Luận án Some problems in pluripotential theory

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
i
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.
ii
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
iii
List of Author’s Related Papers 50
References 51
iv
Table of Notations
N the set of positive integers.
R the set of real numbers.
C the set of complex numbers.
Rn the real vector space of dimension n.
Cn the complex vector space of dimension n.
Bn the unit ball in Rn.
B2n the unit ball in Cn.
∂Bn the unit sphere in Rn.
∂B2n the unit sphere in Cn.
B(x, r) the open ball of center x and radius r in real vector
space or complex vector space.
B(x, r) the closed ball of center x and radius r in real vector
space or complex vector space.
∂B(x, r) the sphere of center x and radius r in real vector
space or complex vector space.
Vn the Lebesgue measure on Rn.
V2n the Lebesgue measure on Cn.
σ the surface measure (in any dimension) on a surface.
∅ the empty set.
‖x‖ the norm of a vector x.
A(Ω) the set of analytic functions on Ω.
C(Ω) the set of continuous functions on Ω.
Ck(Ω) the set of k−times differentiable functions with
derivatives of order k are continuous on Ω.
Ck0 (Ω) the set of k−times differentiable functions with
derivatives of order k are continuous and compact
support on Ω.
C∞(Ω) the set of smooth functions on Ω.
v
C∞0 (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.
L∞loc(Ω) the set of locally bounded functions on Ω.
Lp(Ω) the set of p-th power integrable functions on Ω.
Lploc(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
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 mul-
tivariate 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 sev-
eral 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
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 1. Whether we can conclude that two subharmonic functions on a do-
main 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 1 is devoted to answer Problem 1 completely. For this purpose, we prove
two main theorems with similar assumptions. They concern restrictions of sub-
harmonic 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 re-
sult (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 1 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 2. 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 well-
known result of Ko lodziej (Theorem B in [26]) to some technical assumptions.
These assumptions naturally lead to the following problem.
viii
Problem 3. Find conditions for α such that
v = −(− log(|f1|λ1 + ...+ |fm|λm))α
belongs to the domain of Monge-Ampère operator, where f1, ..., fm are analytic
functions.
Note that if 0 < α < 1
n
, then v ∈ E (see [7], [11]). Our result (Proposition 2.5.1)
is a necessary and sufficient condition where we further assume some conditions
of f1, ..., fm and the non-singularity of their zero-sets.
Chapter 3 is devoted to study the behavior near boundary of the functions
from class F i ... r 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
t→0−
φ(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 ≫ 1 such that k0t < 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∫
Ω
(ddcu)n ≥ ∫
Ω
(ddcuk)
n = kn
∫
Ω
(ddc log |z|)n k→∞−→ ∞.
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(n)
u(φ(z))dµ(φ) =
1
c2n−1|z|2n−1
∫
{|w|=|z|}
u(w)dσ(w),
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)
|z| − 1 = lim|z|→1−
u˜(z)
log |z| <∞.
Hence
lim
r→1−
∫
{|z|=r} |u(z)|dσ(z)
1− r = M <∞.
45
Consequently, we have, for 0 < d≪ 1,
σ({z ∈ B2n : ‖z‖ = 1− d, u(z) < −Ad}) ≤ M + 1
A
, (3.13)
for all A > 0. Note that
V2n({z ∈ B2n : ‖z‖ > 1− d, u(z) < −Ad})
=
d∫
0
σ({z ∈ B2n : ‖z‖ = 1− t, u(z) < −Ad})dt.
Hence, by (3.13), we have, for 0 < d≪ 1,
V2n({z ∈ B2n : ‖z‖ > 1− d, u(z) < −Ad}) ≤
d∫
0
(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 1. 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})
d
= 0,
for some A > 0. Then, do 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 ∈ PSH−(B2n). Assume that there exists A > 0 such that
lim
d→0+
V2n({z ∈ B2n : ‖z‖ > 1− d, u(z) < −Ad})
d
= 0. (3.14)
Then u ∈ F(B2n).
46
Proof. We will find a sequence of functions uj ∈ F(B2n) such that
sup
j≥0
∫
Ω
(ddcuj)
n <∞
and uj converges almost everywhere to u as j →∞. Then, by using Proposition
3.1.4, we will obtain u ∈ F(B2n).
For every 0 < a < 1, we denote Sa = {φ ∈ U(n) : ‖φ− Id‖ < a}.
For every 0 < ϵ, a < 1 and z ∈ B2n1−ϵ := {w ∈ Cn : ‖w‖ < 1− ϵ}, we define
ua,ϵ(z) = (sup{u((1 + r)φ(z)) : φ ∈ Sa, 0 ≤ r ≤ ϵ})∗.
Then ua,ϵ is plurisubharmonic in B2n1−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})∗, (3.16)
where
Ba,ϵ,z =
{
ξ ∈ Cn : ‖ z‖z‖ −
ξ
‖ξ‖‖ < a, ‖z‖ ≤ ‖ξ‖ ≤ (1 + ϵ)‖z‖
}
=
{
tξ : t ∈ [‖z‖, (1 + ϵ)‖z‖], ξ ∈ ∂B2n, ‖ξ − z‖z‖‖ < a
}
.
Denote
Sz/‖z‖,a =
{
ξ ∈ Cn : ‖ξ‖ = 1, ‖ξ − z‖z‖‖ < a
}
.
We have
V2n(Ba,ϵ,z) =
∫
Sz/‖z‖,a
(1+ϵ)‖z‖∫
‖z‖
tdtdS(ξ) =
(2ϵ+ ϵ2)‖z‖2
2
∫
Sz/‖z‖,a
dS(ξ)
=
(2ϵ+ ϵ2)‖z‖2
2
∫
S(0,...,0,1),a
dS(ξ),
47
the last equality holds since the volume of hypersurfaces are preserved under ro-
tations.
We will show that, for all 1/2 > a > 0, there exists a > ϵa > 0 such that for
every ϵa ≥ 3ϵ ≥ 1− ‖z‖ ≥ ϵ > 0,
ua,ϵ(z) ≥ −3Aϵ, (3.17)
whereA > 0 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,
√
1− ‖s‖2).
For each s ∈ B2n−1, we consider the angle α between the vectors e2n = (0, ..., 0, 1)
and p(s). We have
sin(
α
2
) =
‖e2n − p(s)‖
2
and sin(α) = ‖s‖.
Hence,
‖s‖ = ‖e2n − p(s)‖
√
1− ‖e2n − p(s)‖
2
4
.
Then p(B2n−1
a
√
1−a2/4
) = Se2n,a and we have
V2n(Ba,ϵ,z) =
(2ϵ+ ϵ2)‖z‖2
2
∫
Se2n,a
dS(ξ)
=
(2ϵ+ ϵ2)‖z‖2
2
∫
B2n−1
a
√
1−a2/4
√
1 + ‖∇
√
1− ‖ξ‖2‖2dξ
=
(2ϵ+ ϵ2)‖z‖2
2
∫
B2n−1
a
√
1−a2/4
dξ√
1− ‖ξ‖2 .
Therefore, there exist C1, C2 > 0 such that
C1a
2n−1ϵ < V2n(Ba,ϵ,z) < C2a
2n−1ϵ, (3.18)
for every 0 < ϵ, a < 1/2 and 1/2 < ‖z‖ ≤ 1− ϵ.
By (3.14), for every 1/2 > a > 0, there exists a > ϵa > 0 such that, for every
ϵa ≥ 3ϵ > 0,
48
V2n{ξ ∈ B2n : ‖ξ‖ > 1− 3ϵ, u(ξ) < −3Aϵ} < C1a2n−1ϵ,
and therefore, by (3.18), for every 3ϵ ≥ 1− ‖z‖ ≥ ϵ,
Ba,ϵ,z * {ξ ∈ B2n : ‖ξ‖ > 1− 3ϵ, u(ξ) < −3Aϵ}.
Then, by (3.16), for every ϵa ≥ 3ϵ ≥ 1− ‖z‖ ≥ ϵ > 0, we have
ua,ϵ(z) ≥ −3Aϵ. (3.19)
For each 1/2 > a > 0 and ϵa ≥ 3ϵ > 0, we consider the following function
u˜a,ϵ(z) =
3A(−1 + ‖z‖2) if 1− ϵ ≤ ‖z‖ ≤ 1,
max{3A(−1 + ‖z‖2), ua,ϵ(z)− 6Aϵ} if 1− 3ϵ ≤ ‖z‖ ≤ 1− ϵ,
ua,ϵ(z)− 6Aϵ if ‖z‖ ≤ 1− 3ϵ.
By using the gluing theorem (see, for example, [25, Corollary 2.9.15]), we have
u˜a,ϵ ∈ PSH(B2n). For m > 0, we set u˜ma,ϵ = max{u˜a,ϵ,−m}. Then, we have
u˜ma,ϵ ↘ u˜a,ϵ, when m→∞. Moreover, since u˜ma,ϵ = 3A(−1 + ‖z‖2) near ∂B2n, we
have, ∫
B2n
(ddcu˜ma,ϵ)
n =
∫
B2n
(ddc3A(−1 + ‖z‖2))n <∞,
for every m > 0. Then u˜ma,ϵ ∈ E0(B2n). Therefore, u˜a,ϵ ∈ F(B2n). Moreover, by
Theorem 3.1.2, we have∫
B2n
(ddcu˜a,ϵ)
n = lim
m→∞
∫
B2n
(ddcu˜ma,ϵ)
n <∞, (3.20)
for every 1/2 > a > 0 and ϵa ≥ 3ϵ > 0.
For every j ∈ N, we denote uj = u˜2−j ,3−1ϵ
2−j . By (3.15), we have uj converges
pointwise to u as j tends to∞. By (3.20), we have supj
∫
B2n(dd
cuj)
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 ∫
B2n
(ddcu)n = ∞. Then, for every
A > 0,
lim sup
d→0+
V2n({z ∈ B2n : ‖z‖ > 1− d, u(z) < −Ad})
d
> 0.
49
List of Author’s Related Papers
1. Thai Duong Do, A comparison theorem for subharmonic functions, Results
Math. 74 (2019), Paper No. 176, 13pp. (SCI-E)
2. Hoang Son Do, Thai Duong Do, Some remarks on Cegrell’s class F , Ann.
Polon. Math. 125 (2020), 13–24. (SCI-E)
3. 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
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