Table of Contents
International Journal of Partial Differential Equations
Volume 2015, Article ID 947819, 8 pages
http://dx.doi.org/10.1155/2015/947819
Research Article

Weighted Pluricomplex Energy II

Université de Montréal, Pavillon 3744, Rue Jean-Brillant, Montréal, QC, Canada H3C 3J7

Received 31 August 2014; Revised 29 December 2014; Accepted 9 January 2015

Academic Editor: Antonin Novotny

Copyright © 2015 Slimane Benelkourchi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We continue our study of the complex Monge-Ampère operator on the weighted pluricomplex energy classes. We give more characterizations of the range of the classes by the complex Monge-Ampère operator. In particular, we prove that a nonnegative Borel measure is the Monge-Ampère of a unique function if and only if . Then we show that if for some then for some , where is given boundary data. If moreover the nonnegative Borel measure is suitably dominated by the Monge-Ampère capacity, we establish a priori estimates on the capacity of sublevel sets of the solutions. As a 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.

1. Introduction

Let be a bounded hyperconvex domain, that is, a connected, bounded open subset of , such that there exists a negative plurisubharmonic function such that , for all . Such a function is called an exhaustion function. We let denote the cone of plurisubharmonic functions (psh for short) on and let 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, [26] and reference therein. In [7], Cegrell introduced the pluricomplex energy classes and () on which the complex Monge-Ampère operator is well defined. He proved that a measure is the Monge-Ampère of some function if and only if it satisfies where is the cone of all bounded psh functions defined on the domain with finite total Monge-Ampère mass and , for every . Recently, Åhag et al. in [8] proved that, in the case , inequality (1) is equivalent to . In this note, our first objective is to extend this result by showing that, for all positive number , inequality (1) is equivalent to . In fact, we prove some more general result. Given a nondecreasing function , we consider the set of plurisubharmonic functions of finite -weighted Monge-Ampère energy and, in some sense, has boundary values zero. These are the functions for which there exists a decreasing sequence with limit and

Then we have the following characterization of the image of the complex Monge-Ampère acting in the class .

Theorem 1. Let be an increasing convex or homogeneous function such that . The following assertions are equivalent:(1)there exists a unique function such that ;(2).

Next, we extend our previous result to families of functions having prescribed boundary data. Let be a maximal psh function. We define the class to be the class of psh functions such that there exists a function such that Some particular cases of the classes have been studied in [6, 7, 916].

More precisely, we prove the following result.

Theorem 2. Let be a nonnegative measure in , let be an increasing convex or homogeneous function such that , and let be a maximal function. Then, if for some then there exists a unique function 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 and satisfies Write where is nondecreasing.

Such measures dominated by the Monge-Ampère capacity have been extensively studied by Kołodziej in [35]. He proved that if is a continuous function and , then is the Monge-Ampère of a unique function with .

When , we have the following estimate.

Theorem 3. Let be a positive finite measure. Assume, for all compact subsets , Then there exists a unique function such that , and Here is the reciprocal function of .
In particular if then

The paper is organised as follows. In Section 2, we recall the definitions of the energy classes 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 .

2. Energy Classes with Zero Boundary Data

Let us recall some Cegrell’s classes (Cf. [7, 13, 14]). The class is the set of plurisubharmonic functions such that, for all , there exists a neighbourhood of and , a decreasing sequence which converges towards in and satisfies . Cegrell [13] has shown that the operator is well defined on , continuous under decreasing limits, and the class is stable under taking maximum; that is, if and then . This class is the largest class with these properties (Theorem 4.5 in [13]). The class has been further characterized by Błocki [17, 18] and Le Mau et al. in [19].

The class is the global version of : a function belongs to if and only if there exists a decreasing sequence converging towards in all of  , which satisfies . The class has been further characterized in [12, 17].

Let be an increasing sequence of strictly pseudoconvex domains such that . Let be a given psh function and put Then we have and is an increasing sequence. Let . It follows from the properties of that . Note that the definition of is independent of the choice of the sequence and is maximal; that is, . is the smallest maximal psh function above . Define . In fact, this class is the analogous of potentials for subharmonic functions.

Definition 4. Let be a nondecreasing function. We let denote the set of all functions for which there exists a sequence decreasing to in and satisfying

It was proved in [15, 20] that if then In particular, for any function , the complex Monge-Ampère operator is well defined as nonnegative measure. Furthermore, if for all , then The class 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 , a compact subset, its Monge-Ampère capacity relatively to is defined by The following estimates (cf [12]) will be useful later on. For any

Let be a nondecreasing function. Without loss of generality, from now on, we assume that . We define the class

Proposition 5. One has , while Moreover, if is convex, then Here denote the real number satisfying , for all , and , for all .

Proof. Compare with [12, 20].

Theorem 6. Let be an increasing convex function such that . The following conditions are equivalent:(1)there exists a unique function such that ;(2); (3)there exists a constant such that (4)there exists a constant such that (5)there exists a locally bounded function such that and Here denotes the class and .

The equivalences are proved in [11] (Theorem 5.1) and the implication is proved in [12] (Theorem 5.2). For the sake of completeness we include a complete proof.

Proof. We start by the implication . Let . It follows from Proposition 5 that . Hence Now, for the implication , assume that (3) is not satisfied. Then for each we can find a function such that Consider the function Observe that Hence Now, since the weight is convex or homogeneous and using the estimates (14), we get Hence . On the other hand, from (22) we have which yields a contradiction.
Now, we prove that . Let , denote . If , that is, , then If the function defined by Indeed, from the monotonicity of , we have It follows from (18) and the convexity of that Hence we get (19).
For the implication , we consider .
. It follows from [12] (Theorem 4.5) that the class characterizes pluripolar sets in the sense that if is a locally pluripolar subset of then , for some . Then assumption (20) on implies that it vanishes on pluripolar sets. It follows from [13] that there exists a function and such that .
Consider . 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 such that The comparison principle shows that is a decreasing sequence. Set . It follows from (20) that Hence This implies that Then and therefore .
We conclude now by continuity of the complex Monge-Ampère operator along decreasing sequences that . The uniqueness of follows from the comparison principle.

3. The Weighted Energy Class with Boundary Values

Let be a nondecreasing function and let be a maximal psh function. We define the class (resp., , , , and ) to be the class of psh functions such that there exists a function (resp., ) such that Later on, we will use repeatedly the following well known comparison principle from [1] as well as its generalizations to the class (cf. [10, 14]).

Theorem 7 (see [1, 10, 14]). Let be a maximal function and be such that vanishes on all pluripolar sets in . Then Furthermore if then .

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.

Lemma 8. Let be a nondecreasing function such that , for all , and a maximal function. Then for all

Proof. Fix . Let be a compact subset. Then where is the relative extremal function of the compact and . It follows from Theorem 7 that Taking the supremum over all ’s yields the first inequality.

Proposition 9. Let be an increasing function. Then one has In particular, if , then for all and .

Proof. Let . Then there exists a function such that . Therefore . It follows from Lemma 8

Theorem 10. Let be an increasing function which satisfies and a maximal function. Then if there exists a decreasing sequence such that then and .
Conversely, if and then there exists sequence decreasing towards such that

Proof. Assume that the sequence (if necessary, we approximate by a continuous sequence ). For a fixed , let denote the function defined by We claim that(1);(2); (3) on the subset .Then, it follows from statements (1), (2), and (3) that for each Therefore Hence, the function satisfies .
For the converse implication, fix . Then there exists a function such that . Let be a decreasing sequence with limit function . Then for each , consider the function . The sequence decreases towards and where is a constant which depends only on and the proof of the theorem is completed.

Theorem 11. Let be a nonnegative measure in , let be an increasing convex (or homogeneous) function such that , and let be a maximal function. Then there exists a unique function such that if and only if satisfies one of the conditions of Theorem 6.

Proof. Assume that for some . Let be a fundamental sequence of strictly pseudoconvex subsets of . Choose a sequence decreasing towards on and is maximal on . It follows from [13] that there exist a function and a function such that Consider the measure , where denotes the characteristic function of the set . Now, solving the Dirichlet problem in the strictly pseudoconvex domain , we state that there exist functions such that By the comparison principle, we have and are decreasing sequences and Letting we get that . The continuity of the complex Monge-Ampère operator under monotonic sequences yields that . Uniqueness of follows from the comparison principle.

Corollary 12. Let be nonnegative measure in with total finite mass and let be a maximal function. Then there exists a uniquely determined function such that if and only if vanishes on pluripolar subsets.

Proof. It follows from [13] that there exist a function and a function such that By [3], there exists a unique such that . The comparison principle yields that is a decreasing sequence. Let denote by . It follows from Lemma 8 that . Therefore . By the continuity of the complex Monge-Ampère operator under decreasing sequences, we have . Now, since then there exists a convex function with and such that . By Theorem 11, we can find a unique function such that .

4. Measures Dominated by Capacity

Throughout this section, denotes a fixed nonnegative measure of finite total mass . 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 [35]. The main result of his study, achieved in [4], can be formulated as follows. Fix a continuous decreasing function and set . If for all compact subsets and is a continuous function, then for some continuous function with .

The condition means that decreases fast enough towards zero at infinity. This gives a quantitative estimate on how fast , hence decreases towards zero as .

When , it is still possible to show that for some function , but will generally be unbounded. We now measure how far it is from being so.

Theorem 13. Let be a nonnegative finite measure. Assume for all compact subsets Then there exists a unique function such that , and Here is the reciprocal function of .

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 . Hence , . Therefore

Now, we consider the case when is absolutely continuous with respect to Lebesgue measure.

Let denote a generic subspace of that is a real subspace such that , where is the usual complex structure on (cf. [21] for more details). will be endowed with the induced Euclidean structure and the corresponding Lebesgue measure which will be denoted by .

Let be a positive real number. According to [22, 23], the Orlicz space consists of -measurable functions defined on such that On the space , we define the norm The dual space to is the exponential class ; that is, the vector space equipped with the norm Then we have the following Hölder inequality: for and , where is a positive constant depending only on and . By a simple computation, we have

Corollary 14. Let be a measure with nonnegative density . Then there exists a unique bounded function such that and where only depends on , and .

Proof. We claim that there exists a constant such that Indeed, Hölder’s inequality and inequality (66) yield By [21] we have where is a constant which depends only on and .
Inequality (66) follows by combining (67) and (68).
Then we apply Theorem 13 with which yields

Conflict of Interests

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

Acknowledgment

The author wishes to thank the referee for his careful reading and for his remarks which helped to improve the exposition.

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