WEIGHTED PLURICOMPLEX ENERGY II
SLIMANE BENELKOURCHI
1. Introduction
Let Ω ⊂ Cn be a bounded hyperconvex domain, i.e. a connected, bounded
open subset of Cn such that there exists a negative plurisubharmonic function ρ such that {z ∈ Ω; ρ(z) < −c} b Ω, ∀c > 0. Such a function ρ is
called an exhaustion function. We let P SH(Ω) denote the cone of plurisubharmonic functions (psh for short) on Ω and P SH − (Ω) denote the subclass
of negative functions.
Given a nondecreasing function χ : R− → R− , we consider the set Eχ =
(Eχ (Ω)) of plurisubharmonic functions of finite χ-weighted Monge-Ampère
energy and, in a sense, have boundary values zero. These are the functions
u ∈ P SH(Ω) such that there exists a decreasing sequence uj ∈ E0 (Ω) with
limit u and
∫
sup −χ ◦ uj (ddc uj )n < +∞.
j∈N
Ω
Here E0 (Ω) is the cone of all bounded plurisubharmonic functions φ defined
on the domain Ω with finite total Monge-Ampère mass and limz→ζ φ(z) = 0,
for every ζ ∈ ∂Ω.
Let f ∈ P SH(Ω) be a maximal psh function. We define the class Eχ (f )
to be the class of psh functions u such that there exists a function φ ∈ Eχ (Ω)
such that
φ(z) + f (z) ≤ u(z) ≤ f (z),
∀z ∈ Ω.
Some particular cases of the classes Eχ (f ) has been studied in [1], [2],
[6], [7], [12], [13], [14]. One purpose of this paper is to characterize those
measures for which the complex Monge-Ampère equation
(1.1)
(ddc u)n = dµ
has a solution in such a family. More precisely, we prove the following result.
Theorem A. Let µ be a nonnegative measure in Ω, χ : R− → R− be
an increasing convex function such that χ(−∞) = −∞ and f be a bounded
maximal function. Then, if µ = (ddc u)n for some u ∈ Eχ then there exists a
unique function φ ∈ Eχ (f ) such that µ = (ddc φ)n .
Moreover, if µ has a finite total mass: µ(Ω) < ∞. Then there exists a
unique function φ ∈ F a (f ) such that µ = (ddc φ)n if and only if µ vanishes
on all pluripolar sets of Ω.
The second aim concerns the growth of solutions of the equation (1.1)
where nonnegative measures are dominated by the Monge-Ampère capacity.
2000 Mathematics Subject Classification. 32W20, 32U05, 32U15.
Key words and phrases. Complex Monge-Ampère operator,
functions.
1
plurisubharmonic
2
SLIMANE BENELKOURCHI
Given a nonnegative finite Borel measure µ in Ω. We set
Fµ (t) := sup{µ(K), K b Ω Compact; CapΩ (K) ≤ t}, ∀t ≥ 0.
Observe that F := Fµ is a nondecreasing function on R+ which satisfies
µ(K) ≤ F (CapΩ (K)) , for all Borel subsets K ⊂ Ω.
)n
(
Write F (x) = x ε(− ln nx ) where ε : R+ → [0, +∞[ is nondecreasing. Our
second main result is:
Theorem B. Let µ be a nonnegative measure of finite total mass. Assume
for all compact subsets K ⊂ Ω,
(1.2)
(1.3)
µ(K) ≤ Fε (CapΩ (K)) .
Then there exists a unique function φ ∈ F a (f ) such that µ = (ddc φ)n , and
CapΩ ({φ < f − s}) ≤ exp(−nH −1 (s)), for all s > 0,
∫x
Here H −1 is the reciprocal function of H(x) = e 0 ε(t)dt + s0 (µ).
In particular φ ∈ Eχ (f ) with −χ(−t) = exp(nH −1 (t)/2).
For similar results in the case when f ≡ 0 or the compact Kähler manifolds, we refer the reader to [6], [8], [7], [15], [16].
2. Energy classes Eχ
Let recall some U.Cegrell’s classes (Cf. [12], [13] and [14]). The class E(Ω)
is the set of plurisubharmonic functions u such that for all z0 ∈ Ω, there
exists a neighborhood Vz0 of z0 and uj ∈ E0 (Ω)∫ a decreasing sequence which
converges towards u in Vz0 and satisfies supj Ω (ddc uj )n < +∞. U.Cegrell
[13] has shown that the operator (ddc ·)n is well defined on E(Ω), continuous
under decreasing limits and the class E(Ω) is stable under taking maximum
i.e. if u ∈ E(Ω) and v ∈ P SH − (Ω) then max(u, v) ∈ E(Ω). E(Ω) is the
largest class with these properties (Theorem 4.5 in [13]). The class E(Ω) has
been further characterized by Z.Blocki [10], [11].
The class F(Ω) is the “global version” of E(Ω): a function u belongs to
F(Ω) iff there exists a decreasing sequence
uj ∈ E0 (Ω) converging towards
∫
u in all of Ω, which satisfies supj Ω (ddc uj )n < +∞. The class E(Ω) has
been further characterized in [6] and [7].
Let Ωj b Ω be an increasing sequence of strictly pseudoconvex domains
such that Ω = ∪j Ωj . Let u ∈ E(Ω) be a given psh function and put
uΩj := sup {φ ∈ P SH(Ω); φ ≤ u on Ω \ Ωj } .
Then we have uΩj ∈ E(Ω) and uΩj is an increasing sequence. Let ũ :=
(limj uΩj )∗ . It follows from the properties of E(Ω) that ũ ∈ E(Ω). Note that
the definition of ũ is independent of the choice of the sequence Ωj and is
maximal i.e. (ddc ũ)n = 0. ũ is the smallest maximal psh function above u.
Define N (Ω) := {u ∈ E(Ω); ũ = 0}. In fact, this class is the analogous of
potentials for subharmonic functions.
Definition 2.1. Let χ : R− → R− be a nondecreasing function. We let
Eχ (Ω) denote the set of all functions u ∈ P SH(Ω) for which there exists a
sequence uj ∈ E0 (Ω) decreasing to u in Ω and satisfying
∫
sup −χ(uj ) (ddc uj )n < ∞.
j∈N
Ω
WEIGHTED PLURICOMPLEX ENERGY
3
It was proved in [5] that if χ ̸≡ 0. Then
Eχ (Ω) ⊂ E(Ω).
In particular, for any function u ∈ Eχ (Ω), the complex Monge-Ampère operator (ddc u)n is well defined.
Moreover, if χ(−t) < 0 ∀ t > 0, then
∫
Eχ (Ω) = {u ∈ N (Ω);
−χ(u)(ddc u)n < +∞}.
Ω
It was shown in [6], [7], that the classes Eχ (Ω) can be characterized by the
speed of decrease of the capacity of sublevel sets.
Recall that the Monge-Ampère capacity has been introduced and studied
by E.Bedford and A.Taylor in [4]. Given K ⊂ Ω a Borel subset, its MongeAmpère capacity relatively to Ω is defined by
{∫
}
c n
CapΩ (K) := sup
(dd u) ; u ∈ P SH(Ω), −1 ≤ u ≤ 0 .
K
Let χ : R− → R− be a nondecreasing function. We define the class Êχ (Ω)
}
{
∫ +∞
tn χ′(−t)CapΩ ({φ < −t})dt < +∞ .
Êχ (Ω) := φ ∈ P SH − (Ω) /
tχ
Proposition 2.2. We have Êχ (Ω) ⊂ Eχ (Ω), while
Eχ (Ω) ⊂ Êχ̂ (Ω), where χ̂(t) = χ(t/2).
Moreover, if χ :] − ∞, − tχ [→ R− is convex. Then
Eχ (Ω) = Êχ (Ω).
Proof. Cf. [6], [7].
3. (ddc ·)n (Eχ )
Theorem 3.1. Let χ : R− → R− be an increasing convex function such
that χ(−∞) = −∞. The following assertions are equivalent:
(1) there exists a unique function φ ∈ Eχ (Ω) such that µ = (ddc φ)n ;
(2) there exists a constant C1 > 0 such that
∫
(3.1)
−χ(u)dµ ≤ C1 , ∀ u ∈ E˜0 (Ω);
Ω
(3) there exists a constant C2 > 0 such that
∫
(3.2)
( (∫
)1)
n
, ∀ u ∈ E0 (Ω);
−χ(u)dµ ≤ C2 max 1,
−χ ◦ u (ddc u)n
Ω
Ω
(4) there exists a locally bounded function F : R+ → R+ such that
lim supt→+∞ F (t)/t < 1 and
∫
(3.3)
−χ(u)dµ ≤ F (Eχ (u)) , ∀ u ∈ E0 (Ω).
Ω
4
SLIMANE BENELKOURCHI
Here E˜0 (Ω) := {u ∈ E0 (Ω);
denotes the χ-energy of u.
∫
Ω −χ(u)(dd
c u)n
≤ 1} and Eχ (u) :=
∫
c n
Ω −χ(u) (dd u)
The proof of (1) ⇔ (3) ⇔ (4) can be found in [6]. The proof of (1) ⇔ (2)
has been done in [7]. For he convenience of the reader, we include here a
complete proof.
Proof. We start by the implication (1) ⇒ (2). Let u, φ ∈ Eχ (Ω). Observe
that for any s > 0, we have
s
s
(u < −s) ⊂ (u < φ − ) ∪ (φ < − ).
2
2
Hence
∫
∫ ∞
∫
(3.4)
−χ(u)(ddc φ)n =
−χ′ (−s)
(ddc φ)n ds
Ω
∫
≤
∞
′
0
∫
c
χ (−s)
0
∞
∫
≤2
(u<−s)
∞
′
∫
n
(dd φ) ds +
(u<φ− 2s )
χ′ (−2s)
0
∫
(φ<− 2s )
0
∫
∫
(ddc φ)n ds + 2
∞
χ′ (−2s)
0
(u<φ−s)
(ddc φ)n ds
χ (−s)
∫
(ddc φ)n ds.
(φ<−s)
The convexity of χ yields that
χ′ (−2s) ≤ M χ′ (−s), ∀s > 0.
(3.5)
It follows by the comparison principle that, for all s > 0
∫
∫
∫
(3.6)
(ddc φ)n ≤
(ddc u)n ≤
(ddc u)n .
(u<φ−s)
(u<φ−s)
(u<−s)
Together (3.4), (3.5) and (3.6) imply that there exists a constant C independent of u such that
(∫
)
∫
∫
c n
c n
c n
−χ(u)(dd φ) ≤ C
−χ(u)(dd u) +
−χ(φ)(dd φ)
< +∞, ∀u ∈ E˜0 (Ω).
Ω
Ω
Ω
(
)
∫
c φ)n .
To get the estimate (3.1), it’s enough to consider C1 = C 1 + Ω −χ(φ)(dd
∫
Now, we prove that (3) ⇒ (4). Let ψ ∈ E0 (Ω), denote Eχ (ψ) := Ω −χ(ψ)(ddc ψ)n .
If ψ ∈ E˜0 (Ω), i.e. Eχ (ψ) ≤ 1 then
∫
−χ(ψ)dµ ≤ C1 .
Ω
If Eχ (ψ) > 1. The function ψ̃ defined by
ψ̃ :=
ψ
∈ E˜0 (Ω).
Eχ (ψ)1/n
Indeed, from the monotonicity of χ, we have
∫
∫
ψ
ψ
1
c
n
−χ(
)(dd
) ≤
−χ(ψ)(ddc ψ)n = 1.
Eχ (ψ) Ω
Eχ (ψ)1/n
Eχ (ψ)1/n
Ω
It follows from (3.1) and the convexity of χ
∫
∫
ψ
1/n
−χ(ψ)dµ ≤ Eχ (ψ)
−χ(
)dµ
Eχ (ψ)1/n
Ω
Ω
Hence we get (3.2) with C2 = max(1, C1 ).
≤
C1 .Eχ (ψ)1/n .
WEIGHTED PLURICOMPLEX ENERGY
5
For the implication (3) ⇒ (4), we consider F (t) = C2 max(1, t1/n ).
(4) ⇒ (1). It follows from [7] that the class Eχ (Ω) characterizes pluripolar
sets. Then the assumption (3.3) on µ implies in particular that it vanishes
on pluripolar(sets. It )follows from [13] that there exists a function u ∈ E0 (Ω)
and f ∈ L1loc (ddc u)n such that µ = f (ddc u)n .
Consider µj := min(f, j)(ddc u)n . This is a finite measure which is bounded
from above by the complex Monge-Ampère measure of a bounded function.
It follows therefore from [17] that there exist φj ∈ E0 (Ω) such that
(ddc φj )n = min(f, j)(ddc u)n .
The comparison principle shows that φj is a decreasing sequence. Set φ =
limj→∞ φj . It follows from (3.3) that
(∫
)
∫
c
n
c
n
−χ(φj )(dd φj ) ≤ F
−χ(φj )(dd φj ) .
Ω
Ω
∫
Hence
−χ(φj )(ddc φj )n < ∞.
sup
j
Ω
So it follows from Proposition ?? that
∫ +∞
sup
tn χ′ (−t)CapΩ ({φj < −t})dt < +∞,
j
0
which implies that
∫
+∞
tn χ′ (−t)CapΩ ({φ < −t})dt < +∞.
0
Then φ ̸≡ −∞ and therefore φ ∈ Eχ (Ω).
We conclude now by continuity of the complex Monge-Ampère operator
along decreasing sequences that (ddc φ)n = µ. The unicity of φ follows from
the comparison principle.
4. The weighted energy class with boundary data
Let χ : R− → R− be a nondecreasing function and let f ∈ M(Ω) be
a maximal psh function. We define the class Eχ (f ) to be the class of psh
functions u such that there exists a function φ ∈ Eχ (Ω) such that
φ(z) + f (z) ≤ u(z) ≤ f (z),
∀z ∈ Ω.
In the same way, we define N (f ) as the set of psh functions u such that
there exists a function φ ∈ N (Ω) such that
φ(z) + f (z) ≤ u(z) ≤ f (z),
∀z ∈ Ω.
Theorem 4.1. Assume f is bounded maximal function, u ∈ N a (f ) and
v ∈ E(Ω) such that v ≤ f − ε. Then
∫
∫
(ddc v)n ≤
(ddc u)n .
(u<v)
Proof. See [14], [2].
(u<v)
Corollary 4.2. Let u ∈ N a (f ) and v ∈ E(Ω) such that v ≤ f and (ddc u)n ≤
(ddc v)n . Then u ≥ v.
In particular, if (ddc u)n = (ddc v)n with u, v ∈ N a (f ) then u = v.
6
SLIMANE BENELKOURCHI
The following lemma, which gives an estimate of the size of sublevel set
in terms of the mass of Monge-Ampère, will be useful later on.
Lemma 4.3. Let χ : R− → R− be a nondecreasing function such that χ ̸≡ 0
and φ ∈ Eχ (f ). Then for all s > 0 and t > 0,
∫
(4.1)
t CapΩ (φ < −s − t + f ) ≤
n
(ddc φ)n .
(φ<−s+f )
Proof. Fix s, t > 0. Let K ⊂ {φ < f − s − t} be a compact subset. Then
∫
∫
c ∗ n
CapΩ (K) =
(dd uK ) =
(ddc u∗K )n
Ω
{φ<f −s−t}
∫
∫
1
c ∗ n
(dd uK ) ≤ n
(ddc v)n ,
=
∗
t
{φ<f −s+tuK }
{φ<v}
where u∗K is the relative extremal function of the compact K and v :=
f − s + tu∗K . It follows from Theorem 4.1 (see also Theorem 4.3. in [14]) that
∫
∫
1
1
c n
(dd v) = n
(ddc max(φ, v))n ≤
tn {φ<v}
t {φ<max(φ,v)}
∫
∫
∫
1
1
1
n
c n
c
(dd φ) = n
(dd φ)) ≤ n
(ddc φ))n .
n
t {φ<max(φ,v)}
t {φ<f −s+tuK }
t {φ<f −s}
Taking the supremum over all K’s yields the inequality.
Proposition 4.4. Let χ : R− → R− be a increasing function. Then we
have
{
}
∫ +∞
n ′
Eχ (f ) ⊂ u ∈ P SH(Ω); u ≤ f and
s χ (−s)CapΩ (u < f − 2s)ds < +∞ .
0
In particular, if χ(t) < 0 ∀t < 0, then CapΩ (u < f − s) < +∞ for all s > 0
and u ∈ Eχ (f ).
Proof. Let u ∈ Eχ (f ). Then there exists a function φ ∈ Eχ (Ω) such that
φ + f ≤ u. Therefore (u < f − s) ⊂ (φ < −s). It follows from Lemma 4.3
∫ +∞
∫ +∞
sn χ′ (−s)CapΩ (u < f − 2s)ds ≤
sn χ′ (−s)CapΩ (φ < −2s)ds
0
0
∫ +∞
∫
∫
′
c n
≤
χ (−s)
(dd φ) ds =
−χ(φ)(ddc φ)n < ∞.
0
(φ<−s)
Ω
Theorem 4.5. Let µ be a nonnegative measure in Ω, χ : R− → R− be
an increasing convex function such that χ(−∞) = −∞ and f ∈ M∞ be a
bounded maximal function. Then there exists a unique function φ ∈ Eχ (f )
such that µ = (ddc φ)n if µ satisfies one of the conditions of Theorem 3.1.
Proof. Suppose that µ = (ddc v)n for some v ∈ Eχ . Let (Ωj )j∈N be a fundamental sequence of strictly pseudoconvex subsets of Ω and let fj ∈ P SH(Ωj+1 )∩
C(Ωj+1 ) a sequence of maximal functions decreasing towards f. It follows
WEIGHTED PLURICOMPLEX ENERGY
7
from [13] that there exist a function g ∈ E0 and a function θ ∈ L1loc (ddc g)n
such that
µ = g(ddc g)n .
For j ∈ N, put µj := 1Ωj min(θ, j)(ddc g)n , where 1Ωj denotes the characteristic function of the set Ωj . Now solving the Dirichlet Problem in
the strictly pseudoconvex domain Ωj , we state that there exist functions
uj , vj ∈ P SH(Ωj ) ∩ C(Ωj ) such that
(ddc vj )n = (ddc uj )n = µj
and
vj = 0,
uj = fj on ∂Ωj .
Using the comparison principle, we find that the sequences vj and uj are
decreasing and
v + f ≤ vj + fj ≤ uj ≤ fj on Ωj .
Letting j → ∞, we get that u := limj→∞ uj ∈ Eχ (f ). Finally, form the
continuity of the complex Monge-Ampère under monotone sequences, we
have (ddc u)n = µ. The function u is unique follows from the comparison
principle in the classe Eχ (f ) Corollary 4.2.
Corollary 4.6. Let µ be a nonnegative measure in Ω with finite total mass
and f ∈ M∞ be a bounded maximal function. Then there exists a uniquely
determined function φ ∈ F a (f ) suth that (ddc φ)n = µ if and only if µ
vanishes on all pluripolar sets.
Proof. It follows from [13] that there is a function ψ ∈ E0 and a nonnegative
function f ∈ L1 ((ddc ψ)n ) such that µ = f (ddc ψ)n . By [17], there is a unique
function gj ∈ E0 such that (ddc gj )n = min(j, f )(ddc ψ)n . The comparison
principle for bounded functions (see [4]) implies that gj is decreasing sequence. We put g := limj→+∞ gj . It follows from Lemma 4.3 that g ̸≡ −∞.
Therefore g ∈ F. By the continuity of the complex Monge-Ampère operator
under decreasing sequences, we have (ddc g)n = µ. Now, since (cf. [7])
∪
Fa =
Eχ ,
χ convex; χ(0) ̸= 0
χ(−∞) = −∞
then there exists a convex function χ : R− → R− with χ(0) ̸= 0 and
χ(−∞) = −∞ such that g ∈ Eχ . Now using Theorem 4.5, we can find a
function φ ∈ Eχ (f ) ⊂ F a (f ) such that (ddc φ)n = µ. The uniqueness follows
from Theorem 4.1.
5. Mesures dominated by Capacity
Throughout this section, µ denotes a fixed positive Borel measure of finite
total mass µ(Ω) < +∞. We want to solve the Dirichlet problem
(ddc φ)n = µ, with φ ∈ F(f ) and φ|∂Ω = f,
and measure how far the (unique) solution φ is from being bounded, by
assuming that µ is suitable dominated by the Monge-Ampère capacity.
Measures dominated by the Monge-Ampère capacity have been extensively studied by S.Kolodziej in [K 1,2,3]. The main result of his study,
8
SLIMANE BENELKOURCHI
achieved in [K 2], can be formulated as follows. Fix ε : R → [0, ∞[ a continuous decreasing function and set Fε (x) := x[ε(− ln x/n)]n . If for all compact
subsets K ⊂ Ω,
∫ +∞
µ(K) ≤ Fε (CapΩ (K)), and
ε(t)dt < +∞,
then µ = (ddc φ)n for some continuous function φ ∈ P SH(Ω) with φ|∂Ω = 0.
∫ +∞
The condition
ε(t)dt < +∞ means that ε decreases fast enough
towards zero at infinity. This gives a quantitative estimate on how fast
ε(− ln Cap
(K)/n), hence µ(K), decreases towards zero as CapΩ (K) → 0.
∫ Ω+∞
When
ε(t)dt = +∞, it is still possible to show that µ = (ddc φ)n
for some function φ ∈ F (f ), but φ will generally be unbounded. We now
measure how far it is from being so:
Theorem 5.1. Let µ be a nonnegative measure of finite total mass. Assume
for all compact subsets K ⊂ Ω,
(5.1)
µ(K) ≤ Fε (CapΩ (K)) .
Then there exists a unique function φ ∈ F(f ) such that µ = (ddc φ)n , and
CapΩ ({φ < f − s}) ≤ exp(−nH −1 (s)), for all s > 0,
∫x
Here H −1 is the reciprocal function of H(x) = e 0 ε(t)dt + s0 (µ).
In particular φ ∈ Eχ (f ) with −χ(−t) = exp(nH −1 (t)/2).
Proof. It follows from Corollary 4.6 that there exists a function φ ∈ F (f )
such that
µ = (ddc φ)n .
Set
1
α(s) := − log CapΩ ({φ < f − s}), ∀s > 0.
n
The function α is increasing and α(+∞) = +∞, since CapΩ vanishes on
pluripolar sets. It follows from Lemma 4.3 and (5.1) that for all s > 0 and
t > 0,
tn CapΩ (φ < f − s − t) ≤ µ(φ < f − s) ≤ Fε (CapΩ ({φ < f − s})) .
Therefore
(5.2)
log t − log ε ◦ α(s) + α(s) ≤ α(s + t).
We define an increasing sequence (sj )j∈N by induction. Setting
sj+1 = sj + eε ◦ α(sj ), for all j ∈ N.
The choice of s0 . We choose s0 ≥ 0 large enough so that α(s0 ) ≥ 0. We
must insure that s0 = s0 (µ) can chosen to be independent of φ. It follows
from Lemma 4.3
µ(Ω)
CapΩ ({φ < f − s}) ≤ n , ∀s > 0
s
hence α(s) ≥ log s − 1/n log µ(Ω). Therefore α(s0 ) ≥ 0 if s0 ≥ µ(Ω)1/n .
The growth of sj . We can now apply (5.2) and get α(sj ) ≥ j + α(s0 ) ≥ j.
Thus limj α(sj ) = +∞. There are two cases to be considered.
WEIGHTED PLURICOMPLEX ENERGY
9
First case: s∞ = lim sj ∈ R+ , then α(s) ≡ +∞ for s > s∞ , i.e. CapΩ (φ <
f − s) = 0, ∀s > s∞ . Therefore φ is bounded from below by f − s∞ , in
particular φ ∈ Eχ (f ) for all χ.
Assume now ( second case) that sj → +∞. For each s > 0, there exists
N = Ns ∈ N such that sN ≤ s < sN +1 . We can estimate s 7→ Ns ,
s ≤ sN +1 =
N
∑
(sj+1 − sj ) + s0 =
0
≤ e
N
∑
∫
ε(j) + s0 ≤ e
N
∑
e ε ◦ α(sj ) + s0
0
N
ε(t)dt + s̃0 =: H(N ),
0
0
where s̃0 = s0 + e.ε(0). Therefore H −1 (s) ≤ N ≤ α(sN ) ≤ α(s), hence
CapΩ (φ < f − s) ≤ exp(−nH −1 (s)).
Set now g(t) = −χ(−t) = exp(nH −1 (t)/2). Then
∫ +∞
tn g ′ (t)CapΩ (φ < f − t)dt
0
∫
1
n +∞ n
t
exp(−nH −1 (t)/2)dt
≤
−1
2 0
ε(H (t)) + s0
∫ +∞
≤C
(t + 1)n exp(n(α − 1)t)dt < +∞.
0
This shows that φ ∈ Eχ (f ) where χ(t) = − exp(nH −1 (−t)/2).
Let µ > 0 a nonnegative measure and α > 0. The Orlicz space LLog α L(dµ)
consists of those µ−measurable functions f defined on Ω such that
∫
|f |
|f |
logα (1 +
)dµ < ∞ for some λ > 0.
λ
Ω λ
On the space LLog α L(dµ), we define the norm
{
}
∫
|f |
|f |
α
log (1 +
)dµ < 1 .
||f ||LLog1/α L := inf λ > 0;
λ
Ω λ
The dual space to LLog α L(dµ), is the exponential class ExpL1/α : that is
the vector space of µ−measurable functions
(( ) 1 )
}
{
∫
α
|f
|
exp
dµ < ∞
ExpL1/α := f : Ω → R; ∃λ > 0 :
λ
Ω
equipped with the norm
{
||f ||ExpL1/α := inf
(( ) 1 )
)
}
∫ (
|f | α
λ > 0;
exp
− 1 dµ < 1 .
λ
Ω
Then we have the following Hölder inequality
∫
≤ Cn,α ||f ||LLogα L ||g||
(5.3)
f
gdµ
ExpL1/α
Ω
10
SLIMANE BENELKOURCHI
for f ∈ LLog α L and g ∈ ExpL1/α , where Cn,α > 0 is a positive constant
depending only in n and α. Then
1
(5.4)
||1||ExpL1/α (K) =
.
log α (1 + 1/µ(K))
Corollary 5.2. Let µ = f βn be a measure with density 0 ≤ f ∈ L logα L,
where α > n and βn is the Lebesgue measure. Then there exists a unique
bounded function φ ∈ F (f ) ∩ L∞ (Ω) such that (ddc φ)n = µ and
1
n
0 ≤ f − φ ≤ C||f ||LLog
αL,
where C = C(n, α, Ω) > 0 only depends on n, α and Ω.
Proof. We claim that there exists a constant C > 0 such that
(
)1
1
n
α
n
µ(K) ≤ C ||f ||LLogα L
(CapΩ (K)) n , for all Borel sets K ⊂ Ω.
Indeed, Hölder’s inequality (5.3) and the equality (5.4) yield
µ(K) ≤ Cn,α ||f ||LLogα L
1
.
log α (1 + 1/βn (K))
Thus it suffices to estimate the volume βn (K). Combining the Polya’s inequalities which estimate the volume in terms of Alexander-Taylor capacity
(Cf [9]) and the comparison of the two capacities (see [3], we infer
1
log α (1 + 1/βn (K))
1
≤ Cn,α ||f ||LLogα L α
log (1 + 1/βn (K))
≤
µ(K) ≤ Cn,α ||f ||LLogα L
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Laboratoire Emile Picard UMR 5580, Université Paul Sabatier Toulouse 3,
118 route de Narbonne, 31062 TOULOUSE Cedex 09 (FRANCE)
E-mail address: benel@math.ups-tlse.fr