Doctor of philosophy in mathematics: Some problems in pluripotential theory

The dissertation is written on the basis of the paper published in Annales Polonici Mathematici, the paper published in Acta Mathematica Vietnamica and the paper published in Results in Mathematics.
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Doctor of philosophy in mathematics: Some problems in pluripotential theory VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY INSTITUTE OF MATHEMATICS DO THAI DUONGSOME PROBLEMS IN PLURIPOTENTIAL THEORY DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN MATHEMATICS HANOI - 2021VIETNAM 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 - 2021ConfirmationThis dissertation was written on the basis of my research works carried out atInstitute of Mathematics, Vietnam Academy of Science and Technology, underthe supervision of Prof. Dr.Sc. Pham Hoang Hiep and Prof. Dr.Sc. Dinh TienCuong. All the presented results have never been published by others. January 3, 2021 The author Do Thai Duong iAcknowledgments First of all, I am deeply grateful to my academic advisors, Professor Pham HoangHiep 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 supportingmy PhD studies through the IMU Breakout Graduate Fellowship. The wonderful research environment of the Institute of Mathematics, VietnamAcademy of Science and Technology, and the excellence of its staff have helped meto complete this work within the schedule. I would like to thank my colleagues fortheir efficient help during the years of my PhD studies. Especially, I would liketo express my special appreciation to Do Hoang Son for his valuable commentsand suggestions on my research results. I also would like to thank the participantsof the weekly seminar at Department of Mathematical Analysis for many usefulconversations. Furthermore, I am sincerely grateful to Prof. Le Tuan Hoa, Prof. Phung HoHai, Prof. Nguyen Minh Tri, Prof. Le Mau Hai, Prof. Nguyen Quang Dieu,Prof. Nguyen Viet Dung, Prof. Doan Thai Son for their guidance and constantencouragement. Valuable remarks and suggestions of the Professors from the Department-levelPhD Dissertation Evaluation Committee and from the two anonymous indepen-dent referees are gratefully acknowledged. Finally, I would like to thank my family for their endless love and unconditionalsupport. iiContentsTable of Notations vIntroduction xChapter 1. A comparison theorem for subharmonic functions 1 1.1 Some basic properties of subharmonic functions . . . . . . . . . . 1 1.2 Some basic properties of Hausdorff measure . . . . . . . . . . . . . 5 1.3 An extension of the mean value theorem . . . . . . . . . . . . . . 8 1.4 A comparison theorem for subharmonic functions . . . . . . . . . . 13 1.5 Other versions of main results . . . . . . . . . . . . . . . . . . . . 16Chapter 2. Complex Monge-Ampère equation in strictly pseudo- convex 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 pseu- doconvex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.5 A remark on the class E . . . . . . . . . . . . . . . . . . . . . . . 31Chapter 3. Decay near boundary of volume of sublevel sets of plurisub- harmonic 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 iiiList of Author’s Related Papers ...