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
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
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
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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 indepen-
dent referees are gratefully acknowledged.
Finally, I would like to thank my family for their endless love and unconditional
support.
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Contents
Table of Notations v
Introduction x
Chapter 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 . . . . . . . . . . . . . . . . . . . . 16
Chapter 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....................... 31
Chapter 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
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