath∈E0. Then

hdd^c(u1−u2)∧ · · · ∧dd^c(u1−u2)

=

(u1−u2)dd^ch∧dd^c(u1−u2)∧ · · · ∧dd^c(u1−u2)

=

(v1−v2)dd^ch∧dd^c(u1−u2)∧ · · · ∧dd^c(u1−u2)

=

hdd^c(v1−v2)∧dd^c(u1−u2)∧ · · · ∧dd^c(u1−u2), and continuing iteratively we have

hdd^c(u1−u2)∧· · ·∧dd^c(u1−u2)=

hdd^c(v1−v2)∧· · ·∧dd^c(v1−v2).

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Corollary4.6.The following functionals are all continuous onδF:

• The total mass of the Monge-Ampère measure.

• Demailly’s generalized Lelong numbersν(dd^cu, ϕ)for the current dd^cu with weightϕ.

For a definition ofν(T , ϕ)—the Lelong number of the currentT with weight ϕsee [14].

5. Comparison with delta-subharmonic functions

Let us turn our attention to the class of delta-subharmonic functions in domains inCⁿ.

If we have a generating family of seminorms on a Fréchet spaceXand ifK is a closed convex cone inXwe can turnKinto a Fréchet space with topology defined by the seminorms

fj =inf{|g|j+ |h|j ; f =g−h, forgandhinK}, j ∈N, where| · |j are a generating family of seminorms onX.

Deﬁnition5.1. The setδm. Letm()be the set of positive measures that can be written asµ =5ϕ, for someϕ ∈PSH(). We denote the space of differences from this cone byδm()as usual.

Since any open domain⊂Cⁿis para-compact it suffices to define a semi- norm for any compactKand generate the topology from these seminorms.

Using the topology onδPSH, the delta-plurisubharmonic functions, defined in the introduction we have a continuity property of the Laplace operator.

Theorem 5.2.Assume that is pseudoconvex then δm() is a Fréchet space with seminorms defined by

µK =inf

Kµ1+µ2|µ=µ1−µ2, µ1, µ2∈m()

, K.

Furthermore the Laplace operator5:δPSH()!→δm()is continuos.

Proof. Cf. [7]. (By assuming thatis pseudoconvex we don’t have to deal with some homotopy intricacies.)

Deﬁnition5.3. The setδM.Let us denote the set of all positive real Borel measures onbyM(), and the signed real Borel measures asδM(). Then thetotal variation of a measureµ ∈ δM()is by Jordan’s decomposition theorem given as

|µ| =inf

µ1+µ2|µ=µ1−µ2, µ1, µ2∈M()

.

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We will viewδM()as a Banach space with norm defined by the equation above.

Let5denote the Laplacian as a map fromδF toδM. Clearly5is a linear map. Continuity of the map, however, turns out to be more subtle.

Theorem5.4. Supposeis a strict pseudoconvex domain withC^∞-smooth boundary, then the map5:δF →δM is continuous.

Proof. According to [6] the solutionϕ∈PSH()to the Dirichlet prob-

lem: (dd^cϕ)ⁿ =1 on

ϕ= −z² on∂

satisfyϕ∈C^∞()¯ . Thus it follows thatz²+ϕ ∈E0(). Direct calculation gives that

dd^cu∧(dd^cz²)ⁿ⁻¹=4ⁿ⁻¹(n−1)!5u.

Thus we have that 4ⁿ⁻¹(n−1)!

5u=

dd^cu∧(dd^cz²)ⁿ⁻¹

≤

dd^cu∧(dd^c(z²+ϕ))ⁿ⁻¹

≤

(dd^cu)ⁿ 1/n

(dd^c(z²+ϕ))ⁿ

_(n−1)/n

≤C·

(dd^cu)ⁿ 1/n

for some positive constant C. The second inequality above follows from Lemma 2.2.

Takeu ∈ δF and any > 0, then there is a choice ofu¹, u² such that u=u1−u2where

(dd^c(u1+u2))ⁿ <uⁿ+. According to the calculation above we have

5u1+5u2=

5(u1+u2)≤C·

(dd^c(u1+u2))ⁿ 1/n

< Cu+

for some constantC, not depending on. Let→0 and the theorem follows.

Unfortunately, continuity of5does not hold in general, in particular not where the boundary of the domain is “flat”, as can be seen from the following example.

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Example 5.5. Let uk = max(klog|z1|, (1/k)log|z2|). Then there is a constantc, not depending onk, such that

D²5uk ≥c·k,

but

D²(dd^c(uk))²=(2π)⁻².

Proof. Takeχ1, χ2∈C0^∞(D), whereDis the unit disc. Then

D²χ1χ25uk =

D²uk(z1, z2)5(χ1(z1)χ2(z2))

=

D²uk(z1, z2) χ2(z2)51χ1(z1)+χ1(z1)52χ2(z2)

=

D

χ2

D

uk51χ1+

D

χ1

D

uk52χ2≥

D

χ2

D

uk51χ1

=

Dχ2

Dχ151uk.

Takeχ2such thatχ2≡1 onD(1/2). Forz2fixed with|z2|<1/2 we know that 51max(klog|z1|, k⁻¹log|z2|)isktimes the (normalized) Lebesgue measure on the circle{z1 ∈ C; |z1|^k² = |z2|}. Chooseχ1, depending onk, such that χ1≡1 at least where|z1| ≤ ¹₂1/k²

. After making all these choices we have

Dχ2

Dχ151uk =

Dχ2 k

2π dz2∧dz¯2> c·k, for some constantc, independent ofk.

It is well known that

D²(dd^c(uk))²=(2π)⁻².

Remark5.6. Letbe a hyperconvex domain and take a sequence{uk}in F()such that

5ukdiverges. Exhaustwith smooth, strict pseudoconvex domains from inside, then Theorem 5.4 implies that the Laplace mass of the ukhas to be pushed out towards the boundary.

LetU(0, f )be thePerron-Bremermann functionoff, i.e. the largest locally bounded plurisubharmonic function that has boundary values at mostf. (See e.g. [9])

In his Doctoral Thesis, Åhag [1] generalized the notionFp(f )of energy classes with “boundary data”f, from [9], and introducedF(f, ). Assume

that lim

"z→ζU(0, f )=f (ζ)

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for everyζ ∈∂, then we define theF(f, )to be set of plurisubharmonic functions on such that there is aϕ ∈ F such thatU(0, f ) ≥ u ≥ ϕ + U(0, f ).

Example5.7. Let

u(z)=^∞

k=1

max(log|z1|, k⁻⁴log|z2|).

Thenu∈F(D²), but

D²5u= +∞. Furthermore takef = |z2|²−1, then f ∈C^∞(D²)and(dd^cf )²=0 but(dd^c(u+f ))²is not bounded onD².

Proof. Letuk =max(log|z1|, k⁻⁴log|z2|), then

D²(dd^c(uk))²= (2πk²)⁻².

By Lemma 2.5 we get

D²

dd^c

N k=1

uk

2

≤ N

k=1

(dd^cuk)² 1/22

= N

k=1

1 2πk²

2

≤ π² 144, thusu∈F, andu+f ∈F(f ). But we have

dd^cuk∧dd^c(|z2|²−1)=

dd^cuk∧(2i dz2∧dz¯2)=16

51uk > c, where the constantc is independent ofk, by the inequality in Example 5.5 above. Thus

dd^c

f +^N

k=1

uk

2

≥2 N

k=1

dd^cuk∧dd^c(|z2|²−1)≥N, and we get that the total mass ofu+f diverges.

To ensure that ifu∈ F(f )we have

(dd^cu)ⁿ < +∞Åhag introduced the concept ofcompliant functionsf. A continuous functionf : ∂ → R is said to be compliant if the Perron-Bremermann functionU(0, f )satisfies U(0, f )=f on the boundary and

(dd^c(U(0, f )+U(0,−f )))ⁿ<+∞. Åhag proved, using the smoothness result for the Monge-Ampère operator of Caffarelli-Kohn-Nirenberg-Spruck [6], that under the assumption thatis strict pseudoconvex and smooth, any smooth boundary function is compliant.

In relation to this Åhag [2] has posed the following problem:

Problem. Suppose is hyperconvex, f ∈ C^∞, and f = U(0, f ). If u∈F(), is

(dd^c(u+f ))ⁿ <+∞?

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According to Example 5.7 above the answer to this problem is no, not always. To see this it simply suffices to takef anduas in the example.

Since the dual of the spaceδM is well understood it would be nice to pull back(δM)to(δF ). At the moment this does not seems feasible considering the example below.

Example5.8. LetBbe the unit ball inC². The inverse Laplace-operator 5⁻¹:δM(B)!→δF(B)isnotcontinuous. Letu(z1, z2)= −(1− |z1|²)¹^/²+

|z²|. Thenu ∈PSH ∩C(B), andu= 0 on the boundary of the ball. Away from thez1-axis we have that

4∂∂u¯ =

|z1|²

(1− |z1|²)³^/² + 2 (1− |z1|²)¹^/²

dz¹∧dz¯¹+ 1

|z2|dz²∧dz¯². Thus settingr = |z1|andρ= |z2|we calculate

B

5u dV =4(2π)² 1 0

^√1−r² 0

r²

(1−r²)³^/² + 2

(1−r²)¹^/² + 1 ρ

rρ dρ dr

=4π².

In [5] Błocki pointed out that even though(dd^c(−(1−|z¹|²)¹^/²))²=(dd^c|z²|)²

= 0 we still have that

B(dd^cu)ⁿdV = +∞, since for any real number 0 <

a <1, we have

B(dd^cu)ⁿdV ≥ 1 16(2π)²

_a

0

^√1−r² 0

2−r²

(1−r²)³^/²r dρ dr

= π² 4

_a

0

2r−r³

1−r² dr = π²

8 (a²−log(1−a²)) which of course diverges asatends to one.

REFERENCES

1. Åhag, Per,The Complex Monge-Ampère Operator on Bounded Hyperconvex Domains, PhD thesis, Umeå University, 2002.

2. Åhag, Per, Personal communication, 2003.

3. Arsove, Maynard G.,Functions representable as differences of subharmonic functions, Trans.

Amer. Math. Soc. 75 (1953), 327–365.

4. Błocki, Zbigniew,Estimates for the complex Monge-Ampère operator, Bull. Polish Acad. Sci.

Math. 41(2) (1994), 151–157.

5. Błocki, Zbigniew, Personal communication, 1997.

6. Caffarelli, L., Kohn, J. J., Nirenberg, L., and Spruck, J.,The Dirichlet problem for nonlin- ear second-order elliptic equationsII,Complex Monge-Ampère, and uniformly elliptic, equations, Comm. Pure Appl. Math. 38(2) (1985), 209–252.

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7. Cegrell, Urban, Delta-plurisubharmonic functions, Math. Scand. 43(2) (1979), 343–352, 1978.

8. Cegrell, Urban,On the space of delta-convex functions and its dual, Bull. Math. Soc. Sci.

Math. R. S. Roumanie (N.S.) 22(70)(2) (1978), 133–139.

9. Cegrell, Urban,Pluricomplex energy, Acta Math. 180(2) (1998), 187–217.

10. Cegrell, Urban,The general definition of the complex Monge-Ampère operator, Ann. Inst.

Fourier (Grenoble) 54 (2004).

11. Cegrell, Urban, and Persson, Leif,An energy estimate for the complex Monge-Ampère oper- ator, Ann. Polon. Math. 67(1) (1997), 95–102.

12. Cegrell, Urban, and Zeriahi, Ahmed,Subextension of plurisubharmonic functions with boun- ded Monge-Ampère mass, C. R. Math. Acad. Sci. Paris 336(4) (2003), 305–308.

13. Cepedello Boiso, Manuel,Two characterizations of super-reflexive Banach spaces by the behaviour of differences of convex functions, J. Funct. Anal. 191(1) (2002), 1–16.

14. Demailly, Jean-Pierre,Nombres de Lelong généralisés, théorèmes d’intégralité et d’analyti- cité, Acta Math. 159(3-4) (1987), 153–169.

15. Kiselman, Christer O.,Fonctions delta-convexes, delta-sousharmoniques et delta-plurisous- harmoniques, In Séminaire Pierre Lelong (Analyse), année 1975/76, pages 93–107, Lec- ture Notes in Math. 578 (1977).

16. Krein, M., Propriétés fondamentales des ensembles coniques normaux dans l’espace de Banach, C. R. (Doklady) Acad. Sci. URSS (N.S.) 28 (1940), 13–17.

17. Meyer-Nieberg, Peter,Banach Lattices, Universitext, 1991.

18. Schaefer, H. H., and Wolff, M. P.,Topological vector spaces, Graduate Texts in Math. 3, second edition, 1999.

19. Wiklund, Jonas,Topics in Pluripotential Theory, PhD thesis, Umeå University, 2004.

MATEMATISKA INSTITUTIONEN UMEÅ UNIVERSITET

S-901 87 UMEÅ SWEDEN

E-mail:urban.cegrell@math.umu.se, jonas.wiklund@math.umu.se