Monge–Ampère measures of maximal subextensions of plurisubharmonic functions with given boundary values

This article was downloaded by: [Hong Nguyen Xuan] On: 05 March 2015, At: 23:52 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Click for updates Complex Variables and Elliptic Equations: An International Journal Publication details, including instructions for authors and subscription information: Monge–Ampère measures of maximal subextensions of plurisubharmonic functions with given boundary values Nguyen Xuan Honga a Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy Street, Cau Giay District, Ha Noi, Vietnam. Published online: 08 Aug 2014. To cite this article: Nguyen Xuan Hong (2015) Monge–Ampère measures of maximal subextensions of plurisubharmonic functions with given boundary values, Complex Variables and Elliptic Equations: An International Journal, 60:3, 429-435, DOI: 10.1080/17476933.2014.944177 To link to this article: PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. 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Terms & Conditions of access and use can be found at and-conditions D ow nl oa de d by [H on g N gu ye n X ua n] at 23 :52 05 M arc h 2 01 5 Complex Variables and Elliptic Equations, 2015 Vol. 60, No. 3, 429–435, Monge–Ampère measures of maximal subextensions of plurisubharmonic functions with given boundary values Nguyen Xuan Hong∗ Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy Street, Cau Giay District, Ha Noi, Vietnam Communicated by S. Ivashkovich (Received 16 March 2014; accepted 3 July 2014) The aim of the paper is to investigate the Monge–Ampère measures of maximal subextensions of plurisubharmonic functions with given boundary values. As an application, we study the approximation of negative plurisubharmonic function with given boundary values by an increasing sequence of plurisubharmonic functions defined in larger domains. Keywords: Monge–Ampère operator; subextension; approximation; plurisubharmonic functions AMS Subject Classifications: 32U05; 32W20 1. Introduction Let  ⊂ ˜ be domains in Cn and let u be a plurisubharmonic function . A plurisub- harmonic function ϕ in ˜ such that ϕ| ≤ u on  is said to be subextension of a plurisubharmonic function u on ˜. El Mir [1] gave in 1980 an example of a plurisubharmonic function on the unit bidisc for which the restriction to any smaller bidisc admits no plurisubharmonic subextension to the whole space. Later, Wiklund [2] proved in 2006 that for any bounded hyperconvex domain  inCn , there is a plurisubharmonic function u ∈ E() such that u has no plurisubharmonic subextension. It appears natural to put additional assumptions u to guarantee that u has plurisubhar- monic subextension. The first result in this direction is the theorem of Cegrell and Zeriahi [3] about subextension of plurisubharmonic functions with bounded Monge–Ampère mass. Later, Hiep [4] proved in 2008 that if u ∈ F() with  be bounded hyperconvex domain, then the maximal subextensions u˜ to any larger hyperconvex domain ˜ belong to F(˜) and (ddcu˜)n ≤ 1(ddcu)n in ˜. On the other hand, Cegrell and Hed [5] studied in 2008 the subextension problem concerning to boundary values in bounded hyperconvex domains. Recently, Hai and Hong [6] proved that plurisubharmonic functions with uniformly bounded Monge–Ampère mass on a bounded hyperconvex domain always admit a plurisubharmonic subextension without changing the Monge–Ampère measures. ∗Email: xuanhongdhsp@yahoo.com © 2014 Taylor & Francis D ow nl oa de d by [H on g N gu ye n X ua n] at 23 :52 05 M arc h 2 01 5 430 N.X. Hong The aim of this paper is to study the Monge–Ampère measures of maximal subextensions of plurisubharmonic functions with given boundary values. Namely, we prove the following. Theorem 1.1 Let  ⊂ ˜ be bounded hyperconvex domains in Cn and let f ∈ E(), g ∈ E(˜) with f ≥ g on . Assume that u ∈ F(, f ) with ∫  (ddcu)n < +∞. Put u˜ := sup{ϕ ∈ PSH−(˜) : ϕ ≤ g on ˜, and ϕ ≤ u on }. Then, u˜ ∈ F(˜, g) and (ddcu˜)n ≤ 1(ddcu)n + (ddcg)n on ˜. Moreover, (ddcu˜)n = 0 on {˜u < min(u, g)} ∩ . The organization of the paper is as follows. In Section 2, we recall some notions of pluripotential theory which are necessary for the next results of the paper. In Section 3, we prove the main result of the paper. The purpose of Section 4 is to apply the above result to investigating the approximation of negative plurisubharmonic functions. 2. Preliminaries Some elements of pluripotential theory that will be used throughout the paper can be found in [1–22]. 2.1. Bounded hyperconvex domains in Cn Let n ∈ N∗. An open set  ⊂ Cn is bounded hyperconvex domain if it is a bounded, connected and there exists a bounded plurisubharmonic function ρ such that {z ∈  : ρ(z) < c}  , for every c ∈ (−∞, 0). 2.2. Cegrell’s classes Assume that  be an open set inCn . We denote by PSH(), the family of plurisubharmonic functions defined on . The set of negative plurisubharmonic functions is denoted by PSH−(). Now, we recall some Cegrell’s classes of plurisubharmonic functions (see [11]). E0() = { ϕ ∈ PSH−() ∩ L∞() : lim z→∂ ϕ(z) = 0, ∫  (ddcϕ)n < ∞ } , F() = { ϕ ∈ PSH−() : ∃ E0  ϕ j ↘ ϕ, sup j ∫  (ddcϕ j )n < ∞ } and E() = {ϕ ∈ PSH−() : ∀G  , ∃ uG ∈ F(), u = uG on G}. It is easy to see that E0() ⊂ F() ⊂ E(). 2.3. Maximal plurisubharmonic functions A plurisubharmonic function u on  is said to be maximal plurisubharmonic (briefly, u ∈ MPSH()) if for every compact set K   and for every v ∈ PSH(), if v ≤ u on  \ K then, v ≤ u on . It is well known if u ∈ E() then, u ∈ MPSH(), iff (ddcu)n = 0 in . D ow nl oa de d by [H on g N gu ye n X ua n] at 23 :52 05 M arc h 2 01 5 Complex Variables and Elliptic Equations 431 2.4. Cegrell’s classes with given boundary values Now, we recall classes of plurisubharmonic functions with generalized boundary values in the class E(). Assume that f ∈ E() and K ∈ {E0,F , E}. Then we say that a plurisubharmonic function u defined on  belongs to K(, f ) if there exists a function ϕ ∈ K() such that ϕ + f ≤ u ≤ f on . 3. Monge–Ampère measure of maximal subextensions First, we give the proof of Theorem 1.1. The idea of the proof is taken from Hiep paper [4]. The Proof of Theorem 1.1 Let ψ ∈ F() such that f + ψ ≤ u ≤ f on . By Lemma 4.5 in [4], there exists ψ˜ ∈ F(˜) such that ψ˜ ≤ ψ on  and (ddcψ˜)n ≤ 1(ddcψ)n . It is clear that ψ˜ + g ≤ u˜ ≤ g on ˜. Hence, u˜ ∈ F(˜, g). By Theorem 2.1 in [11], there exists {g j } ⊂ E0(˜) ∩ C(˜) be such that g j ↘ g on ˜ and there exists {u j } ⊂ E0() ∩ C() be such that u j ↘ u in . Put h j = { min(u j , g j ) on  g j on ˜\ and u˜ j := (sup{ϕ ∈ PSH−(˜) : ϕ ≤ h j in ˜})∗. It is easy to see that u˜ j ∈ PSH−(˜) ∩ L∞(˜) and u˜ j ↘ u˜ in ˜ as j ↗ +∞. Since h j ∈ C(˜) so by Corollary 9.2 in [8], we have (ddcu˜ j )n = 0 on {˜u j < h j }. (1) We claim that (ddcu˜ j )n ≤ (ddcg j )n + 1(ddcu j )n on ˜. (2) Indeed, let K ⊂ {˜u j = g j } be a compact set. Since K ⊂ {˜u j + 1k > g j } for every k ∈ N∗ so by Theorem 4.1 in [20], we have∫ K (ddcu˜ j )n = lim k→+∞ ∫ K ( ddc max ( u˜ j + 1k , g j ))n ≤ ∫ K ( ddc max (˜ u j , g j ))n = ∫ K ( ddcg j )n . Therefore, (ddcu˜ j )n ≤ (ddcg j )n on {˜u j = g j }. (3) Similarly, if K ⊂ {˜u j = u j } ∩  is a compact set then by Theorem 4.1 in [20], we have∫ K (ddcu˜ j )n = lim k→+∞ ∫ K ( ddc max ( u˜ j + 1k , u j ))n ≤ ∫ K (ddc max(˜u j , u j ))n = ∫ K (ddcu j )n . Hence, (ddcu˜ j )n ≤ (ddcu j )n on {˜u j = u j } ∩ . (4) D ow nl oa de d by [H on g N gu ye n X ua n] at 23 :52 05 M arc h 2 01 5 432 N.X. Hong Since {˜u j ≥ h j } ⊂ {˜u j = g j } ∪ ({˜u j = u j } ∩ ) so from (1), (3) and (4), we get (2). This proves the claim. Now, we prove that (ddcu˜)n = 0 on {˜u < min(u, g)} ∩ . (5) Fix a ∈ Q and k ∈ N∗. For every j ≥ k, since {u > a} ∩ {g > a} ∩ {˜uk < a} ∩ ⊂ {˜u j < a < h j } ∩  so by (1), we have max(u − a, 0) max(g − a, 0)(ddcu˜ j )n = 0 on {˜uk < a} ∩ . Let j → +∞ and k → +∞, we obtain max(u − a, 0) max(g − a, 0)(ddcu˜)n = 0 on {˜u < a} ∩ . It follows that (ddcu˜)n = 0 on {˜u < min(u, g)} ∩ . Next, we prove that (ddcu˜)n ≤ (ddcg)n + 1(ddcu)n on ˜. (6) Indeed, by [13], we have (ddcg j )n → (ddcg)n weakly in ˜ and (ddcu j )n → (ddcu)n weakly in . Hence, by (2) we get 1˜\∂(ddcu˜)n ≤ 1˜\∂(ddcg)n + 1(ddcu)n on ˜. (7) Let K ⊂ (∂) ∩ ˜ is a compact set. Let ε > 0 and G   such that∫ \G (ddcu)n < ε. Choose χ ∈ C∞0 (˜\G) such that 0 ≤ χ ≤ 1 in ˜\G and χ = 1 on K . Since 1K is upper semicontinuous function so there exists {h j } ⊂ C(˜) such that 0 ≤ h j ≤ 1 and h j ↘ 1K in ˜. Fix k ≥ 1. By (2) we have∫ K (ddcu˜)n ≤ ∫ ˜ χhk(ddcu˜)n = limj→+∞ ∫ ˜ χhk(ddcu˜ j )n ≤ lim sup j→+∞ ∫ ˜ χhk(ddcg j )n + lim sup j→+∞ ∫ \G (ddcu j )n ≤ ∫ ˜ χhk(ddcg)n + lim sup j→+∞ ∫  (ddcu j )n − lim infj→+∞ ∫ G (ddcu j )n ≤ ∫ ˜ χhk(ddcg)n + ∫ \G (ddcu)n ≤ ∫ ˜ χhk(ddcg)n + ε. Let k → +∞ and let ε → 0, we obtain∫ K (ddcu˜)n ≤ ∫ K (ddcg)n . Therefore, (ddcu˜)n ≤ (ddcg)n on (∂) ∩ ˜. D ow nl oa de d by [H on g N gu ye n X ua n] at 23 :52 05 M arc h 2 01 5 Complex Variables and Elliptic Equations 433 Combining this with (7), we obtain (6). The proof is complete.  Corollary 3.1 Let  ⊂ ˜ be bounded hyperconvex domains in Cn. Assume that u ∈ E(), v ∈ E(˜) and define w := sup{ϕ ∈ PSH−(˜) : ϕ ≤ v on ˜, and ϕ ≤ u on } ∈ E(˜). Then, (ddcw)n ≤ (ddcu)n + (ddcv)n on . Proof Let { j } be a sequence of bounded hyperconvex domains such that  j   j+1   and  = ⋃+∞j=1  j . Put u j := sup{ϕ ∈ PSH−() : ϕ ≤ u on  j }. By [4], we have u j ∈ F( j ), u j = u on  j and u j ↘ u on . Put w j := sup{ϕ ∈ PSH−(˜) : ϕ ≤ v on ˜, and ϕ ≤ u j on }. We have w j ↘ w in ˜ as j ↗ +∞. By Theorem 1.1, we have w j ∈ E(˜) and (ddcw j )n ≤ (ddcu j )n + (ddcv)n on . Moreover, by [13] we have (ddcw j )n → (ddcw)n weakly and (ddcu j )n → (ddcu)n weakly in . Therefore, let j → +∞, we get (ddcw)n ≤ (ddcu)n + (ddcv)n on . The proof is complete.  4. Applications The purpose of this section is to prove a generalization of Cegrell and Hed theorem from [5]. Proposition 4.1 Let   Cn be bounded hyperconvex domains and let { j }+∞j=1 be a sequence of bounded hyperconvex domains such that  ⊂  j+1 ⊂  j for every j ≥ 1. Then, the following statements are equivalent. (a) There exists u ∈ F() and an increasing sequence of functions u j ∈ F( j ) such that u j ↗ u a.e. in  as j ↗ +∞. (b) For every u ∈ F(), there exists an increasing sequence of functions u j ∈ F( j ) such that u j ↗ u a.e. in  as j ↗ +∞. (c) For every f ∈ E() ∩ MPSH(), f j ∈ E( j ), j = 1, 2, . . . such that f j ↗ f a.e. in  as j ↗ +∞ and for every u ∈ F(, f ) such that ∫  (ddcu)n < +∞, there exists an increasing sequence of functions u j ∈ F( j , f j ) such that u j ↗ u a.e. in  as j ↗ +∞. Proof (a)⇔(b): see [5]. (c)⇒(b) is obvious. We prove (b)⇒(c). Put u j := sup{ϕ ∈ PSH−( j ) : ϕ ≤ f j on  j , and ϕ ≤ u on }. By Theorem 1.1, we have u j ∈ F( j , f j ) and (ddcu j )n ≤ 1(ddcu)n + (ddc f j )n on  j . (8) It is easy to see that u j ≤ u j+1 on  j+1. Put v = (lim j→+∞ u j )∗ in . D ow nl oa de d by [H on g N gu ye n X ua n] at 23 :52 05 M arc h 2 01 5 434 N.X. Hong We claim that v ∈ F(, f ). Indeed, let w ∈ F() such that w + f ≤ u ≤ f on . By (b), there exists w j ∈ F( j ) such that w j ↗ w a.e. in  as j ↗ +∞. Since w j + f j ∈ F( j , f j ), w j + f j ≤ f j in  j and w j + f j ≤ w + f ≤ u in  so w j + f j ≤ u j ≤ u ≤ f in . Moreover, since f j ↗ f and w j ↗ w a.e. in  as j ↗ +∞, so w j + f j ↗ w + f a.e. in  as j ↗ +∞. Therefore, w + f ≤ v ≤ f in . Hence, v ∈ F(, f ). This proves the claim. Now, since u j ↗ v a.e. in  as j ↗ +∞, so u j → v in Cn-capacity in . Since u j ≥ w1 + f1 in  and w1 + f1 ∈ E() so by [13], we get (ddcu j )n → (ddcv)n weakly in . Moreover, since f j ↗ f in  as j ↗ +∞ so by [13], we get (ddc f j )n → (ddc f )n = 0 weakly in . Hence, by (8) we get (ddcv)n ≤ (ddcu)n in . Since u, v ∈ F(, f ) and v ≤ u so by Theorem 3.6 in [7], we obtain u = v in . Therefore, u j ↗ u a.e. in  as j ↗ +∞. The proof is complete.  Acknowledgements The author would like to thank the referees for valuable remarks which lead to the improvements of the exposition of the paper. References [1] El Mir H. Fonctions plurisousharmoniques et ensembles pluripolaires. Séminaire Lelong-Skoda. Vol. 822, Lecture notes in math. Springer-Verlag. 1980. p. 61–76. [2] Wiklund J. On subextension of pluriharmonic and plurisubharmonic functions. Ark. Mat. 2006;44:182–190. [3] Cegrell U, ZeriahiA. 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