Subextension of plurisubharmonic functions with boundary values in weighted pluricomplex energy classes

Full Terms & Conditions of access and use can be found at Download by: [Hong Nguyen Xuan] Date: 19 September 2015, At: 03:33 Complex Variables and Elliptic Equations An International Journal ISSN: 1747-6933 (Print) 1747-6941 (Online) Journal homepage: Subextension of plurisubharmonic functions with boundary values in weighted pluricomplex energy classes Le Mau Hai, Nguyen Xuan Hong & Trieu Van Dung To cite this article: Le Mau Hai, Nguyen Xuan Hong & Trieu Van Dung (2015) Subextension of plurisubharmonic functions with boundary values in weighted pluricomplex energy classes, Complex Variables and Elliptic Equations, 60:11, 1580-1593, DOI: 10.1080/17476933.2015.1036244 To link to this article: Published online: 06 May 2015. Submit your article to this journal Article views: 14 View related articles View Crossmark data Complex Variables and Elliptic Equations, 2015 Vol. 60, No. 11, 1580–1593, Subextension of plurisubharmonic functions with boundary values in weighted pluricomplex energy classes Le Mau Haia∗, Nguyen Xuan Honga and Trieu Van Dungb aDepartment of Mathematics, Hanoi National University of Education, Hanoi, Vietnam; bHung Vuong Gifted High School, Viettri City, Phutho, Vietnam Communicated by S. Krantz (Received 16 May 2014; accepted 27 March 2015) In this paper, we investigate subextension of plurisubharmonic functions in the weighted pluricomplex energy class Eχ (", f ). Moreover, we show the equality of the weighted Monge–Ampère measures of subextension and the given function. Keywords: weighted pluricomplex energy classes; complex Monge–Ampère operator; subextension of plurisubharmonic functions AMS Subject Classifications: 32U05; 32U15; 32W20 1. Introduction Subextension of unbounded plurisubharmonic functions with estimation on total Monge– Ampère mass of subextension was investigated by many authors in the last ten years. Let " ⊂ "˜ be domains in Cn and let u be a plurisubharmonic function on " (briefly, u ∈ P SH(")). A function u˜ ∈ P SH("˜) is subextension of u if for all z ∈ ", u˜(z) ≤ u(z). On bounded hyperconvex domains in Cn , Cegrell and Zeriahi investigated the subex- tension problem for the class F(") ( for more details see Section 2). In [1], the authors proved that if " ! "˜ are bounded hyperconvex domains in Cn and u ∈ F("), then there exists u˜ ∈ F("˜) such that u˜ ≤ u on " and∫ "˜ (ddcu˜)n ≤ ∫ " (ddcu)n . In the class Ep("), p > 0, introduced and investigated early in [2], the subextension problem was investigated by Hiep. He proved in [3] that if " ⊂ "˜ ! Cn are bounded hyperconvex domains and u ∈ Ep("), then there exists a function u˜ ∈ Ep("˜) such that u˜ ≤ u on " and ∫ "˜ (−u˜)p(ddcu˜)n ≤ ∫ " (−u)p(ddcu)n . Recently a weighted pluricomplex energy class Eχ ("), which is generalization of the classes Ep(") and F(") was introduced and investigated by Benelkourchi, Guedj and ∗Corresponding author. Email: mauhai@fpt.vn © 2015 Taylor & Francis Do wn loa de d b y [ Ho ng N gu ye n X ua n] at 03 :33 19 Se pte mb er 20 15 Complex Variables and Elliptic Equations 1581 Zeriahi [4]. In [5], Benelkourchi studied subextension for the class Eχ ("). Theorem 6.2 in [5] claimed that if " ⊂ "˜ are hyperconvex domains in Cn and χ : R− −→ R+ is a decreasing function with χ(−∞) = +∞ then for every u ∈ Eχ (") there exists u˜ ∈ Eχ ("˜) such that u˜ ≤ u on " and (ddcu˜)n ≤ (ddcu)n on " and∫ "˜ χ (˜u)(ddcu˜)n ≤ ∫ " χ(u)(ddcu)n . The subextension problem in the classes with boundary values was considered in recent years. Namely, in 2008, in [6], the authors showed that if " and "˜ are two bounded hyperconvex domains such that" ⊂ "˜ ⊂ Cn, n ≥ 1 and u ∈ F(", f ) with f ∈ E(") has subextension v ∈ F("˜, g) with g ∈ E("˜) ∩ M P SH("˜), and∫ "˜ (ddcv)n ≤ ∫ " (ddcu)n, under the assumption that f ≥ g on ", where M P SH("˜) denotes the set of maximal plurisubharmonic functions on "˜. It should be noticed that in results above only estimation of total Monge–Ampère mass of subextension was obtained. In our recent paper (see [7]), we investigated subextension in the classF(", f ) and we proved that the Monge–Ampère measure of subextension does not change. Namely, let " ⊂ "˜ be bounded hyperconvex domains in Cn and let f ∈ E(") and g ∈ E("˜) ∩ M P SH("˜) with f ≥ g on ", then for every u ∈ F(", f ) with∫ " (ddcu)n < +∞, there exists u˜ ∈ F("˜, g) such that u˜ ≤ u on " and (ddcu˜)n = 1"(ddcu)non "˜. In this paper, we extend this result to the class Eχ (", f ). Our main theorem is the following. Theorem 3.3 Let " ! "˜ be bounded hyperconvex domains in Cn and let f ∈ E(")∩ M P SH("), g ∈ E("˜) ∩ M P SH("˜) with f ≥ g on ". Assume that χ : R− −→ R+ is a decreasing continuous function such that χ(t) > 0 for all t < 0. Then for every u ∈ Eχ (", f ) such that ∫ " [χ(u)− ρ](ddcu)n < +∞, for some ρ ∈ E0("), there exists u˜ ∈ Eχ ("˜, g) such that u˜ ≤ u on " and χ (˜u)(ddcu˜)n = 1"χ(u)(ddcu)n on "˜. Note that the proof of the above theorem is different than traditional proofs of the subextension problem in the other classes, for example, F(", f ) or Eχ (") because the class Eχ (", f ) does not have good properties as the class F(", f ) and the comparison between the measure 1"χ(u)(ddcu)n , u ∈ Eχ (", f ) with the measure of subextension is not simple. Hence, in order to prove the above result we have to find a new approach for the subextension problem in the class Eχ (", f ). The paper is organized as follows. Beside the introduction, the paper has two sections. In Section 2, we recall some elements of pluripotential theory needed for the proof of the main result. Section 3 is devoted to proving the main result of the paper. Do wn loa de d b y [ Ho ng N gu ye n X ua n] at 03 :33 19 Se pte mb er 20 15 1582 L.M. Hai et al. 2. Preliminairies Some elements of pluripotential theory that will be used throughout the paper can be found in [8–12]. Now, we recall some Cegrell classes of plurisubharmonic functions (see [13–15]) and classes of plurisubharmonic functions with generalized boundary values concerning Cegrell classes. Let " be an open set in Cn . By P SH−("), we denote the set of negative plurisubharmonic functions on ". 2.1. Cegrell classes Now, we assume that " is a bounded hyperconvex domain in Cn . This means that " is a bounded domain in Cn and there exists a plurisubharmonic function ϕ : " −→ (−∞, 0) such that for every c < 0 the set "c = {z ∈ " : ϕ(z) < c} ! ". As in [13], we define the following subclasses of P SH−("): E0(") = { ϕ ∈ P SH−(") ∩ L∞(") : lim z→∂"ϕ(z) = 0, ∫ " (ddcϕ)n <∞ } , F(") = { ϕ ∈ P SH−(") : ∃ E0(") ∋ ϕ j ↘ ϕ, sup j ∫ " ( ddcϕ j )n <∞ } , and E(") = { ϕ ∈ P SH−(") : ∀z0 ∈ ", ∃ a neighbourhood U ∋ z0, E0(") ∋ ϕ j ↘ ϕ on U, sup j ∫ " ( ddcϕ j )n <∞ } . The following inclusions are obvious: E0(") ⊂ F(") ⊂ E("). 2.2. Maximal plurisubharmonic functions Since in this paper we also need the class of maximal plurisubharmonic functions, we recall the following definition given in [16]. Definition 2.1 A plurisubharmonic function u on " is said to be maximal (briefly, u ∈ M P SH(")) if for every compact et K ! " and for every v ∈ P SH("), if v ≤ u on"\ K then v ≤ u on ". It is well known (see, e.g. [10]) that locally bounded plurisubharmonic functions are maximal if and only if they satisfy the homogeneous Monge–Ampère equation (ddcu)n = 0. In [17], Błocki extended the above result for the class E("). 2.3. The classN (!) We recall the class N (") introduced in [14]. Let " be a hyperconvex domain in Cn and {" j } j≥1 a fundamental sequence of ". This is an increasing sequence of strictly pseudoconvex subsets {" j } j≥1 of " such that " j ! " j+1 and ⋃+∞j=1 " j = ". Let ϕ ∈ P SH−("). For each j ≥ 1, put Do wn loa de d b y [ Ho ng N gu ye n X ua n] at 03 :33 19 Se pte mb er 20 15 Complex Variables and Elliptic Equations 1583 ϕ j = sup {u : u ∈ P SH("), u ≤ ϕ on "\" j} . As in [14], the function (lim j→∞ ϕ j )∗ ∈ M P SH("). Set N (") = { ϕ ∈ E(") : ϕ j ↑ 0 } . It is easy to see that F(") ⊂ N (") ⊂ E("). 2.4. The class Eχ (!) Next, we recall the class Eχ (") (see Definition 3.1 in [4]) and the relation between this class and the classes Ep("),F(") and N ("). Definition 2.2 Let χ : R− −→ R+ be a decreasing function and " be a hyperconvex domain in Cn . We say that the function u ∈ P SH−(") belongs to Eχ (") if there exists a sequence {u j } ⊂ E0(") decreasing to u on " and satisfying sup j ∫ " χ(u j )(ddcu j )n < +∞. If we take χ(t) = (−t)p, p > 0 then the class Eχ (") coincides with the class Ep("). If χ(t) is bounded and χ(0) > 0 then Eχ (") is the class F("). Proposition A in [5] and, more generally, Corollary 3.3 in [18] claims that if χ ̸≡ 0 then Eχ (") ⊂ E(") and, hence, in this case the Monge–Ampère operator is well defined on Eχ ("). Corollary 3.3 in [18] shows that if χ(t) > 0 for t < 0 then Eχ (") ⊂ N ("). Moreover, Theorem 2.7 in [19] implies that Eχ (") = ⎧⎨⎩u ∈ N (") : ∫ " χ(u)(ddcu)n < +∞ ⎫⎬⎭ . Below we need the following result. Proposition 2.3 Let χ : R− −→ R+ be a decreasing continuous function such that χ(t) > 0 for all t < 0 and " be a bounded hyperconvex domain in Cn. Assume that µ is a positive Radon measure which vanishes on pluripolar sets of " and u, v ∈ E(") are such that χ(u)(ddcu)n ≥ µ and χ(v)(ddcv)n ≥ µ. Then χ(max(u, v))(ddc max(u, v))n ≥ µ. Proof From the hypothesis, we have (ddcu)n ≥ 1{u=v} µ χ(u) = 1{u=v} µ χ(max(u, v)) . Similarly, (ddcv)n ≥ 1{u=v} µ χ(v) = 1{u=v} µ χ(max(u, v)) . Proposition 4.3 in [20] implies that (ddc(max(u, v))n ≥ 1{u=v} µ χ(max(u, v)) , Do wn loa de d b y [ Ho ng N gu ye n X ua n] at 03 :33 19 Se pte mb er 20 15 1584 L.M. Hai et al. or χ(max(u, v))(ddc(max(u, v))n ≥ 1{u=v}µ. The above inequality together with Theorem 4.1 in [20] give us that χ(max(u, v))(ddc max(u, v))n = 1{u>v}χ(max(u, v))(ddc max(u, v))n + 1{u=v}χ(max(u, v))(ddc max(u, v))n + 1{u<v}χ(max(u, v))(ddc max(u, v))n = 1{u>v}χ(u)(ddcu)n + 1{u=v}χ(max(u, v))(ddc max(u, v))n + 1{u<v}χ(v)(ddcv)n ≥ 1{u>v}µ+ 1{u=v}µ+ 1{u<v}µ = µ, and the proof is complete. " 2.5. Cegrell classes with boundary values We recall classes of plurisubharmonic functions with generalized boundary values in the class E("). Definition 2.4 LetK ∈ {E0("),F("),N ("), Eχ ("), E(")} and let f ∈ E("). Then we say that a plurisubharmonic function u defined on" is inK(", f ) if there exists a function ϕ ∈ K such that ϕ + f ≤ u ≤ f, on ". By Ka(", f ), we denote the set of plurisubharmonic functions u ∈ K(", f ) such that (ddcu)n vanishes on all pluripolar sets of ". For systematic and complete study of classes of plurisubharmonic functions with gen- eralized boundary values in other classes, we refer readers to the paper of [21]. Note that functions in K(", f ) not necessarily have finite total Monge–Ampère mass (see [22]). We need the following proposition which will be used in the main result of the paper. Proposition 2.5 Let " be a bounded hyperconvex domain in Cn and let f ∈ E(") ∩ M P SH("). Then for every u ∈ N (", f ) such that∫ {u=−∞}∩" (ddcu)n < +∞, there exists v ∈ F(", f ) such that v ≥ u and (ddcv)n = 1{u=−∞}(ddcu)n . Proof Let ϕ ∈ N (") such that ϕ + f ≤ u ≤ f on ". We may assume that ϕ ≥ u on ". Indeed, put ψ = max(ϕ, u). Then ψ ∈ N ("),ψ ≥ u on ". It is easy to see that ψ+ f ≤ u ≤ f on". By replacing ϕ byψ , the desired conclusion follows. Since (ddc f )n vanishes on pluripolar sets of " and 0 ≤ ϕ − u ≤ − f so by Lemma 4.12 in [8], we get 1{u=−∞}(ddcu)n = 1{ϕ=−∞}(ddcϕ)n ≤ (ddcϕ)n . Do wn loa de d b y [ Ho ng N gu ye n X ua n] at 03 :33 19 Se pte mb er 20 15 Complex Variables and Elliptic Equations 1585 Hence, Theorem 4.14 in [8] implies that there existsw ∈ N (", f ) such thatϕ+ f ≤ w ≤ f on " and (ddcw)n = 1{u=−∞}(ddcu)n . We have ∫ "(dd cw)n = ∫{u=−∞}∩"(ddcu)n < +∞ then Proposition 2.2 in [6] implies that w ∈ F(", f ). Put ψ1 = ( sup{ψ ∈ P SH−(") : ψ + f ≤ w on "} )∗ . Then ψ1 ∈ P SH−("), ϕ ≤ ψ1 and ψ1 + f ≤ w on ". From Theorem 2.1 in [14], we have ∫ " (ddcψ1)n ≤ ∫ " (ddcw)n . (2.1) Since ψ1 + f ≤ w on " then by Lemma 4.1 in [8], we have 1{w=−∞}(ddcw)n ≤ 1{ψ1+ f =−∞}(ddc(ψ1 + f ))n = 1{ψ1=−∞} ( ddcψ1 )n , where the second equality follows from |ψ1 + f − ψ1| = − f by Lemma 4.12 in [8]. But (ddcw)n is carried by a pluripolar set then we have the following: (ddcw)n = 1{w=−∞}(ddcw)n ≤ 1{ψ1+ f =−∞} ( ddc (ψ1 + f ))n (2.2) = 1{ψ1=−∞}(ddcψ1)n ≤ ( ddcψ1 )n . Combining (2.1) and (2.2) we get (ddcw)n = 1{ψ1=−∞}(ddcψ1)n . Hence 1{ψ1=−∞}(ddcψ1)n = 1{u=−∞}(ddcu)n . Put v = max(w, u). We have ϕ + f ≤ u ≤ v ≤ f . Hence, v ∈ N (", f ), v ≥ u. We shall prove that (ddcv)n = 1{u=−∞}(ddcu)n and v ∈ F(", f ) and the proof of Proposition 2.5 will be finished. First, we note that v ≤ ψ1 on ". Indeed, because u + f ≤ ϕ + f ≤ w andw+ f ≤ w then v+ f = max(w+ f, u+ f ) ≤ w and by the definition ofψ1 the desired conclusion follows. On the other hand, since v,w ∈ N (", f ), v ≥ w, ∫"(ddcw)n < +∞ then Lemma 3.3 in [8] implies that∫ " (ddcv)n ≤ ∫ " (ddcw)n . (2.3) From u ≤ v ≤ ψ1 then by Lemma 4.1 in [8], we have 1{ψ1=−∞}(ddcψ1)n ≤ 1{v=−∞}(ddcv)n ≤ 1{u=−∞}(ddcu)n . But we have proved that 1{ψ1=−∞}(ddcψ1)n = 1{u=−∞}(ddcu)n then 1{v=−∞}(ddcv)n = 1{u=−∞}(ddcu)n = (ddcw)n . Hence, (ddcv)n ≥ (ddcw)n . (2.4) Now (2.3) and (2.4) give us that∫ " (ddcw)n ≤ ∫ " (ddcv)n ≤ ∫ " (ddcw)n = ∫ {u=−∞}∩" (ddcu)n < +∞. Do wn loa de d b y [ Ho ng N gu ye n X ua n] at 03 :33 19 Se pte mb er 20 15 1586 L.M. Hai et al. Therefore, (ddcv)n = (ddcw)n = 1{u=−∞}(ddcu)n . Moreover, by Proposition 2.2 in [6] we infer that v ∈ F(", f ). The proof is complete. " 3. Proof of the main theorem In this section, we give the proof of the main result of the paper. However, to arrive at the desired result we need some auxiliary lemmas. Lemma 3.1 Let " ⊂ "˜ be bounded hyperconvex domains in Cn and let f ∈ E("), g ∈ E("˜) ∩ M P SH("˜) with f ≥ g on ". Assume that u ∈ F(", f ) is such that (a) (ddcu)n is carried by a pluripolar set. (b) ∫"(ddcu)n < +∞. Then the function u˜ := (sup {ϕ ∈ F("˜, g) : ϕ ≤ u on "})∗ belongs to F("˜, g) and (ddcu˜)n = 1"(ddcu)n on "˜. Proof Without loss of generality we may assume that (ddcu)n is carried by the set {u = −∞}. By [6], there exists v ∈ F ("˜, g) such that v ≤ u on " and∫ "˜ (ddcv)n ≤ ∫ " (ddcu)n . Since v ≤ u˜ ≤ g in "˜ so u˜ ∈ F("˜, g). By Lemma 3.3 in [8], we have∫ "˜ ( ddcu˜ )n ≤ ∫ "˜ (ddcv)n ≤ ∫ " (ddcu)n . Moreover, since u˜ ≤ u in " so Lemma 4.1 in [8] implies that 1{˜u=−∞} ( ddcu˜ )n ≥ 1{u=−∞}(ddcu)n on ". Therefore, since (ddcu)n is carried by the set {u = −∞} so( ddcu˜ )n ≥ 1"(ddcu)n on "˜. Now, we have ∫ "˜ (ddcu˜)n ≤ ∫ " (ddcu)n = ∫ "∩{u=−∞} (ddcu)n ≤ ∫ "∩{˜u=−∞} (ddcu˜)n ≤ ∫ "˜ (ddcu˜)n, and it follows that (ddcu˜)n = 1"(ddcu)n on "˜. The proof is complete. " The following lemma is an essential tool in the proof of the main result. Do wn loa de d b y [ Ho ng N gu ye n X ua n] at 03 :33 19 Se pte mb er 20 15 Complex Variables and Elliptic Equations 1587 Lemma 3.2 Let" be a bounded hyperconvex domain inCn and letµ be a positive Radon measure which vanishes on pluripolar sets of" with µ(") < +∞. Let χ : R− → R+ be a bounded decreasing continuous function such that χ(t) > 0 for all t < 0 and χ(−∞) = 1. Assume that f ∈ E(") ∩ M P SH(") and v ∈ F(", f ) such that (ddcv)n is carried by a pluripolar set and ∫ " (ddcv)n < +∞. Then the function u defined by u := (sup {ϕ ∈ E(") : ϕ ≤ v and χ(ϕ)(ddcϕ)n ≥ µ})∗ belongs to N (", f ) and χ(u)(ddcu)n ≥ µ+ (ddcv)n . Moreover, if supp(ddcv)n ! " and ∫"(−ρ)(ddcu)n < +∞ for some ρ ∈ E0(") then χ(u)(ddcu)n = µ+ (ddcv)n . Proof First, we prove that u ∈ N (", f ). Indeed, since µ vanishes on all pluripolar sets of " and µ(") < +∞ so Theorem 4.10 in [18] implies that there exists '0 ∈ N (") such that χ('0)(ddc'0)n = µ. We have '1 := '0 + v ∈ N (", f ) and χ('1)(ddc'1)n ≥ χ('0)(ddc'0)n = µ. Hence, we have u ≥ '1 and the desired conclusion follows. Next, we claim that χ(u)(ddcu)n ≥ µ. Indeed, by Choquet’s lemma and Proposition 2.3 we can choose a sequence {ϕ j } ⊂ E(") such that ϕ j ≥ '1, ϕ j ↗ u a.e. in " and χ(ϕ j )(ddcϕ j )n ≥ µ. Fix k and let j ≥ k. Since χ(ϕ j ) ≤ χ(ϕk) so χ(ϕk)(ddcϕ j )n ≥ χ(ϕ j )(ddcϕ j )n ≥ µ. It follows that (ddcϕ j )n ≥ 1 χ(ϕk) µ for every j ≥ k. Letting j → +∞, by [23] we get (ddcu)n ≥ 1 χ(ϕk) µ. Since ϕk ↗ u as k ↗∞ outside a pluripolar set of " and µ vanishes on pluripolar sets of " so by letting k → +∞, we get (ddcu)n ≥ 1 χ(u) µ, as we wanted. Do wn loa de d b y [ Ho ng N gu ye n X ua n] at 03 :33 19 Se pte mb er 20 15 1588 L.M. Hai et al. Next, we prove thatχ(u)(ddcu)n ≥ µ+(ddcv)n . Indeed, sinceµ vanishes on pluripolar sets of " and χ(u)(ddcu)n ≥ µ so 1{u ̸=−∞}χ(u)(ddcu)n ≥ µ. Moreover, since u ≤ v so Lemma 4.1 in [8] implies that 1{u=−∞}χ(u)(ddcu)n = 1{u=−∞}(ddcu)n ≥ 1{v=−∞}(ddcv)n . Therefore, χ(u)(ddcu)n ≥ µ+ (ddcv)n . Now, assume that supp(ddcv)n ! U ! " and ∫"(−ρ)(ddcu)n < +∞. Note that by Theorem 2.4 in [8] we can assume that ρ ∈ E0(") ∩ C("). By Proposition 2.5 in [8] we can choose a sequence {v j } ∈ E0(", f ) such that supp(ddcv j )n ! U ! " and v j ↘ v as j ↗∞. Since 0 ≤ χ ≤ 1 and v ≤ v j so by Lemma 3.3 in [8], we have∫ " χ(v j )(ddcv j )n ≤ ∫ " (ddcv j )n ≤ ∫ " (ddcv)n < +∞. Moreover, since µ(") < +∞ so by Theorem 4.10 in [18] there exists a function w j ∈ N (", f ) such that χ(w j )(ddcw j )n = µ+ χ(v j )(ddcv j )n . (3.1) Since χ(w j )(ddcw j )n ≥ χ(v j )(ddcv j )n so Theorem 4.8 in [18] implies that w j ≤ v j on ". Put u j := (sup {ϕ ∈ E(") : ϕ ≤ v j and χ(ϕ)(ddcϕ)n ≥ µ})∗ . We have u j ≥ w j on ". Since u j+1 ≤ u j so u j+k ≤ u j for every k ≥ 0. Hence, w j ≤ ψ j := ( sup k∈N w j+k )∗ ≤ u j . Therefore, ψ := lim j→+∞ ψ j ≤ u := lim j→+∞ u j . Moreover, since (ddc('0 + v j ))n ≥ (ddc'0)n + (ddcv j )n, and χ('0 + v j ) ≥ χ('0),χ('0 + v j ) ≥ χ(v j ) and by (3.1) we infer that χ('0 + v j ) (ddc ('0 + v j ))n ≥ χ('0 + v j ) ((ddc'0)n + (ddcv j )n) = χ('0 + v j ) (ddc'0)n + χ('0 + v j ) (ddcv j )n ≥ χ('0) (ddc'0)n + χ(v j ) (ddcv j )n = µ+ χ(v j ) (ddcv j )n = χ(w j ) (ddcw j )n Do wn loa de d b y [ Ho ng N gu ye n X ua n] at 03 :33 19 Se pte mb er 20 15 Complex Variables and Elliptic Equations 1589 so by Theorem 4.8 in [18] we havew j ≥ '0+v j ≥ '0+v. It follows thatψ,ψ j ∈ N (", f ) and ψ j ↘ ψ as j −→∞. Hence, by Lemma 3.3 and Corollary 3.4 in [8], we have∫ " (−ρ)(ddcu)n ≤ ∫ " (−ρ)(ddcψ)n = lim j→∞ ∫ " (−ρ)(ddcψ j )n (3.2) ≤ lim sup j→∞ ∫ " (−ρ)(ddcw j )n ≤ lim sup j→∞ ∫ " −ρ χ(ψ j ) [µ+ (ddcv j )n] ≤ lim sup j→∞ ∫ " −ρ χ(ψ j ) µ+ lim sup j→∞ ∫ " −ρ χ(ψ j ) (ddcv j )n ≤ ∫ " −ρ χ(ψ) µ+ lim sup j→∞ ∫ U −ρ χ(ψ j ) (ddcv j )n, where the first term in the (3.2) follows from the monotone convergence theorem and for the second by invoking supp(ddcv j )n ! U ! ". We have v j ↘ v ∈ E(") then Corollary 3.3 in [23] implies that−ρ(ddcv j )n is convergent weakly to−ρ(ddcv)n . Moreover, 1χ(ψ j ) is upper-semicontinuous and decreasing to 1χ(ψ) . Hence, we have the following estimation for the second term of (3.2): lim sup j→∞ ∫ U −ρ χ(ψ j ) (ddcv j )n ≤ ∫ U −ρ χ(ψ) (ddcv)n (3.3) ≤ ∫ " −ρ χ(ψ) (ddcv)n ≤ ∫ " −ρ χ(u) (ddcv)n . Combining (3.2) and (3.3) we get∫ " (−ρ)(ddcu)n ≤ ∫ " −ρ χ(u) [µ+ (ddcv)n]. On the other hand, since ∫ "(−ρ)(ddcu)n < +∞ and −ρ(ddcu)