THE COMPLEX MONGEAMPERE EQUATION IN ` DOMAIN OF C n

For a C 2 smooth plurisubharmonic function u defined on an open subset Ω of C n , the complex MongeAmp`ere operator is defined by (ddcu) n = n4n det  ∂ 2u ∂zj∂zk  dV, where dV is the volume form. The MongeAmp`ere operator cannot be well defined as a nonnegative Radon measure for an arbitrary plurisubharmonic function. Bedford and Taylor 3 proved in 1982 that the complex MongeAmp`ere operator is defined for locally bounded plurisubharmonic functions. Cegrell 5 has shown that the complex MongeAmp`ere operator is well de fined on the subclass of unbounded plurisubharmonic functions in hyperconvex domains. B locki 4 has defined in 2006 the class D(Ω) of plurisubharmonic functions in an open Ω ⊂ C n , for which the complex MongeAmp`ere operator can be well defined as a nonnegative Radon measure and it is continuous on decreasing sequences of plurisubharmonic functions in this class. Recently, El Kadiri and Wiegerinck 10 defined in 2014 the complex MongeAmp`ere operator for finite Fplurisubharmonic functions on an Fdomain Ω.

` THE COMPLEX MONGE-AMPERE EQUATION IN n DOMAIN OF C NGUYEN XUAN HONG Abstract. In this paper, we prove a generalization of Kolodziej’s subsolution theorem. 1. Introduction For a C 2 -smooth plurisubharmonic function u defined on an open subset Ω of Cn , the complex Monge-Amp`ere operator is defined by (ddc u)n = n!4n det ∂2u ∂zj ∂z k dV, where dV is the volume form. The Monge-Amp`ere operator cannot be well defined as a non-negative Radon measure for an arbitrary plurisubharmonic function. Bedford and Taylor [3] proved in 1982 that the complex MongeAmp`ere operator is defined for locally bounded plurisubharmonic functions. Cegrell [5] has shown that the complex Monge-Amp`ere operator is well defined on the subclass of unbounded plurisubharmonic functions in hyperconvex domains. Blocki [4] has defined in 2006 the class D(Ω) of plurisubharmonic functions in an open Ω ⊂ Cn , for which the complex Monge-Amp`ere operator can be well defined as a non-negative Radon measure and it is continuous on decreasing sequences of plurisubharmonic functions in this class. Recently, El Kadiri and Wiegerinck [10] defined in 2014 the complex MongeAmp`ere operator for finite F-plurisubharmonic functions on an F-domain Ω. Let K(Ω) ⊂ D(Ω) and µ is a non-negative measure in Ω. The Dirichlet problem is to find a plurisubharmonic function u ∈ K(Ω) such that (ddc u) = µ. Bedford and Taylor [2] showed in 1976 that the Dirichlet problem is solvable if µ = f dV with f ∈ C(Ω). Kolodziej [14] proved in 1995 that if there exists a subsolution for the Dirichlet problem then it is solvable. Cegrell [5] proved in 2004 that the Dirichlet problem is solvable if µ vanishes on pluripolar sets and Ω is hyperconvex domains. Recently, ˚ Ahag, Cegrell, Czy˙z and Hiep [1] studied in 2009 the Dirichlet problem for non-negative measures carry by a pluripolar set. The purpose of this paper is to study the Dirichlet problem for domain of Cn . We will investigate this problem by using the F-maximal F-plurisubharmonic functions. Namely, we prove the following. 2010 Mathematics Subject Classification: 32U05 Key words and phrases: Plurisubharmonic function, Monge-Amp`ere equation 1 2 NGUYEN XUAN HONG Theorem 1.1. Let Ω be a domain in Cn and let w ∈ D(Ω), f ∈ D(Ω) ∩ M P SH(Ω) with f w in Ω. Then for every non-negative measure µ in c n Ω with µ (dd w) there exists u ∈ D(Ω) such that w u f and (ddc u)n = µ in Ω. In the following, we give an example to show that there exists a bounded domain Ω and the non-negative measure µ in Cn such that the MongeAmp`ere equation (ddc u)n = µ is solvable in Ω, but is not solvable in every larger hyperconvex domains. n j=2 {|zj | Example 1.2. Let n 2 and Ω := {0 < |z1 | < 1} × Ω is not hyperconvex domain in Cn . Let < n1 }. Then, w(z) := (− log |z1 |)1/n (|z2 |2 + · · · + |zn |2 − 1). Since w ∈ P SH(Ω) ∩ C ∞ (Ω) so w ∈ D(Ω). By computation we get 1 (ddc w)n = +∞, ∀δ ∈ (0, ). n B(0,δ)\{z1 =0} If U be bounded hyperconvex domain containing Ω then B(0, δ) every δ ∈ (0, n1 ). It follows that the Monge-Amp`ere equation (ddc u)n = 1B(0,δ)\{z1 =0} (ddc w)n U for (1.1) is not solvable in U . However, by Theorem 1.1 the equation (1.1) is solvable in Ω. The organization of the paper is as follows. In Section 2 we recall some notions of pluripotential theory which is necessary for the next results of the paper. In Section 3 we prove the main result of the paper. 2. Preliminaries Some elements of pluripotential theory that will be used throughout the paper can be found in [1]-[15]. Let Ω be a domain in Cn . We denote by P SH(Ω) (P SH − (Ω)) the family of plurisubharmonic (negative plurisubharmonic) functions. By M P SH(Ω), we denote the set of maximal plurisubharmonic functions. 2.1. Cegrell’s classes We recall some Cegrell’s classes of plurisubharmonic functions. Let Ω be an bounded hyperconvex domain in Cn . Put E0 (Ω) = ϕ ∈ P SH − (Ω) ∩ L∞ (Ω) : lim ϕ(z) = 0, F(Ω) = z→∂Ω ϕ ∈ P SH − (Ω) : ∃E0 ϕj (ddc ϕ)n < ∞ , Ω (ddc ϕj )n < ∞ ϕ, sup j Ω and E(Ω) = ϕ ∈ P SH − (Ω) : ∀G Ω, ∃uG ∈ F(Ω), u = uG on G . ` THE COMPLEX MONGE-AMPERE EQUATION 3 If f ∈ E(Ω) then we say that a plurisubharmonic function u defined on Ω belong to F(Ω, f ) if there exists a function ϕ ∈ F(Ω) such that ϕ+f u f on Ω. 2.2. The classes D(Ω) Let Ω be a domain in Cn . Following [4], Blocki has defined in 2006 the class D(Ω) as follows: a plurisubharmonic function u belongs to D(Ω) if there exists a non-negative Radon measure µ on Ω such that if U ⊂ Ω is open and a sequence uj ∈ P SH ∩ C ∞ (U ) decreases to u in U then (ddc uj )n tends weakly to µ in U . The Monge-Amp`ere measure µ we then denote by (ddc u)n . Note that D(Ω) = {u ∈ P SH(Ω) : ∀B(a; r) Ω, ∃c > 0, u − c ∈ E(B(a, r))}. Moreover, if Ω be bounded hyperconvex domain then E(Ω) = D(Ω) ∩ P SH − (Ω). 2.3. The F-maximal F-plurisubharmonic functions The plurifine topology F on open subsets of Cn is the weakest topology in which all plurisubharmonic functions are continuous. Notions pertaining to the plurifine topology are indicated with the prefix F and notions pertaining to the fine topology are indicated with Cn . For a set A ⊂ Cn we write A F for the closure of A in the one point compactification of Cn , A for the Fclosure of A and ∂F A for the F-boundary of A. We denote by F-P SH(Ω) the set of F-plurisubharmonic functions on an F-open set Ω. We say that u is F-maximal if for every bounded F-open set G of Cn such that G ⊂ Ω, and for every function v ∈ F-P SH(G) that is bounded from F above on G and extends F-upper semicontinuously to G , the following holds: v u on ∂F G imply v u on G. Denote the class of all F-maximal F-plurisubharmonic functions in an F-open set Ω by F-M P SH(Ω). 3. Proof of Theorem 1.1 First, we need the following auxiliary result. The idea of the proof is used the F-maximal F-plurisubharmonic functions. Lemma 3.1. Let Ω ⊂ Cn be a domain and let D Ω be a hyperconvex domain. Assume that f ∈ D(Ω) ∩ M P SH(Ω) and v ∈ F(D) is carried by pluripolar set of U with U D. If c is non-negative constant such that f c in D and the function u = (sup{ϕ ∈ P SH(Ω) : ϕ f in Ω and ϕ F v + c in U })∗ belong to D(Ω) then (ddc u)n = 1D (ddc v)n . Proof. Without loss of generality we can assume that c = 0. First, we claim that u is F-maximal F-plurisubharmonic function in {−j < u < v} ∩ D for all j 1. Indeed, let G be a bounded F-open set such that G ⊂ {−j < u < v} ∩ D and let ψ ∈ F-P SH(G) be bounded from above on G, F-upper 4 NGUYEN XUAN HONG semicontinuous on G the function F such that ψ ϕ= u on ∂F G. By Proposition 2.3 in [9], max(ψ, u) u on G, on Ω\G belong to F-P SH(Ω). Hence, by Proposition 2.14 in [8] we have ϕ ∈ P SH(Ω). Since (ddc v)n = 0 in {v > −j} ∩ D so by Theorem 4.11 in [9] we have v is F-maximal F-plurisubharmonic function in {v > −j} ∩ D. Therefore, ψ v in G, and hence, ϕ v in U . Similarly, we also have ϕ f in Ω. It follows that ϕ u in Ω. Hence, ψ u in G. Thus, u is F-maximal F-plurisubharmonic function in {−j < u < v} ∩ D. This proves the claim. Therefore, by Theorem 4.8 in [9] we get (ddc u)n = 0 in {−j < u < v} ∩ D, and hence, (ddc u)n = 0 in {−∞ < u v} ∩ D. Since F F u v in U so (ddc u)n = 0 in {u > −∞} ∩ U . Analysis similar shows F F that u ∈ F-M P SH(Ω\U ). Since U ⊂ U so from Proposition 2.4 in [9] we have u ∈ M P SH(Ω\U ). Therefore, (ddc u)n = 0 in (Ω\D) ∩ ({u > −∞} ∩ D). Now, let k0 ∈ N∗ such that D B(0, k0 ). For k uk = (sup{ϕ ∈ P SH(Ω∩B(0, k)) : ϕ k0 , put f in Ω∩B(0, k) and ϕ F v in U })∗ . Since uk u in Ω ∩ B(0, k) so uk ∈ D(Ω ∩ B(0, k)). By Theorem 3.4 in [11] there exists vk ∈ F(B(0, k)) such that (ddc vk )n = 1D (ddc v)n in B(0, k). Since (ddc uk )n = 0 in ((Ω ∩ B(0, k))\D) ∪ ({uk > −∞} ∩ D), vk + f uk in Ω ∩ B(0, k) and uk v in U so from Lemma 4.1 in [1] and Lemma 4.12 in [1] we get 1D (ddc v)n = 1{v=−∞}∩U (ddc v)n 1{uk =−∞}∩U (ddc uk )n (ddc uk )n = 1{uk =−∞} (ddc uk )n 1{vk =−∞} (ddc vk )n = 1{v=−∞}∩D (ddc v)n = 1D (ddc v)n in Ω∩B(0, k). Therefore, (ddc uk )n = 1D (ddc v)n in Ω∩B(0, k). Since uk so from [7] we obtain (ddc u)n = 1D (ddc v)n in Ω. The proof is complete. u Lemma 3.2. Let Ω be bounded hyperconvex domain in Cn . Assume that uj ∈ E(Ω), j = 1, . . . , m such that (ddc uj )n be carried by pluripolar sets of Ω, (ddc uj )n = 0 in {uj = −∞} ∩ {uk = −∞} for any j = k and (ddc (u1 + · · · + um ))n < +∞. Ω Then 1{u1 +···+um =−∞} (ddc (u1 + · · · + um ))n = 1{u1 =−∞} (ddc u1 )n + · · · + 1{um =−∞} (ddc um )n . Proof. The statement is clear for m = 1. Assume that the statement has proved for m − 1. Put v = u1 + · · · + um−1 . Since (ddc v)n = (ddc u1 )n + · · · + (ddc um−1 )n in {v = −∞} so (ddc v)n = (ddc um )n = 0 in {v = −∞} ∩ {um = −∞}. Hence, from Lemma 4.4 in [1] we have (ddc (v + um ))n {v+um =−∞} ` THE COMPLEX MONGE-AMPERE EQUATION (ddc v)n + = {v=−∞}\{um >−∞} 5 (ddc um )n {um =−∞}\{v>−∞} [(ddc v)n + (ddc um )n ] {v+um =−∞} Moreover, since (ddc (v + um ))n ddc v)n + (ddc um )n so (ddc (v + um ))n = c n c n (dd v) + (dd um ) in {v + um = −∞}. Therefore, 1{u1 +···+um =−∞} (ddc (u1 + · · · + um ))n = 1{v+um =−∞} (ddc (v + um ))n = 1{v+um =−∞} (ddc v)n + 1{v+um =−∞} (ddc um )n = 1{u1 =−∞} (ddc u1 )n + · · · + 1{um =−∞} (ddc um )n . The proof is complete. We now give the proof of Theorem 1.1. Proof of Theorem 1.1. We consider two cases. Case 1. µ is carried by pluripolar set of Ω. Choose {aj } ⊂ Ω and {rj } of positive real number such that B(aj , 2rj ) Ω and Ω = ∞ j=1 B(aj , rj ). Choose cj 0 such that f < cj in B(aj , 2rj ). From Theorem 4.14 in [1] there exists vj ∈ F(B(aj , 2rj )) such that vj w − cj and (ddc vj )n = 1B(aj ,rj )\ j−1 k=1 B(ak ,rk ) µ in B(aj , 2rj ). Put ψj = (sup{ϕ ∈ P SH(Ω) : ϕ Since w f in Ω and ϕ F vj + cj in B (aj , 2rj )})∗ . f in Ω so ψj ∈ D(Ω). From Lemma 3.1 we have ψj c (dd ψj )n = 1B(aj ,2rj ) (ddc vj )n = 1B(aj ,rj )\ j−1 k=1 B(ak ,rk ) µ (3.1) in Ω. Put uj = (sup{ϕ ∈ P SH(Ω) : ϕ ψk , k = 1, 2, . . . , j})∗ . Since w uj f in Ω so uj ∈ D(Ω). Analysis similar to that in the proof of Lemma 3.1 shows that (ddc uj )n = 0 in {uj > −∞}. On the other hand, since ψ1 + · · · + ψj Lemma 4.1 in [1] and Lemma 3.2 we get 1{ψk =−∞} (ddc ψk )n uj (3.2) ψk , k = 1, . . . , j so from 1{uj =−∞} (ddc uj )n 1{ψ1 +···+ψj =−∞} (ddc (ψ1 + · · · + ψj ))n = 1{ψ1 =−∞} (ddc ψ1 )n + · · · + 1{ψj =−∞} (ddc ψj )n . Combining this with (3.1) and (3.2) we obtain 1B(ak ,rk )\ k−1 h=1 B(ah ,rh ) µ (ddc uj )n 1 for all k = 1, . . . , j. Therefore, (ddc uj )n = 1 j k=1 B(ak ,rk ) µ. j h=1 B(ah ,rh ) µ 6 NGUYEN XUAN HONG Put u = limj→+∞ uj . Since w uj f so w Moreover, since uj u so from [7] we get (ddc u)n = 1 ∞ k=1 B(ak ,rk ) µ u f and u ∈ D(Ω). = µ in Ω. Case 2. µ is arbitrary measure. Since 1{w=−∞} µ is carried by pluripolar set {w = −∞} of Ω so case 1 there exists v ∈ D(Ω) such that w v f c n and (dd v) = 1{w=−∞} µ in Ω. Put u = (sup{ϕ ∈ D(Ω) : ϕ 1{w>−∞} µ})∗ . v and (ddc ϕ)n We have u ∈ D(Ω) and w u v. We claim that (ddc u)n µ in Ω. Indeed, since the measure 1{w>−∞} µ vanishes on pluripolar sets of Ω so by using the Choquet lemma and Proposition 4.3 in [13] we can choose a increasing sequence {ϕj } ∈ P SH(Ω) such that w ϕj v, (ddc ϕj )n 1{w>−∞} µ and ϕj u a.e. in Ω. Since c n ϕj → u in Cn -capacity in Ω so by [7] we have (dd ϕj ) → (ddc u)n weakly in Ω. Thus, (ddc u)n 1{w>−∞} µ in Ω. It follows that 1{w>−∞} (ddc u)n Moreover, since u 1{w>−∞} µ in Ω. (3.3) v so from Lemma 4.1 in [1] we infer 1{u=−∞} (ddc u)n 1{v=−∞} (ddc v)n = 1{v=−∞} µ. Combining this with (3.3) we obtain (ddc u)n claim. Next, assume that B(a, r) Ω and define ϕa,r = sup{ϕ ∈ P SH(Ω) : ϕ µ in Ω. This proves the u in Ω\B(a, r)}. Then, ϕa,r ∈ D(Ω) ∩ M P SH(B(a, r)), ϕa,r u in Ω and ϕa,r = u in Ω\B(a, r). Without loss of generality we assume that ϕa,r < 0 in B(a, r). From Theorem 2.1 in [6] we have u|B(a,r) ∈ F(B(a, r), ϕa,r ). Since 1{u=−∞} (ddc u)n 1{v=−∞} (ddc v)n in B(a, r) so from the proof of Proposition 5.1 in [12] there exists ψa,r ∈ F(B(a, r), ϕa,r ) such that u ψa,r v in B(a, r) and (ddc ψa,r )n = µ in B(a, r). Put ua,r = ψa,r u in B(a, r), in Ω\B(a, r). Since ψa,r = u in ∂B(a, r) so ua,r ∈ P SH(Ω). We have u = ua,r in Ω\B(a, r), u ua,r v in Ω, (ddc ua,r )n µ in Ω\B(a, r) and (ddc ua,r )n = µ in B(a, r). Now, we prove that (ddc u)n = µ in Ω. Indeed, fix B(a, r) Ω. Choose {rj } be decreasing sequence of positive real numbers that converge to r such that B(a, rj+1 ) B(a, rj ) Ω for all j 1. We claim that {ua,rj } is decreasing sequence. Indeed, since ua,rj+1 = u ua,rj in ∂B(a, rj+1 ) so max(ua,rj+1 , ua,rj ) = ua,rj in ∂B(a, rj+1 ). Moreover, from Lemma 4.1 in [1] and Theorem 4.1 in [13] we have (ddc max(ua,rj+1 , ua,rj ))n (ddc ua,rj )n in B(a, rj+1 ). ` THE COMPLEX MONGE-AMPERE EQUATION 7 Hence, by Lemma 3.3 in [1] and Lemma 3.6 in [1] we get max(ua,rj+1 , ua,rj ) = ua,rj in B(a, rj+1 ). Thus, ua,rj+1 ua,rj in B(a, rj+1 ). This proves the claim. Put ga,r = limj→+∞ ua,rj . We have u g v in Ω. Let k ∈ N∗ . Since (ddc ua,rj )n µ in Ω\B(a, rk ) for all j k so (ddc ga,r )n µ in Ω\B(a, rk ). Let k → +∞ we get (ddc ga,r )n µ in Ω\B(a, r). Moreover, since (ddc ua,rj )n = µ in B(a, r) so (ddc ga,r )n = µ in B(a, r) and (ddc ga,r )n µ in ∂B(a, r). Therefore, we have (ddc ga,r )n µ in Ω. Since u ga,r v so u ga,r in Ω. Moreover, since u ua,rj so u ga,r . Hence, u = ga,r in Ω. It follows that (ddc u)n = (ddc ga,r )n = µ in B(a, r). Therefore, (ddc u)n = µ in Ω. The proof is complete. Acknowledgment. This work was written during visits of the author at the Vietnam Institute for Advanced Study in Mathematics. He wishes to thank this institutions for their kind hospitality and support. He also thank Pham Hoang Hiep for many useful discussions. References [1] P. ˚ Ahag, U. Cegrell, R. Czy˙z and P. H. Hiep, Monge-Amp`ere measures on pluripolar sets, J. Math. Pures Appl., 92 (2009), 613–627. [2] E. Bedford and B. A. Taylor, The Dirichlet problem for a complex Monge-Amp`ere operator, Invent. Math., 37 (1976), 1–44. [3] E. Bedford and B. A. Taylor, A new capacity for plurisubharmonic functions. Acta Math., 149 (1982), 1–41. [4] Z. Blocki, The domain of definition of the complex Monge-Amp`ere operator, Amer. J. Math., 128 (2006), 519–530. [5] U. Cegrell, The general definition of the complex Monge-Amp`ere operator, Ann. Inst. 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Kolodziej, The complex Monge-Amp`ere equation and pluripotential theory, Memoirs of AMS., 840 (2005). 8 NGUYEN XUAN HONG Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy Street, Caugiay District, Hanoi, Vietnam E-mail address: xuanhongdhsp@yahoo.com  ... the plurifine topology are indicated with the prefix F and notions pertaining to the fine topology are indicated with Cn For a set A ⊂ Cn we write A F for the closure of A in the one point compactification... 2.3 The F-maximal F-plurisubharmonic functions The plurifine topology F on open subsets of Cn is the weakest topology in which all plurisubharmonic functions are continuous Notions pertaining... can choose a increasing sequence {ϕj } ∈ P SH(Ω) such that w ϕj v, (ddc ϕj )n 1{w>−∞} µ and ϕj u a.e in Ω Since c n ϕj → u in Cn -capacity in Ω so by [7] we have (dd ϕj ) → (ddc u )n weakly in