Weighted Pluricomplex energy II

We continue our study of the Complex Monge-Amp\`ere Operator on the Weighted Pluricomplex energy classes. We give more characterizations of the range of the classes $\mathcal E_ \chi$ by the Complex Monge-Amp\`ere Operator. In particular, we prove that a non-negative Borel measure $\mu $ is the Monge-Amp\`ere of a unique function $\varphi \in \mathcal E_\chi$ if and only if $\chi(\mathcal E_\chi ) \subset L^1(d\mu ).$ Then we show that if $\mu = (dd^c \varphi )^n $ for some $\varphi \in \mathcal E_\chi $ then $\mu = (dd^c u )^n $ for some $u \in \mathcal E_\chi (f) $ where $f$ is a given boundary data. If moreover, the non-negative Borel measure$\mu $ is suitably dominated by the Monge-Amp\`ere capacity, we establish a priori estimates on the capacity of sub-level sets of the solutions. As consequence, we give a priori bounds of the solution of the Dirichlet problem in the case when the measure has a density in some Orlicz space.


Introduction
Let Ω ⊂ C be a bounded hyperconvex domain, that is, a connected, bounded open subset of C , such that there exists a negative plurisubharmonic function such that { ∈ Ω; ( ) < − } ⋐ Ω, for all > 0. Such a function is called an exhaustion function. We let PSH(Ω) denote the cone of plurisubharmonic functions (psh for short) on Ω and let PSH − (Ω) denote the subclass of negative functions.
As known (see [1,2]), the complex Monge-Ampère operator ( ⋅ ) is well defined, as a nonnegative measure, on the set of locally bounded plurisubharmonic functions. Therefore the question of describing the measures which are the Monge-Ampère of bounded psh functions is very important for pluripotential theory, complex dynamic, and complex geometry. This problem has been studied extensively by various authors; see, for example, [2][3][4][5][6] and reference therein. In [7], Cegrell introduced the pluricomplex energy classes E (Ω) and F (Ω) ( ≥ 1) on which the complex Monge-Ampère operator is well defined. He proved that a measure is the Monge-Ampère of some function ∈ E (Ω) if and only if it satisfies where E 0 (Ω) is the cone of all bounded psh functions defined on the domain Ω with finite total Monge-Ampère mass and lim → ( ) = 0, for every ∈ Ω. Recently,Åhag et al. in [8] proved that, in the case = 1, inequality (1) is equivalent to E 1 (Ω) ⊂ 1 ( ). In this note, our first objective is to extend this result by showing that, for all positive number , inequality (1) is equivalent to E (Ω) ⊂ ( ). In fact, we prove some more general result. Given a nondecreasing function : R − → R − , we consider the set E (Ω) of plurisubharmonic functions of finite -weighted Monge-Ampère energy and, in some sense, has boundary values zero. These are the functions ∈ PSH(Ω) for which there exists a decreasing sequence ∈ E 0 (Ω) with limit and Then we have the following characterization of the image of the complex Monge-Ampère acting in the class E (Ω). (1) there exists a unique function ∈ E (Ω) such that = ( ) ; (2) (E (Ω)) ⊂ 1 ( ).

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International Journal of Partial Differential Equations Next, we extend our previous result to families of functions having prescribed boundary data. Let ∈ PSH(Ω) be a maximal psh function. We define the class E ( ) to be the class of psh functions such that there exists a function ∈ E (Ω) such that Some particular cases of the classes E ( ) have been studied in [6,7,[9][10][11][12][13][14][15][16].
More precisely, we prove the following result.

Theorem 2.
Let be a nonnegative measure in Ω, let : R − → R − be an increasing convex or homogeneous function such that (−∞) = −∞, and let be a maximal function. Then, if = ( ) for some ∈ E (Ω) then there exists a unique function ∈ E ( ) such that = ( ) .
Moreover, when the nonnegative measure is dominated by the Monge-Ampère capacity, we give an estimate of the growth of solutions of the equation ( ) = . As in [12], let us consider the function Then := is a nondecreasing function on R + and satisfies ( ) ≤ (cap ( )) , ∀Borel subsets ⊂ Ω.
In particular if ∫ The paper is organised as follows. In Section 2, we recall the definitions of the energy classes E (Ω) and some classes of psh functions introduced by Cegrell [7,13,14] and we prove Theorem 1. In Section 3, we prove Theorem 2. As a consequence, we generalize the main result in the paper [9]. In Section 4, we prove Theorem 3. As application, we give a priori bound of the solution of Dirichlet problem in the case when the measure = , where belongs to some Orlicz space log .

Energy Classes with Zero Boundary Data E
Let us recall some Cegrell's classes (Cf. [7,13,14]). The class E(Ω) is the set of plurisubharmonic functions such that, for all 0 ∈ Ω, there exists a neighbourhood 0 of 0 and ∈ E 0 (Ω), a decreasing sequence which converges towards in 0 and satisfies sup ∫ Ω ( ) < +∞. Cegrell [13] has shown that the operator ( ⋅ ) is well defined on E(Ω), continuous under decreasing limits, and the class E(Ω) is stable under taking maximum; that is, if ∈ E(Ω) and V ∈ PSH − (Ω) then max( , V) ∈ E(Ω). This class is the largest class with these properties (Theorem 4.5 in [13]). The class E(Ω) has been further characterized by Błocki [17,18] and Le Mau et al. in [19].
The class F(Ω) is the global version of E(Ω): a function belongs to F(Ω) if and only if there exists a decreasing sequence ∈ E 0 (Ω) converging towards in all of Ω, which satisfies sup ∫ Ω ( ) < +∞. The class E(Ω) has been further characterized in [12,17].
Let Ω ⋐ Ω be an increasing sequence of strictly pseudoconvex domains such that Ω = ⋃ Ω . Let ∈ E(Ω) be a given psh function and put Then we have Ω ∈ E(Ω) and Ω is an increasing sequence. Let̃:= (lim Ω ) * . It follows from the properties of E(Ω) that̃∈ E(Ω). Note that the definition of̃is independent of the choice of the sequence Ω and is maximal; that is, (̃) = 0.̃is the smallest maximal psh function above . Define N(Ω) := { ∈ E(Ω);̃= 0}. In fact, this class is the analogous of potentials for subharmonic functions.
We let E (Ω) denote the set of all functions ∈ PSH(Ω) for which there exists a sequence ∈ E 0 (Ω) decreasing to in Ω and satisfying It was proved in [15,20] In particular, for any function ∈ E (Ω), the complex Monge-Ampère operator ( ) is well defined as nonnegative measure. Furthermore, if (− ) < 0 for all > 0, then The class E (Ω) has been characterized by the speed of decrease of the capacity of sublevel sets [11,12].
Recall that the Monge-Ampère capacity has been introduced and studied by Bedford and Taylor in [1]. Given ⊂ Ω, a compact subset, its Monge-Ampère capacity relatively to Ω is defined by The following estimates (cf [12]) will be useful later on. For Let : R − → R − be a nondecreasing function. Without loss of generality, from now on, we assume that (0) = 0. We define the classÊ (Ω) Here denote the real number satisfying ( ) < 0, for all < − , and ( ) = 0, for all ≥ − .
Proof. We start by the implication (1) ⇒ (2). Let , ∈ E (Ω). It follows from Proposition 5 that + ∈ E ( ). Hence Now, for the implication (2) ⇒ (3), assume that (3) is not satisfied. Then for each ∈ N we can find a function Consider the function Observe that Hence Now, since the weight is convex or homogeneous and using the estimates (14), we get 4 International Journal of Partial Differential Equations Hence ∈ E (Ω). On the other hand, from (22) we have which yields a contradiction. Now, we prove that (3) ⇒ (4). Let ∈ E 0 (Ω), denote If ( ) > 1 the functioñdefined bỹ Indeed, from the monotonicity of , we have It follows from (18) and the convexity of that Hence we get (19). For the implication (4) ⇒ (5), we consider ( ) = max(1, 1/ ).
(5) ⇒ (1). It follows from [12] (Theorem 4.5) that the class E (Ω) characterizes pluripolar sets in the sense that if is a locally pluripolar subset of Ω then ⊂ {V = −∞}, for some V ∈ E (Ω). Then assumption (20) on implies that it vanishes on pluripolar sets. It follows from [13] that there exists a function ∈ E 0 (Ω) and ∈ 1 loc (( ) ) such that = ( ) . Consider := min( , )( ) . This is a finite measure which is bounded from above by the complex Monge-Ampère measure of a bounded function. It follows therefore from [3] that there exist ∈ E 0 (Ω) such that The comparison principle shows that is a decreasing sequence. Set = lim → ∞ . It follows from (20) that This implies that Then ̸ ≡ −∞ and therefore ∈ E (Ω). We conclude now by continuity of the complex Monge-Ampère operator along decreasing sequences that ( ) = . The uniqueness of follows from the comparison principle.

The Weighted Energy Class with Boundary Values
Let : R − → R − be a nondecreasing function and let ∈ M(Ω) be a maximal psh function. We define the class E ( ) (resp., N( ), F( ), N ( ), and F ( )) to be the class of psh functions such that there exists a function ∈ E (Ω) (resp., N, F, N , F ) such that Later on, we will use repeatedly the following well known comparison principle from [1] as well as its generalizations to the class N( ) (cf. [10,14]).
The following lemma, which gives an estimate of the size of sublevel set in terms of the mass of Monge-Ampère measure, will be useful shortly.
International Journal of Partial Differential Equations 5 where * is the relative extremal function of the compact and V := − + * . It follows from Theorem 7 that Taking the supremum over all 's yields the first inequality.
Then, it follows from statements (1), (2), and (3) that for each Hence, the function := lim → ∞ satisfies + ≤ ≤ . For the converse implication, fix ∈ E ( ). Then there exists a function ∈ E such that + ≤ ≤ . Let ∈ E 0 ∩ (Ω) be a decreasing sequence with limit function . Then for each ∈ N, consider the function := max( + , ) ∈ E ( ). The sequence decreases towards and where is a constant which depends only on and the proof of the theorem is completed. Proof. Assume that = ( V) for some V ∈ E . Let (Ω ) be a fundamental sequence of strictly pseudoconvex subsets of Ω. Choose a sequence ∈ PSH(Ω) ∩ (Ω) decreasing towards on Ω and is maximal on Ω +1 . It follows from [13] that there exist a function ∈ E 0 and a function ∈ International Journal of Partial Differential Equations Consider the measure = ⊩ Ω min( , )( ) , where ⊩ Ω denotes the characteristic function of the set Ω . Now, solving the Dirichlet problem in the strictly pseudoconvex domain Ω , we state that there exist functions , V ∈ PSH(Ω ) ∩ (Ω ) such that By the comparison principle, we have and V are decreasing sequences and Letting → +∞ we get that := lim → ∞ ∈ E ( ). The continuity of the complex Monge-Ampère operator under monotonic sequences yields that ( ) = . Uniqueness of follows from the comparison principle. Proof. It follows from [13] that there exist a function ∈ E 0 and a function ∈ 1 loc ( ) such that By [3], there exists a unique ℎ ∈ E 0 such that ( ℎ ) = min( , )( ) . The comparison principle yields that ℎ is a decreasing sequence. Let denote by ℎ := lim → ∞ ℎ . It follows from Lemma 8 that ℎ ̸ ≡ −∞. Therefore ℎ ∈ F . By the continuity of the complex Monge-Ampère operator under decreasing sequences, we have ( ℎ) = . Now, since then there exists a convex function : R − → R − with (0) ̸ = 0 and (−∞) = −∞ such that ℎ ∈ E . By Theorem 11, we can find a unique function ∈ E ( ) ⊂ F ( ) such that ( ) = .

Measures Dominated by Capacity
Throughout this section, denotes a fixed nonnegative measure of finite total mass (Ω) < +∞. We want to solve the Dirichlet problem and measure how far the distance between the solution and the given boundary data is from being bounded, by assuming that is suitable dominated by the Monge-Ampère capacity.
Measures dominated by the Monge-Ampère capacity have been extensively studied by Kołodziej in [3][4][5]. The main result of his study, achieved in [4], can be formulated as follows. and : Ω → R is a continuous function, then = ( ) for some continuous function ∈ PSH(Ω) with | Ω = .
When ∫ +∞ 0 ( ) = +∞, it is still possible to show that = ( ) for some function ∈ F(Ω), but will generally be unbounded. We now measure how far it is from being so.
Then there exists a unique function ∈ F ( ) such that = ( ) , and Here −1 is the reciprocal function of ( ) = ∫ 0 ( ) The proof is almost the same as that of Theorem 5.1 in [12], except that we use Corollary 12 for the existence of the solution and Lemma 8 to estimate the capacity of sublevel set.
Observe that if ∫ ∞ ( ) < ∞ then is bounded by ∫ Now, we consider the case when = is absolutely continuous with respect to Lebesgue measure.
Let G ⊂ C denote a generic subspace of C that is a real subspace such that G + G = C , where is the usual complex structure on C (cf. [21] for more details). G will be endowed with the induced Euclidean structure and the corresponding Lebesgue measure which will be denoted by G .
International Journal of Partial Differential Equations 7 The dual space to log + is the exponential class Exp 1/ + ; that is, the vector space Then we have the following Hölder inequality: for ∈ log + and ∈ Exp 1/ + , where , > 0 is a positive constant depending only on and . By a simple computation, we have ‖1‖ Exp 1/ + ( ) = 1 log + (1 + 1/ G ( )) .
where > 0 is a constant which depends only on Ω and G.

Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.