Monge–Ampère measures associated with semi-exhaustive functions

In this paper, we study the current T∧ddcψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T \wedge \text {{dd}}^{c}\psi $$\end{document} for positive currents T and semi-exhaustive, not necessarily plurisubharmonic, functions ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi $$\end{document}. The study leads to new definitions of capacity and Lelong–Demailly numbers with respect to the weight ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi $$\end{document}.

(1.1) A. K. Abdulaali (B) The Department of Mathematics and Statistics, College of Science, King Faisal University, 380, 31982 Al-Ahsaa, Saudi Arabia E-mail: aalabdulaaly@kfu.edu.sa Moreover, Theorem 3.8 shows that the previous formula remains true when T is positive (or negative) plurisubharmonic and ψ is plurisubharmonic of class C 1 . These results generalize some classical conclusions of [2,5,8]. As a consequence of these formulas, one can obtain the Lelong-Demailly number ν(T, ψ) with respect to the weight ψ for positive plurisubharmonic current T and plurisubharmonic function ψ of class C 1 .
The second part is devoted to study the Monge-Ampère measure T ∧ dd c ψ. Namely, the contribution of this section is stated as follows.
Theorem. (Theorem 4.1) Let T be a positive current. If ψ is of class C 1 and d c ψ ∧ T is well defined on S ψ (r ) for all 0 < r < R ψ . Then we have (1.2) If, in addition, T is plurisuperharmonic, then The above inequalities make possible to introduce different capacities, each originating from a different source.

Preliminaries and notations
Let D p,q ( , k) be the space of C k compactly supported differential forms of bi-degree ( p, q) on . A form ϕ ∈ D p, p ( , k) is said to be strongly positive form if ϕ can be written as where γ j ≥ 0 and α s, j ∈ D 1,0 ( , k). Then, D p, p ( , k) admits a basis consisting of strongly positive forms. The dual space D p,q ( , k) is the space of currents of bi-dimension ( p, q) or bi-degree (n − p, n − q) and of order k. If T ∈ D p, p ( , k), then it can be written as where the coefficients T I,J are distributions on . If these coefficients are measures, then T is called of order zero. Remember that when T and dd c T are of order zero, then T is called C-normal. The current T ∈ D p, p ( , k) is said to be positive if T, ϕ ≥ 0 for all forms ϕ ∈ D p, p ( , k) that are strongly positive. For such currents T , the mass is denoted by T and defined by |T I,J |, where |T I,J | are the total variations of the measures T I,J . Let β = dd c |z| 2 be the Kähler form on C n ( where d = ∂ + ∂ and d c = i(−∂ + ∂), thus dd c = 2i∂∂), then for each open subset 1 ⊂ , there exists a constant C > 0 depends only on n and p such that

Lelong-Jensen Formula
We start this section with some basic facts that will be used frequently in this paper.
where j * t : S(t) → is the canonical injection.

Lemma 3.3
Let ϕ be a function of class C 1 . If T and γ are two C 1 -form of bi-degree (n − p, n − p) and Proof Take a test form ϕ in and let (u j ) j∈N be a sequence of smooth functions converges in C 1 ( ) to u. Then, Hence, by a simple computation, one can deduce that This shows that lim j→∞ dd c u j ∧ T exists as the right-hand side terms of the previous equality are convergent.
Lemma 3.5 Let u 1 , ..., u q , 1 ≤ q ≤ p be plurisubharmonic functions of class C 1 on . If T is positive (or negative) plurisubharmonic, then the current T ∧ dd c u 1 ∧ · · · ∧ dd c u q is well defined.
Proof By the precedent lemma, T ∧ dd c u j is well defined for all j ∈ {1, . . . , q}. Now, the result is induced by induction and the fact that each T ∧ dd c u j is positive (or negative) plurisubharmonic.
Theorem 3.6 (See [6]) Let T be an (n − p, n − p)-form of class C 2 on . Then for all 0 < r 1 < r 2 < R ψ , we have Proof By Stokes' theorem, we have Now, (3.3) follows by applying Lemma 3.2.
Notice that the previous formula is obtained without constraint on dT as required in [8] and [6].
Proof We first assume that T of class C 2 . Then by the previous lemma, one has Thus, the result is verified for C 2 currents T by combining the latter equalities. Now, for C-normal currents T , set E T = {r ∈ R, ||T || S(r ) * ||dd c T || S(r ) = 0}. By the assumptions of T and dd c T , it is clear that ||T || K and ||dd c T || K are bounded for all compact subset K of . Hence, the set E T is countable. Consider a regularization ρ ε . Then for all t ∈ R \ E T , we have 2 ) j∈N increasing to r 2 so that r ( j) k ∈ R \ E T . The result is achieved by taking the limits. This result generalizes the formulas in [2] to the case of C 1 functions.

Theorem 3.8 If T is positive (or negative) plurisubharmonic current and ψ is plurisubharmonic and of class
Proof By regularizing ψ, one can assume that ψ is smooth. Now the result follows by applying, first, Theorem 3.7 and, second, Lemma 3.5.

Capacity related to semi-exhaustive functions
In this section, we study the current dd c ψ ∧ T . From now on, we relax the classification of ψ to C 1 .

Theorem 4.1 If T is positive and d c ψ ∧ T is well defined on S ψ (r ) for all
If, in addition, T is plurisuperharmonic, then Proof Notice first that dψ∧d c ψ∧T is a positive current. Hence, the function f (r ) = B ψ (r ) dψ ∧d c ψ ∧T ∧β p−1 is non decreasing. So, f (r ) ≥ 0. But Now, assume that dd c T ≤ 0. By Stokes' formula, we have This shows that

Remark 4.2
If T is C-normal on , then the current d c ψ ∧ T is well defined on S ψ (r ). Indeed, the wedge product dd c ψ ∧ T is achieved by Lemma 3.4. Hence, we set As shown above, semi-exhaustive functions have things in common with plurisubharmonic functions. Despite this, we must be cautious once we deal with these semi-exhaustive functions as some important properties of Psh are not applicable to this type of functions. For example, if ψ is plurisubharmonic, then it is This fact is not valid when the plurisubharmonicity is omitted. The following example shows this.

Example 4.3
In C, set = B(0, 1) and put T = 1. Now, take ψ(z) = sin ( π 2 |z| 2 ). Clearly, ψ is an semiexhaustive function on where R ψ = 1. By a simple computation, we have Notice that dd c ψ tends to π 2 β when |z| → 0, while dd c ψ tends to −( π 2 ) 2 β as |z| → 1 − . Let us recall a very fundamental fact about currents. When g is a locally bounded plurisubharmonic function on and T is positive and closed, the current gT is well defined. The exterior derivatives lead to the current dd c g ∧ T as it is defined by dd c (gT ).

Proposition 4.4 Let T be a positive closed current of bi-dimension ( p, p) on and g be a locally bounded plurisubharmonic function. Then,
If g is positive, then Proof First, we show that the quantity dψ ∧ d c g ∧ T ∧ β p−1 is non-negative and increasing in r . By Now, assume that g is positive. Then the current gT is C-normal. Hence by Lemma 3.1 and Theorem 4.1, we have In virtue of [7], if g ∈ Psh( ) ∩ L ∞ loc ( \ K ) for some compact subset K of , then dd c g ∧ T is well defined. Therefore, by following a similar technique as in [1], the current gT in this case can be deduced. Indeed, take neighborhoods V and W so that K V W ⊂ , and χ ∈ C ∞ 0 (W ) such that χ = 1 on V . Now, construct a decreasing sequence of smooth plurisubharmonic functions (g j ) converges point-wise to g on . Then, This implies that, The previous discussion yields to the fulfillment of Proposition 4.4 in the case of unbounded functions g.
Clearly, the set C 2 ∩ Psh( ) ⊆ P p (T, ). Moreover, the early studies of currents lead to many cases where the previous inclusion is proper. For instant, if T is positive and closed, then we already know that Psh( ) ∩ L ∞ loc ( ) ⊂ P p (T, ). Also, the above study shows that the C 1 semi-exhaustive function ψ ∈ P 1 (T, B ψ (r )). Definition 4.6 Let S be a positive current of bi-dimension ( p, p) on . We define the capacity C q S (O, ) for all Borel set O by Observe that for positive and closed currents S, the capacity C S , which is introduced in [4], is dominated by C p S . This is an obvious inclusion from the fact that Psh( ) ∩ L ∞ loc ( ) ⊂ P p (T, ). We give an example Example 4.7 In C 1 , set = B(0, 1) and S = 1. From [3] it is very well known that C S (B(0, Now construct a positive smooth semi-exhaustive function ψ on B(0, 1) so that ψ(z) = 2 3 |z| 2 on B(0, 1 2 ), and ψ(z) = sin ( π 2 |z| 2 ) on an appropriate neighborhood of {|z| = 1}. Clearly, This show that C S (B(0, ).
The above definitions together with Proposition 4.4 yield to the next properties. Proposition 4.9 Let S be a positive closed current of bi-dimension ( p, p) on . Then for all 0 < r < r < r < R ψ , we have (1) C S (dψ, r, r ) ≤ C S (dψ, r , r ).
Proposition 4.10 Let K be a compact subset of so that B ψ (r ) K . If T is positive and plurisuperharmonic, then there exists a constant C K (r ) > 0 such that (4.12) Proof By similar arguments as above, one can assume that ψ is of class (4.14) But Chern-Lieven-Nirenberg shows that