Vietnam Journal of Mathematics 33:3 (2005) 335–342 A Remark on the Dirichlet Problem * Pham Hoang Hiep Department of Mathematics, Hanoi University of Education, 136 Xuan Thuy Str eet, C au Giay, Hanoi, Vietnam Received October 06, 2004 Revised March 03, 2005 Abstract. Given a positive measure μ on a strongly p seudoconvex domain in C n .We study the Dirichlet problem (dd c u) n = μ in a new class of plurisubharmonic function. This class includes the classes E p (p ≥ 1) introduced by Cegrell in [5]. 1. Introduction. Let Ω be a bounded domain in C n .ByPSH(Ω) we denote the set of plurisub- harmonic (psh) functions on Ω. By the fundamental work of Bedford and Taylor [1, 2], the complex Monge-Ampere operator (dd c ) n is well defined over the class PSH(Ω) ∩ L ∞ loc (Ω) of locally bounded psh functions on Ω, more precisely, if u ∈ PSH(Ω) ∩ L ∞ loc (Ω) is a positive Borel measure. Furthermore, this operator is continuous with respect to increasing and decreasing sequences. Later, De- mailly has extended the domain of definition of the operator (dd c u) n to the class of psh functions which are locally bounded near ∂Ω. Recently in [5, 6], Cegrell introduced the largest class of upper bounded psh functions on a bounded hyper- convex domain Ω such that the operator (dd c u) n can be defined on it. In these papers, he also studied the Dirichlet problems for the classes F p (see Sec. 2 for details). The aim of our work is to investigate the Dirichlet problem for a new class of psh function. This class consist, in particular, the sum of a function in the class E p and a function in B a loc (see Sec. 2 for the definitions of these classes). Now we are able to formulate the main result of our work ∗ This work was supported by the National Research Program for Natural Science, Vietnam 336 Pham Hoang Hiep Main theorem. (i) Let Ω be a bounded strongly pseudoconvex domain in C n and let μ be a positive measure on Ω, h ∈ C(∂Ω) such that there exists v ∈ E p + B a loc (resp. F p + B a loc ) with (dd c v) n ≥ μ Then there exists u ∈ E p + B a loc (resp. F p + B a loc ) such that (dd c u) n = μ and lim z→ξ u(z)=h(ξ), ∀ξ ∈ ∂Ω. (ii) There exists f ∈ L 1 (Ω) such that there exists no function u ∈ E p + B a loc which satisfying fdλ ≤ (dd c u) n . For the definitions of E p + B a loc and F p + B a loc see Sec. 2. Note that the main theorem for the subclass B of B a loc consisting of psh functions which are bounded near ∂Ω was proved by Xing in [13] and for the classes E p and F p ,p≥ 1 by Cegrell in [5]. The key element in the proof of our main theorem is a comparison principle (Theorem 3.1), which is an extension of Lemma 4.4, Theorem 4.5 in [5]. 2. Preliminaries In this section we recall some elements and results of pluripotential theory that will be used through out the paper. All this can be found in [2, 3, 5, 6, 11 ]. 2.0. Unless otherwise specified, Ω will be a bounded hyperconvex domain in C n meaning that there exists a negative exhaustive psh function for Ω . 2.1. Let Ω be a bounded domain in C n .TheC n -capacity in the sense of Bedford and Taylor on Ω is the set function given by C n (E)=C n (E,Ω) = sup E (dd c u) n : u ∈ PSH(Ω), −1 ≤ u ≤ 0 for every Borel set E in Ω. 2.2. According to Xing (see [13]), a sequence of positive measures {μ j } on Ω is called uniformly absolutely continuous with respect to C n in a subset E of Ω if ∀>0, ∃δ>0:F ⊂ E, C n (F ) <δ⇒ μ j (F ) <, ∀j ≥ 1 We write μ j C n in E uniformly for j ≥ 1. 2.3. By B a loc = B a loc (Ω) we denote the set of upper bounded psh functions u which are locally bounded near ∂Ω such that (dd c u) n C n in every E ⊂⊂ Ω. 2.4. The following classes of psh functions were introduced by Cegrell in [5] and [6] E 0 = E 0 (Ω) = ϕ ∈ PSH(Ω) ∩ L ∞ (Ω) : lim z→∂Ω ϕ(z)=0, Ω (dd c ϕ) n < +∞ , E p = E p (Ω) = ϕ ∈ PSH(Ω) : ∃E 0 ϕ j ϕ, sup j≥1 Ω (−ϕ j ) p (dd c ϕ j ) n < +∞ , A Remark o n the Dirichlet Problem 337 F p = F p (Ω) = ϕ ∈ PSH(Ω) : ∃E 0 ϕ j ϕ, sup j≥1 Ω (−ϕ j ) p (dd c ϕ j ) n , < + ∞, sup j≥1 Ω (dd c ϕ j ) n < +∞ E = E(Ω) = ϕ ∈ PSH(Ω) : ∀z 0 ∈ Ω ∃ a neighborhood ω z 0 , E 0 ϕ j ϕ on ω, sup j≥1 Ω (dd c ϕ j ) n < +∞}. The following inclusions are obvious E 0 ⊂F p ⊂E p ⊂E.Itisalsoknown that these inclusion are strict (see [5, 6]). The interesting theorem below was proved by Cegrell (see [6]) Theorem 2.5. The class E has the following properties 1. E is a convex cone. 2. If u ∈E,v∈ PSH − (Ω) = {ϕ ∈ PSH(Ω) : ϕ ≤ 0},thenmax(u, v) ∈E. 3. If u ∈E,PSH(Ω) ∩L ∞ loc (Ω) u j u, then (dd c u j ) n is weakly convergent. Conversely if K⊂PSH − (Ω) satisfies 2 and 3,thenK⊂E Since B − loc = B a loc ∩ PSH − (Ω) satisfies 2 and 3 we have by [8] B − loc ⊂E. 2.6. Cegrell also studied the following Dirichlet problem: Given μ a positive measure on Ω, find u ∈F p such that (dd c u) n = μ. He gave a necessary and sufficient condition for this problem to have a solution (Theorem 5.2 in [5]). 2.7. We define E p + B a loc = u ∈ PSH(Ω) : ∃ ϕ ∈E p ,f∈B a loc : ϕ + f ≤ u ≤ sup Ω u<+∞ , F p + B a loc = u ∈ PSH(Ω) : ∃ ϕ ∈F p ,f∈B a loc : ϕ + f ≤ u ≤ sup Ω u<+∞ . It follows that if ϕ + f ≤ u<sup Ω u<+∞,ϕ∈E,f∈B a loc then u − c =max(u − c, ϕ + f − c) ∈E, because ϕ +(f − c) ∈E, where c =max(sup Ω f, sup Ω u). Thus we can define (dd c u) n for u ∈ E p + B a loc . 2.8. The aim of this work is to study a Dirichlet problem similar to the one considered by Cegrell but for the classes E p + B a loc and F p + B a loc . Namely, given a positive measure μ on Ω and h ∈ C(∂Ω), find u ∈ E p + B a loc (resp. F p + B a loc ) such that (dd c u) n = μ and lim z→ξ u(z)=h(ξ) ∀ξ ∈ ∂Ω. 2.9. Let μ be a positive measure on Ω and h ∈ C(∂Ω). Following Cegrell, we define 338 Pham Hoang Hiep B(μ, h)={v ∈ PSH(Ω) ∩ L ∞ loc (Ω) : (dd c v) n ≥ μ, lim z→ξ v(z) ≤ h(ξ)}, U(μ, h)(z)=sup{v(z):v ∈ B(μ, h)} ,z∈ Ω. Observe that B(μ, h) = ∅ implies that μ vanishes on pluripolar sets. The function U (μ, h) plays a crucial role in solving the Dirichlet problem. 3. The Comparison Principle for E p + B a loc In order to prove the main theorem, in this section we prove the following comparison principle Theorem 3.1. Let u, v be functions in E p + B a loc satisfying lim z→∂Ω [u(z) − v(z)] ≥ 0. Then {u<v} (dd c v) n ≤ {u<v} (dd c u) n . We need the following result Lemma 3.2. Let PSH(Ω) ∩ L ∞ (Ω) u j u. Assume that lim s→+∞ s n C n ({u<−s}∩D)=0, ∀D ⊂⊂ Ω. Then (dd c u j ) n C n in every D ⊂⊂ Ω uniformly for j ≥ 1. Proof. Given D ⊂⊂ Ω. Without loss of generality we may assume that D is hyperconvex and u j ≤ 0onD. By [6] for each j ≥ 1thereexistsu k j ∈ PSH(D) ∩ C( ¯ D) such that u k j u j on D and u k j =0on∂D.Asin[9]for every k, j ≥ 1ands>0 put D kj (s)={u k j < −s}∩D, D j (s)={u j < −s}∩D, D(s)={u<− s}∩D, a kj (s)=C n (D kj (s)),a j (s)=C n (D j (s)),a(s)=C n (D(s)), b kj (s)= D kj (s) (dd c u k j ) n ,b j (s)= D j (s) (dd c u j ) n . For 0 <s<twe have max(u k j , −t)=u k j on {u k j > −t} an open neighborhood of ∂D kj (s). It follows that a kj (s) ≥ t −n D kj (s) (dd c max(u k j , −t)) n = t −n D kj (s) (dd c u k j ) n = t −n b kj (s). A Remark o n the Dirichlet Problem 339 Letting t s we get s n a kj (s) ≥ b kj (s)fork,j ≥ 1ands>0. (1) Given >0. By the hypothesis there exists s 0 > 0 such that s n 0 a(s 0 ) <. (2) Let E ⊂ D with C n (E) < s n 0 . TakeanopenneighborhoodG of E such that C n (G) < s n 0 .Since(dd c u k j ) n → (dd c u j ) n weakly as k →∞we have E (dd c u j ) n ≤ G (dd c u j ) n ≤ lim k→∞ G (dd c u k j ) n ≤ lim k→∞ [ D kj (s 0 ) (dd c u k j ) n + G\D kj (s 0 ) (dd c u k j ) n ] ≤ lim k→∞ [s n 0 a kj (s 0 )+s n 0 C n (G)] ≤ s n 0 a(s 0 )+<2 for j ≥ 1. Hence (dd c u j ) n C n in D uniformly for j ≥ 1. Proof of Theorem 3.1. We may assume that u, v ≤ 0 and lim z→∂Ω [u(z) − v(z)] > δ>0. By hypothesis u, v ∈ E p + B a loc it is easy to find ϕ ∈E p ,g∈B − loc such that ϕ + g ≤ min(u, v). Let ϕ j ϕ be a sequence decreasing to ϕ as in the definition of E p .Foreachj ≥ 1 put g j =max(g, −j),u j =max(u, ϕ j + g j ),v j =max(v, ϕ j + g j ). It follows that g j ,u j ,v j are bounded and g j g, u j u, v j v.Bythe comparison principle for bounded psh functions we have {u j <v k } (dd c v k ) n ≤ {u j <v k } (dd c u j ) n for k ≥ j ≥ 1. On the other hand, since s n C n ({u<−s}∩D) ≤ s n C n ({ϕ<− s 2 }∩D)+s n C n g<− s 2 ∩ D → 0 as s → +∞ (see [5, 9]) By Lemma 3.2 (dd c u j ) n +(dd c v j ) n C n in every D ⊂⊂ Ω uniformly for j ≥ 1. Thus by the quasicontinuity of psh functions as Theorem 2.2.6 in [4] we obtain {u<v} (dd c v) n ≤ {u≤v} (dd c u) n . By replacing u by u + δ, δ > 0andthenletδ 0, we have 340 Pham Hoang Hiep {u<v} (dd c v) n ≤ {u<v} (dd c u) n . This is the desired conclusion. From Theorem 3.1, as Corollary 2.2.8 in [4], we get the following dominant principle. Corollary 3.3. Assume that u and v are as in Theorem 3.1 and (dd c u) n ≤ (dd c v) n .Thenu ≥ v. 4.ProofoftheMainTheorem (i) We can assume v ≤ 0. Since (dd c v) n vanishes on every pluripolar set in Ω, by Theorem 6.3 in [5] we can find ψ ∈E 0 and 0 ≤ f ∈ L 1 loc ((dd c ψ) n ) such that μ = f (dd c ψ) n .Putμ k =min(f,k)(dd c ψ) n .Thenμ k ≤ (dd c k 1 n ψ) n . By Theorem 2 in [13] there exists ω k ∈E 0 such that (dd c ω k ) n = μ k .The comparison principle implies that 0 ≥ ω k ω ≥ v. Hence ω ∈ E p + B a loc and (dd c ω) n = μ. We show that lim z→ξ ω(z)=0forξ ∈ ∂Ω. Assume the contrary, then lim z→ξ 0 ω(z) < − for some ξ 0 ∈ ∂Ω,>0. Take δ>0 such that ω(z) < − for z ∈ B(ξ 0 ,δ) ∩ Ω. Let τ ∈ C(∂Ω) such that τ| B(ξ 0 , δ 2 )∩∂Ω = , suppτ ⊂ B(ξ 0 ,δ) ∩ ∂Ω. By [2] there exists φ ∈ PSH(Ω) ∩ C( ¯ Ω) such that (dd c φ) n =0andφ| ∂Ω = τ. Since lim z→ξ [ω k (z) − (ω(z)+φ(z))] ≥ 0forξ ∈ ∂Ω and (dd c ω k ) n = μ k ≤ μ =(dd c ω) n ≤ (dd c (ω + φ)) n ,wehaveω k ≥ ω + φ on Ω for k ≥ 1. Thus ω ≥ ω + φ on Ω. Hence φ ≤ 0onΩ\{ω = −∞}.Sinceφ is plurisubharmonic, φ ≤ 0 on Ω. This is impossible, because φ(ξ)=τ(ξ)= for ξ ∈ B(ξ 0 , δ 2 ) ∩ ∂Ω. Hence lim z→ξ ω(z)=0forξ ∈ ∂Ω. From the relations U((dd c (ω k + U (0,h))) n ,h)=ω k + U (0,h), (dd c (ω + U(0,h))) n ≥ μ k , and from Theorem 8.1 in [5] it follows that (dd c U(μ k ,h)) n = μ k , U(0,h) ≥ U(μ k ,h) ≥ ω k + U (0,h). Theorem 3.1 implies that U (μ k ,h) u ∈ E p + B a loc with (dd c u) n = μ and U(0,h) ≥ u ≥ ω + U(0,h). Thus for ξ ∈ ∂Ωwehave h(ξ)= lim z→ξ U(0,h) ≥ lim z→ξ u(z) ≥ lim z→ξ [ω(z)+U (0,h)(z)] = lim z→ξ ω(z) + lim z→ξ U(0,h)(z)=h(ξ). Consequently u ∈ E p + B a loc such that (dd c u) n = μ and lim z→ξ u(z)=h(ξ) ∀ξ ∈ ∂Ω. A Remark o n the Dirichlet Problem 341 (ii) Let {Ω j } be an increasing exhaustion sequence of strongly pseudoconvex subdomains of Ω. For each j ≥ 1 take a sequence of distinguished points z jm ⊂ Ω j \Ω j−1 converging to ξ j ∈ ∂Ω j as m →∞and a sequence s j 0such that B(z jm ,s jm )⊂ Ω j \Ω j−1 and B(z jm ,s jm ) ∩ B(z jt ,s jt )=∅ for m = t.Let a jm > 0with ∞ j,m=1 a jm < ∞.Put f = j,m≥1 a jm d n r 2n jm χ B(z jm ,r jm ) , where 0 <r jm <s jm are chosen such that 1 a jm (C n (B(z jm ,r jm ), Ω)) p n+p → 0asm →∞, for j ≥ 1andd n is the volume of the unit ball in C n . Assume that fdλ ≤ (dd c u) n for some u ∈ E p + B a loc .Takeϕ ∈E p ,g∈B a loc such that ϕ+g ≤ u ≤ sup Ω u<+∞. We may assume that g and u are negative. Let j 0 ≥ 2andM>0 such that g>−M on Ω j 0 \Ω j 0 −1 . Put ˜g =max(g, Ah Ω j 0 )whereA = − M sup ¯ Ω j 0 h Ω j 0 > 0. It follows that ˜g ∈E 0 , ˜g = g on Ω j 0 \Ω j 0 −1 . Let ˜u =max(u, ϕ +˜g). Since ϕ +˜g ≤ ˜u ≤ 0andϕ +˜g ∈E p + E 0 = E p ,by [5] we have ˜u ∈E p . Moreover ˜u = u on Ω j 0 \Ω j 0 −1 .ThusforB m = B(z j 0 m ,r j 0 m )wehave a j 0 m = B m fdλ = B m (dd c u) n = B m (dd c ˜u) n . Let ˜u k ˜u as in the definition of E p .Then(dd c ˜u k ) n → (dd c ˜u) n weakly (see [5]). Applying the Holder inequality (see [7]) we have a j 0 m = B m (dd c ˜u) n ≤ lim k→∞ B m (dd c ˜u k ) n = lim k→∞ B m (−h B m ) p (dd c ˜u k ) n ≤ α 1 lim k→∞ [ Ω (−h B m ) p (dd c h B m ) n ] p n+p [ Ω (−˜u k ) p (dd c ˜u k ) n ] n n+p ≤ α 2 [ Ω (dd c h B m ) n ] p n+p = α 2 [C n (B m , Ω)] p n+p , where α 2 = α 1 [sup k≥1 Ω (−˜u k ) p (dd c ˜u k ) n ] n n+p < +∞. This is impossible, because 342 Pham Hoang Hiep lim m→∞ [C n (B m , Ω)] p p+n a j 0 m =0. Remark. Using Theorem 7.5 in [1] we can find u ∈F a such that (dd c u) n = fdλ where f is constructed as in (ii). Hence, there exists a function u in F a \ ( ξ p + B a loc ). Acknowledgements. The author is grateful to Professor Nguyen Van Khue for sug- gesting the problem and for many helpful discussion during the preparation of this work. References 1. P. ˚ Ahag, The Complex Monge–Ampere Operator on Bounded Hyperconvex Do- mains, Ph. D. 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Hiep, A characterization of bounded plurisubharmonic functions, Annales Polonici Math. 85(2005) 233–238. 11. N. V. Khue, P. H. Hiep, Complex Monge-Ampere measures of plurisubharmonic functions which are locally bounded near the boundary, Preprint 2004. 12. M. Klimek, Pluripotential Theory, Oxford, 1990. 13. Y. Xing, Complex Monge-Ampere measures of pluriharmonic functions with bounded values near the boundary, Cand. J. Math. 52 (2000) 1085–1100. . bounded psh functions on a bounded hyper- convex domain Ω such that the operator (dd c u) n can be defined on it. In these papers, he also studied the Dirichlet problems for the classes F p (see. for details). The aim of our work is to investigate the Dirichlet problem for a new class of psh function. This class consist, in particular, the sum of a function in the class E p and a function in. vanishes on pluripolar sets. The function U (μ, h) plays a crucial role in solving the Dirichlet problem. 3. The Comparison Principle for E p + B a loc In order to prove the main theorem, in