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VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY INSTITUTE OF MATHEMATICS DO THAI DUONG SOME PROBLEMS IN PLURIPOTENTIAL THEORY DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN MATHEMATICS HANOI - 2021 luan an VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY INSTITUTE OF MATHEMATICS DO THAI DUONG SOME PROBLEMS IN PLURIPOTENTIAL THEORY Speciality: Mathematical Analysis Speciality code: 9460102 (62 46 01 02) DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN MATHEMATICS Supervisor: Prof Dr.Sc PHAM HOANG HIEP Prof Dr.Sc DINH TIEN CUONG HANOI - 2021 luan an Confirmation This dissertation was written on the basis of my research works carried out at Institute of Mathematics, Vietnam Academy of Science and Technology, under the supervision of Prof Dr.Sc Pham Hoang Hiep and Prof Dr.Sc Dinh Tien Cuong All the presented results have never been published by others January 3, 2021 The author Do Thai Duong i luan an Acknowledgments First of all, I am deeply grateful to my academic advisors, Professor Pham Hoang Hiep and Professor Dinh Tien Cuong, for their invaluable help and support I am sincerely grateful to IMU (The International Mathematical Union), FIMU (Friends of the IMU) and TWAS (The World Academy of Sciences) for supporting my PhD studies through the IMU Breakout Graduate Fellowship The wonderful research environment of the Institute of Mathematics, Vietnam Academy of Science and Technology, and the excellence of its staff have helped me to complete this work within the schedule I would like to thank my colleagues for their efficient help during the years of my PhD studies Especially, I would like to express my special appreciation to Do Hoang Son for his valuable comments and suggestions on my research results I also would like to thank the participants of the weekly seminar at Department of Mathematical Analysis for many useful conversations Furthermore, I am sincerely grateful to Prof Le Tuan Hoa, Prof Phung Ho Hai, Prof Nguyen Minh Tri, Prof Le Mau Hai, Prof Nguyen Quang Dieu, Prof Nguyen Viet Dung, Prof Doan Thai Son for their guidance and constant encouragement Valuable remarks and suggestions of the Professors from the Department-level PhD Dissertation Evaluation Committee and from the two anonymous independent referees are gratefully acknowledged Finally, I would like to thank my family for their endless love and unconditional support ii luan an (Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi Contents Table of Notations v Introduction x Chapter A comparison theorem for subharmonic functions 1.1 Some basic properties of subharmonic functions 1.2 Some basic properties of Hausdorff measure 1.3 An extension of the mean value theorem 1.4 A comparison theorem for subharmonic functions 13 1.5 Other versions of main results 16 Chapter Complex Monge-Ampère equation in strictly pseudoconvex domains 18 2.1 Some properties of plurisubharmonic functions 19 2.2 Domain of Monge-Ampère operator and notions of Cegrell classes 21 2.3 Some basic properties of relative capacity 25 2.4 Dirichlet problem for the Monge-Ampère equation is strictly pseudoconvex 27 A remark on the class E 31 2.5 Chapter Decay near boundary of volume of sublevel sets of plurisubharmonic functions 36 3.1 Some properties of the class F 37 3.2 An integral theorem for the class F 39 3.3 Some necessary conditions for membership of the class F 42 3.4 A sufficient condition for membership of the class F 46 iii (Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi luan an (Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi List of Author’s Related Papers 50 References 51 iv (Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi luan an (Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi Table of Notations N R C Rn Cn Bn B2n ∂ Bn ∂ B2n B(x, r) B(x, r) ∂ B(x, r) Vn V2n σ ∅ ∥x∥ A(Ω) C(Ω) C k (Ω) C0k (Ω) C ∞ (Ω) the set of positive integers the set of real numbers the set of complex numbers the real vector space of dimension n the complex vector space of dimension n the unit ball in Rn the unit ball in Cn the unit sphere in Rn the unit sphere in Cn the open ball of center x and radius r in real vector space or complex vector space the closed ball of center x and radius r in real vector space or complex vector space the sphere of center x and radius r in real vector space or complex vector space the Lebesgue measure on Rn the Lebesgue measure on Cn the surface measure (in any dimension) on a surface the empty set the norm of a vector x the set of analytic functions on Ω the set of continuous functions on Ω the set of k−times differentiable functions with derivatives of order k are continuous on Ω the set of k−times differentiable functions with derivatives of order k are continuous and compact support on Ω the set of smooth functions on Ω v (Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi luan an (Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi C0∞ (Ω) the set of smooth functions with compact support on Ω E0 (Ω), E(Ω), F(Ω), N (Ω) Cegrell’s classes on Ω H(Ω) the set of harmonic functions on Ω USC(Ω) the set of upper semicontinuous functions on Ω ∞ L (Ω) the set of bounded functions on Ω ∞ Lloc (Ω) the set of locally bounded functions on Ω Lp (Ω) the set of p-th power integrable functions on Ω p Lloc (Ω) the set of locally p-th power integrable functions on Ω SH(Ω) the set of subharmonic functions on Ω PSH(Ω) the set of plurisubharmonic functions on Ω − PSH (Ω) the set of negative plurisubharmonic functions on Ω MPSH(Ω) the set of maximal plurisubharmonic functions on Ω OX,z the space of germs of holomorphic functions at a point z ∈ X u∗v the convolution of u and v vi (Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi luan an (Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi Introduction It has been known since the 19th century that gravity and electrostatic forces are two fundamental forces of nature They were believed to be derived from the use of functions so-called “potentials”, that satisfied Laplace’s equation The term “(classical) potential theory” arose to describe a linear theory associated to the Laplacian This theory focused on harmonic functions, subharmonic functions, the Dirichlet problem, harmonic measure, Green’s functions, potentials and capacity in several real variables The potential theory in two dimensional space, which is always considered as the potential theory in the complex plane, has attracted considerable interest since it is closely related to complex analysis In particular, there is a connection between Laplace’s equation and analytic functions While the real and imaginary parts of analytic functions of a complex variable satisfy the Laplace’s equation in two dimensions, the solution to Laplace’s equation is the real part of an analytic function In general, some techniques of complex analysis, particularly conformal mapping, can be used to simplify proofs of some results in the potential theory, while some theorems in the potential theory have analogies and applications in complex analysis In the 20th century, pluripotential theory was developed as the complex multivariate analogue of the classical potential theory in the complex plane This theory is highly non-linear and associated to complex Monge-Ampère operators The basic objects are plurisubharmonic functions of several complex variables that were defined in 1942 by Kiyoshi Oka and Pierre Lelong This class is the natural counterpart of the class of subharmonic functions of one complex variable The plurisubharmonic functions are also considered as subharmonic functions on several real variables which are invariant with respect to all biholomorphic coordinate systems In this dissertation, we study some specific problems in the pluripotential theory and the potential theory vii (Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi luan an (Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi In Chapter 1, motivated by the fact that two subharmonic functions which agree almost everywhere on a domain with respect to Lebesgue measure must coincide everywhere on that domain, we are interested in the following problem Problem Whether we can conclude that two subharmonic functions on a domain of Rn which agree almost everywhere on a m−dimensional submanifold with respect to m-dimensional Hausdorff measure must coincide everywhere on that submanifold? Chapter is devoted to answer Problem completely For this purpose, we prove two main theorems with similar assumptions They concern restrictions of subharmonic functions in Ω to a Borel subset K ⊂ Ω which, together with a measure µ, is subject to some technical assumptions These allow K to have co-dimension one (and a little more, but not two), with µ being more or less a corresponding Hausdorff measure The first main result (Theorem 1.3.3) is an extension of the mean value theorem It states that the mean value theorem in an infinitesimal form still holds when restricted to K , and with respect to µ The second main result (Theorem 1.4.1) is a comparison theorem for subharmonic functions It states that a comparison between an upper semicontinuous function and a subharmonic function which holds almost everywhere (with respect to µ) on K actually holds at every point of K By these theorems, we prove that Problem has a positive answer in the case of hypersurfaces We also provide a counterexample (Example 1.4.4) in the case of subspaces of higher co-dimension In addition, we apply the main theorems to Ahlfors-David regular sets to obtain some consequences, and prove other versions of the main results in terms of measure densities In Chapter 2, we study the Dirichlet problem for the complex Monge-Ampère equation We are interested in the following problem Problem Find conditions for µ such that the solution u of Dirichlet problem for complex Monge-Ampère equation is continuous outside an analytic set but u may not be continuous in Ω This problem arises from the fact that there are some plurisubharmonic functions which are not continuous in the whole domain, though they are continuous outside an analytic set For example, u(z) = −(− log ∥z∥)1/2 is not continuous in the whole unit ball B2n , but it is continuous in B2n \{0} In studying this problem, we prove a sufficient condition (Theorem 2.4.8) which relaxes assumptions of a wellknown result of Kolodziej (Theorem B in [26]) to some technical assumptions These assumptions naturally lead to the following problem viii (Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi luan an (Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi F(H) = {v ∈ E : f + H ≤ v ≤ H for some f ∈ F} The following result was shown in [16] Theorem 3.1.10 Let Ω be a bounded hyperconvex domain Suppose u ∈ E(Ω) ∫ and Ω (ddc u)n < ∞ Then, u ∈ F (¯ u) By Corollary 3.1.8, Theorem 3.1.10 and Corollary 3.1.3, we can see that ∫ F(Ω) = {u ∈ N (Ω) : (ddc u)n < ∞} (3.1) Ω The class F can be used to characterize the boundary behavior in the Dirichlet problem for the Monge-Ampère equation in the class E Formally, the problem can be stated as follows: given a positive Borel measures µ on Ω and a maximal plurisubharmonic function H ∈ E ∩ MPSH(Ω), find (if it exists) u ∈ E such that (ddc u)n = µ and u ¯ = H By Theorem 3.1.10, if µ has finite mass then the solutions of the Dirichlet problem belong to F(H) It is inconvenient to recall here all results about the existence of solution In order to give the readers some insight into the problem, we combine some results in [1, 2, 14] to state the following theorem: Theorem 3.1.11 Let µ1 , µ2 be positive Borel measures on such that ã = à1 + à2 has finite mass; ã à1 vanishes on every pluripolar set; • (ddc f )n ≥ µ2 for some f ∈ F Then, for every H ∈ E ∩MPSH(Ω), there exists u ∈ F (H) such that (ddc u)n = µ Moreover, if µ2 = then u is unique 3.2 An integral theorem for the class F First, we introduce an integral theorem for plurisubharmonic functions which is a direct consequence of Theorem 2.6.5 in [25] and the definition of plurisubharmonic functions Lemma 3.2.1 Let Ω ⊂ Cn be a connected open set and (X, µ) be a σ−finite measure space Suppose that u : Ω × X → [−∞, +∞) is a measurable function such that 39 (Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi luan an (Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi (i) For every a ∈ X , u(·, a) ∈ PSH(Ω) (ii) There exists g ∈ L1 (µ) such that for every z ∈ Ω, a ∈ X , we have u(z, a) ≤ g(a) Then the function ∫ u˜(z) = u(z, a)dµ(a) X is either plurisubharmonic in Ω or identically −∞ Next, we establish an integral theorem for the class F which will be used to prove a necessary condition for membership of the class F in case of unit ball Lemma 3.2.2 Let Ω ⊂ Cn be a bounded hyperconvex domain and (X, µ) be a totally bounded metric probability space Let u : Ω×X → [−∞, 0) is a measurable function such that (i) For every a ∈ X , u(·, a) ∈ F (Ω) and ∫ (ddc u(z, a))n < M , Ω where M > is a constant (ii) For every z ∈ Ω, the function u(z, ·) is upper semicontinuous in X ∫ Then u ˜(z) = u(z, a)dµ(a) ∈ F (Ω) Moreover X ∫ (ddc u˜)n ≤ M Ω Proof By Lemma 3.2.1, we have that either u ˜ ∈ PSH− (Ω) or u˜ ≡ −∞ We need to find a sequence of functions u ˜j ∈ F (Ω) such that u˜j is decreasing to u˜ as j → ∞ and ∫ (ddc u˜j )n ≤ M sup j Ω Since X is totally bounded, there exists a finite cover {Xk }m k=1 of X such that the diameter of each Xk is at most 1/2 Denote −1 U1,1 = X1 , U1,2 = X2 \ X1 , , U1,m1 = Xm1 \ (∪m l=1 Xl ) Then {U1,k }m k=1 is a finite cover of X such that its elements are pairwise disjoint and of diameter at most 1/2 Moreover, U1,k is totally bounded for every k 40 (Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi luan an (Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi By using induction, for every j ∈ Z+ , we can divide X into a finite pairwise mj disjoint collection {Uj,k }k=1 of sets of diameter at most 2−j satisfying: for every ≤ k ≤ mj+1 , there exists ≤ l ≤ mj such that Uj+1,k ⊂ Uj,l For every j ∈ Z+ , we define uj (z) = mj ∑ µ(Uj,k ) sup u(z, a) u˜j = (uj )∗ and a∈Uj,k k=1 Then u ˜j ∈ F (Ω) Moreover, by using Corollary 3.1.7 for u˜j and (with ak ∈ Uj,k ) and by applying Proposition 3.1.5, we have ∫ ∫ (dd u˜j ) ≤ c n Ω c (dd ( Ω ∑ = k1 + +kmj ∑ ≤ k1 + +kmj µ(Uj,k )u(z, ak ) k=1 µ(Uj,k )u(z, ak )))n k=1 ∏ n! µ(Uj,l )kl k ! kmj ! l=1 =n mj ∫ (ddc u(z, a1 ))k1 ∧ ∧ (ddc u(z, amj ))kmj Ω mj ∫ ∏ ∏ n! µ(Uj,l )kl ( (ddc u(z, al ))n )kl /n k ! kmj ! l=1 =n l=1 mj Ω ∑ ≤M mj ∑ mj ∑ k1 + +kmj ∏ n! µ(Uj,l )kl k ! kmj ! l=1 =n mj = M (µ(Uj,1 ) + + µ(Uj,kmj ))n = M, for all j ∈ Z+ We will show that u ˜j is decreasing to u˜ and use Proposition 3.1.4 to prove that u˜ ∈ F (Ω) For every z ∈ Ω, a ∈ X and j ∈ Z+ , we define ϕj (z, a) = mj ∑ χUj,k (a) sup u(z, a) = k=1 a∈Uj,k sup u(z, ξ), ξ∈Uj,k(j,a) where χUj,k is the characteristic function of Uj,k and k(j, a) is given by a ∈ Uj,k(j,a) Then, we have ∫ ∫ uj (z) = ϕj (z, a)dµ(a) ≥ u(z, a)dµ(a) = u˜(z), (3.2) X X for every z ∈ Ω and j ∈ Z+ 41 (Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi luan an (Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi Note that Uj+1,k(j+1,a) ∩ Uj,k(j,a) ̸= ∅ Then, it follows from the construction of the sets Uj,k that Uj+1,k(j+1,a) ⊂ Uj,k(j,a) Hence ∫ ∫ uj (z) = ϕj (z, a)dµ(a) ≥ ϕj+1 (z, a)dµ(a) = uj+1 (z), (3.3) X X for every z ∈ and j ∈ Z + By the semicontinuity of u(z, ·), we have, u(z, a) ≥ lim (sup{u(z, ξ) : |ξ − a| ≤ 2−j }) ≥ lim ϕj (z, a), j→∞ j→∞ (3.4) for every z ∈ Ω and a ∈ X Integrating both sides of (3.4) with respect to a and using Fatou’s lemma, we get u˜(z) ≥ lim uj (z), (3.5) j→∞ for every z ∈ Ω Combining (3.2), (3.3) and (3.5), we get that uj is decreasing to u ˜ as j → ∞ Note that, by Theorem 2.1.3, uj = u ˜j almost everywhere and then lim u˜j = u˜ j→∞ almost everywhere Since lim u ˜j is either plurisubharmonic or identically −∞, j→∞ we have lim u ˜j = u˜ everywhere Therefore, u˜j is decreasing to u˜ as j → ∞ j→∞ By Proposition 3.1.4, max{˜ u, −k} ∈ F(Ω) for k > and it implies that u˜ is not identically −∞ Then, by using Proposition 3.1.4 for u ˜, we get that u˜ ∈ F (Ω) Moreover, since the sequence u ˜j is decreasing, we have ∫ ∫ (ddc u˜)n ≤ lim inf (ddc u˜j )n ≤ M Ω 3.3 j→∞ Ω Some necessary conditions for membership of the class F In this section, we estimate the size of sublevel sets of the class F For convenience, we denote Wd = {z ∈ Ω|dist(z, ∂Ω) < d} Theorem 3.3.1 Assume that Ω is a bounded strictly pseudoconvex domain in Cn and u ∈ F (Ω) Then, there exists C > depending only on Ω, n and u such that 42 (Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi luan an (Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi C.dn+1−na V2n ({z ∈ Wd , u(z) < −ϵ}) ≤ , (1 − a)an ϵn for any < ϵ, a < and d > (3.6) ¯ [−1, 0]) Proof Since Ω is bounded strictly pseudoconvex, there exists ρ ∈ C (Ω, such that Ω = {z : ρ(z) < 0} and ¯ |Dρ| > C1 in Ω, (3.7) ddc ρ ≥ C2 ddc |z|2 = C2 ω, (3.8) and where C1 , C2 > are constants By (3.7), there exist C3 , C4 > depending only on Ω and ρ such that C3 dist(z, ∂Ω) ≤ −ρ(z) ≤ C4 dist(z, ∂Ω), (3.9) for every z ∈ Ω For every a ∈ (0, 1) and z ∈ Ω, we have ddc ρa (z) := ddc (−(−ρ(z))a ) = a(1 − a)(−ρ)a−2 dρ ∧ dc ρ + a(−ρ)a−1 ddc ρ Then (ddc ρa )n ≥ an (1 − a)(−ρ)na−n−1 dρ ∧ dc ρ ∧ (ddc ρ)n−1 (3.10) Hence, by (3.7), (3.8) and (3.9), there exists ≫ d0 > depending only on Ω and ρ such that, for every < d < d0 and z ∈ Wd , (ddc ρa )n ≥ C5 (1 − a)an dna−n−1 ω n , (3.11) where C5 > depends only on n and ρ Since u ∈ F (Ω), there exists {uj }∞ j=1 ⊂ E0 (Ω) such that uj ↘ u and ∫ (ddc uj )n < C6 , (3.12) Ω for every j ∈ Z , where C6 > depends only on u By using (3.11), (3.12) and the Bedford-Taylor comparison principle [5, 6] (see also [25]), we have, for every j ∈ Z+ , ϵ, d > and a ∈ (0, 1), ∫ ∫ + (ddc uj )n ≥ C6 > {uj such that V2n ({z ∈ B2n : ∥z∥ > − d, u(z) < −Ad}) C lim sup < , d A d→0+ for every A > 44 (Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi luan an (Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi The proof of this theorem needs the following lemma which gives a necessary and sufficient condition for a radial plurisubharmonic function to be in the class F Note that if u is a radial plurisubharmonic function then u(z) = ϕ(log |z|) for some convex, increasing function ϕ Lemma 3.3.4 Let u = ϕ(log |z|) be a radial plurisubharmonic function in B2n Then, u ∈ F (B2n ) iff the following conditions hold (i) lim− ϕ(t) = 0; t→0 (ii) lim− t→0 ϕ(t) < ∞ t Proof By Theorem 3.3.1, the condition (i) is a necessary condition for u ∈ F (B2n ) We need to show that, when (i) is satisfied, the condition u ∈ F (B2n ) is equivalent to (ii) If (ii) is satisfied then there exists k0 ≫ such that k0 t < ϕ(t) Hence u(z) > k0 log |z| ∈ F(B2n ) Thus, u ∈ F (B2n ) Conversely, if (ii) is not satisfied, we consider the functions uk = max{u, k log |z|} Then, for every k , uk > u near ∂ B2n Hence ∫ ∫ ∫ c n c n n c n k→∞ (dd log |z|) −→ ∞ (dd u) ≥ (dd uk ) = k Ω Ω Ω Thus u ∈ / F (B2n ) The proof is completed Proof of Theorem 3.3.3 Denote by µ the unique invariant probability measure on the unitary group U (n) For every z ∈ B2n , we define u˜(z) = ∫ u(ϕ(z))dµ(ϕ) = U (n) ∫ u(w)dσ(w), c2n−1 |z|2n−1 {|w|=|z|} where c2n−1 is the (2n−1)-dimensional volume of ∂ B2n By Lemma 3.2.2, we have u˜ ∈ F (B2n ) Since u˜ is radial, we have, by Lemma 3.3.4, lim− |z|→1 u˜(z) u˜(z) = lim− < ∞ |z| − |z|→1 log |z| ∫ Hence lim− r→1 {|z|=r} |u(z)|dσ(z) 1−r 45 = M < ∞ (Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi luan an (Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi Consequently, we have, for < d ≪ 1, σ({z ∈ B2n : ∥z∥ = − d, u(z) < −Ad}) ≤ M +1 , A (3.13) for all A > Note that V2n ({z ∈ B2n : ∥z∥ > − d, u(z) < −Ad}) ∫d σ({z ∈ B2n : ∥z∥ = − t, u(z) < −Ad})dt = Hence, by (3.13), we have, for < d ≪ 1, V2n ({z ∈ B 2n : ∥z∥ > − d, u(z) < −Ad}) ≤ ∫d (M + 1) Ad/t dt = (M + 1)d 2A Thus we get the last assertion of Theorem 3.3.3 The proof is completed 3.4 A sufficient condition for membership of the class F Our second purpose is to find a sharp sufficient condition for u to belong to F(Ω) based on the near-boundary behavior of u We are interested in the following question: Question Let Ω be a bounded strictly pseudoconvex domain Assume that u is a negative plurisubharmonic function in Ω satisfying lim+ d→0 V2n ({z ∈ Wd : u(z) < −Ad}) = 0, d for some A > Then, we have u ∈ F (Ω)? In this section, we answer this question for the case where Ω is the unit ball Theorem 3.4.1 Let u ∈ P SH − (B2n ) Assume that there exists A > such that lim+ d→0 V2n ({z ∈ B2n : ∥z∥ > − d, u(z) < −Ad}) = d Then u ∈ F (B2n ) 46 (Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi luan an (3.14) (Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi Proof We will find a sequence of functions uj ∈ F (B2n ) such that ∫ (ddc uj )n < ∞ sup j≥0 Ω and uj converges almost everywhere to u as j → ∞ Then, by using Proposition 3.1.4, we will obtain u ∈ F (B2n ) For every < a < 1, we denote Sa = {ϕ ∈ U (n) : ∥ϕ − Id∥ < a} n For every < ϵ, a < and z ∈ B2n 1−ϵ := {w ∈ C : ∥w∥ < − ϵ}, we define ua,ϵ (z) = (sup{u((1 + r)ϕ(z)) : ϕ ∈ Sa , ≤ r ≤ ϵ})∗ Then ua,ϵ is plurisubharmonic in B2n 1−ϵ (see [25, Corollary 2.9.5] and [25, Theorem 2.9.14]) and, by the semicontinuity of u, we have lim max{a,ϵ}→0+ ua,ϵ (z) = u(z), (3.15) for every z ∈ B2n Moreover, for z ̸= 0, ua,ϵ (z) = (sup{u(ξ) : ξ ∈ Ba,ϵ,z })∗ , where (3.16) } { Ba,ϵ,z = { = ξ z − ∥ < a, ∥z∥ ≤ ∥ξ∥ ≤ (1 + ϵ)∥z∥ ξ ∈ Cn : ∥ ∥z∥ ∥ξ∥ } z tξ : t ∈ [∥z∥, (1 + ϵ)∥z∥], ξ ∈ ∂ B2n , ∥ξ − ∥ 0, there exists a > ϵa > such that for every ϵa ≥ 3ϵ ≥ − ∥z∥ ≥ ϵ > 0, ua,ϵ (z) ≥ −3Aϵ, (3.17) where A > is the constant in the condition (3.14) Consider the parameterization p : B2n−1 → ∂ B2n ∩ {z ∈ Cn = R2n : yn > 0} √ s = (s1 , , s2n−1 ) 7−→ p(s) = (s, − ∥s∥2 ) For each s ∈ B2n−1 , we consider the angle α between the vectors e2n = (0, , 0, 1) and p(s) We have ∥e2n − p(s)∥ α sin( ) = 2 sin(α) = ∥s∥ and Hence, √ ∥e2n − p(s)∥2 ∥s∥ = ∥e2n − p(s)∥ − √ Then p(B2n−1 a 1−a2 /4 ) = Se2n ,a and we have (2ϵ + ϵ2 )∥z∥2 V2n (Ba,ϵ,z ) = ∫ dS(ξ) Se2n ,a ∫ (2ϵ + ϵ2 )∥z∥2 = √ √ + ∥∇ − ∥ξ∥2 ∥2 dξ √ B2n−1 a 2 (2ϵ + ϵ )∥z∥ = 1−a2 /4 ∫ √ √ B2n−1 a dξ − ∥ξ∥2 1−a2 /4 Therefore, there exist C1 , C2 > such that C1 a2n−1 ϵ < V2n (Ba,ϵ,z ) < C2 a2n−1 ϵ, (3.18) for every < ϵ, a < 1/2 and 1/2 < ∥z∥ ≤ − ϵ By (3.14), for every 1/2 > a > 0, there exists a > ϵa > such that, for every ϵa ≥ 3ϵ > 0, 48 (Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi luan an (Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi V2n {ξ ∈ B2n : ∥ξ∥ > − 3ϵ, u(ξ) < −3Aϵ} < C1 a2n−1 ϵ, and therefore, by (3.18), for every 3ϵ ≥ − ∥z∥ ≥ ϵ, Ba,ϵ,z * {ξ ∈ B2n : ∥ξ∥ > − 3ϵ, u(ξ) < −3Aϵ} Then, by (3.16), for every ϵa ≥ 3ϵ ≥ − ∥z∥ ≥ ϵ > 0, we have ua,ϵ (z) ≥ −3Aϵ (3.19) For each 1/2 > a > and ϵa ≥ 3ϵ > 0, we consider the following function u˜a,ϵ (z) = 3A(−1 + ∥z∥ ) if − ϵ ≤ ∥z∥ ≤ 1, max{3A(−1 + ∥z∥2 ), ua,ϵ (z) − 6Aϵ} ua,ϵ (z) − 6Aϵ if if − 3ϵ ≤ ∥z∥ ≤ − ϵ, ∥z∥ ≤ − 3ϵ By using the gluing theorem (see, for example, [25, Corollary 2.9.15]), we have ua,ϵ , −m} Then, we have u˜a,ϵ ∈ PSH(B2n ) For m > 0, we set u˜m a,ϵ = max{˜ 2n u˜m ˜a,ϵ , when m → ∞ Moreover, since u˜m a,ϵ ↘ u a,ϵ = 3A(−1 + ∥z∥ ) near ∂ B , we have, ∫ ∫ (ddc 3A(−1 + ∥z∥2 ))n < ∞, n (ddc u˜m a,ϵ ) = B2n B2n 2n ˜a,ϵ ∈ F (B2n ) Moreover, by for every m > Then u ˜m a,ϵ ∈ E0 (B ) Therefore, u Theorem 3.1.2, we have ∫ ∫ c n n (dd u˜a,ϵ ) = lim (ddc u˜m (3.20) a,ϵ ) < ∞, B2n m→∞ B2n for every 1/2 > a > and ϵa ≥ 3ϵ > For every j ∈ N, we denote uj = u ˜2−j ,3−1 ϵ2−j By (3.15), we have uj converges ∫ pointwise to u as j tends to ∞ By (3.20), we have supj B2n (ddc uj )n < ∞ Then, by using Proposition 3.1.4, we have u ∈ F (B2n ) The proof is completed By (3.1) and Theorem 3.4.1, we get the following as a direct consequence ∫ Corollary 3.4.2 Let u ∈ N (B2n ) such that (ddc u)n = ∞ Then, for every B2n A > 0, V2n ({z ∈ B2n : ∥z∥ > − d, u(z) < −Ad}) > lim sup d d→0+ 49 (Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi luan an (Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi List of Author’s Related Papers Thai Duong Do, A comparison theorem for subharmonic functions, Results Math 74 (2019), Paper No 176, 13pp (SCI-E) Hoang Son Do, Thai Duong Do, Some remarks on Cegrell’s class F , Ann Polon Math 125 (2020), 13–24 (SCI-E) Hoang Son Do, Thai Duong Do, Hoang Hiep Pham, Complex Monge-Ampère equation in strictly pseudoconvex domains, Acta Math Vietnam 45 (2020), 93–101 (SCOPUS) 50 (Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi luan an (Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi Bibliography [1] P ˚ Ahag A Dirichlet problem for the complex Monge-Ampère operator in F(f ), Michigan Math J 55 (2007), 123–138 [2] P ˚ Ahag, U Cegrell R Czy˙z, Hoang Hiep Pham, Monge-Ampère measures on pluripolar sets, J Math Pures Appl 92 (2009), 613–627 [3] H Alexander, B A Taylor, Comparison of two capacities in Cn , Math Z 186 (1984), 407–417 [4] D Armitage, S Gardiner, Classical potential theory, Springer-Verlag London, London, 2001 [5] E Bedford, B A Taylor, The Dirichlet problem for a complex Monge-Ampère equation, Invent Math 37 (1976), 1–44 [6] E Bedford, B A Taylor, A new capacity for plurisubharmonic functions, Acta Math 149 (1982), 1–40 [7] E Bedford, Survey of pluri-potential theory, Several Complex Variables: Proceedings of the Mittag-Leffler Institute 1987-88, 48–97, Princeton University Press, 1993 [8] S Benelkourchi, A note on the approximation of plurisubharmonic functions, C R Math Acad Sci Paris 342 (2006), 647–650 [9] Z Blocki, On the definition of the Monge-Ampère operator in Cn , Math Ann 328 (2004), 415–423 [10] Z Blocki, The domain of definition of the complex Monge-Ampère operator, Amer J Math 128 (2006), 519–530 [11] Z Blocki, Remark on the definition of the complex Monge-Ampère operator, American Mathematical Society, Providence, RI, 2009 [12] U Cegrell, Sums of continuous plurisubharmonic functions and the complex Monge-Ampère operator in Cn , Math Z 193 (1986), 373–380 51 (Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi luan an (Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi [13] U Cegrell, Pluricomplex energy, Acta Math 180 (1998), 187–217 [14] U Cegrell, The general definition of the complex Monge-Ampère operator, Ann Inst Fourier (Grenoble) 54 (2004), 159–179 [15] U Cegrell, S Kolodziej, A Zeriahi, Subextension of plurisubharmonic functions with weak singularities, Math Z 250 (2005), 7–22 [16] U Cegrell, A general Dirichlet problem for the Complex Monge-Ampère operator, Ann Polon Math 94 (2008), 131–147 [17] G David, S Semmes, Analysis of and on Uniformly Rectifiable Sets, American Mathematical Society, Providence, RI, 1993 [18] J.P Demailly, Monge-Ampère operators, Lelong numbers and intersection theory, Complex Analysis and Geometry, 115–193, Springer, Boston, MA, 1993 [19] Hoang Son Do, Thai Duong Do, Some remarks on Cegrell’s class F , Ann Polon Math 125 (2020), 13–24 [20] Hoang Son Do, Thai Duong Do, Hoang Hiep Pham, Complex Monge-Ampère equation in strictly pseudoconvex domains, Acta Math Vietnam 45 (2020), 93–101 [21] Thai Duong Do, A comparison theorem for subharmonic functions, Results Math 74 (2019), Paper No 176, 13pp [22] L Hăormander, Notion of Convexity, Birkhăauser Boston, 1994 [23] D Huybrechts, Complex geometry - An Introduction, Springer-Verlag, Berlin, 2005 [24] C O Kiselman, Sur la définition de l’opérateur de Monge-Ampère complexe, Analyse Complexe, 139–150, Springer-Verlag, Berlin, 1984 [25] M Klimek, Pluripotential theory, Oxford University Press, New York, 1991 [26] S Kolodziej, The complex Monge-Ampère equation, Acta Math 180 (1998), 69–117 [27] S Kolodziej, The complex Monge-Ampère equation and pluripotential theory, American Mathematical Society, Providence, RI, 2005 [28] S.G Krantz, Function Theory of Several Complex Variables, American Mathematical Society, Providence, RI, 2001 52 (Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi luan an (Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi (Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi(Luan.an.tien.si).mot.so.van.de.trong.ly.thuyet.da.the.vi
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