Articulo 15.dvi


CUBO A Mathematical Journal
Vol.12, No¯ 02, (235–259). June 2010

On Semisubmedian Functions and Weak Plurisubharmonicity

Chia-chi Tung1

Dept. of Mathematics and Statistics,

Minnesota State University, Mankato,

Mankato, MN 56001, USA

email: chia.tung@mnsu.edu

ABSTRACT

In this note subharmonic and plurisubharmonic functions on a complex space are studied

intrinsically. For applications subharmonicity is characterized more effectually in terms

of properties that need be verified only locally off a thin analytic subset; these include

the submean-value inequalities, the spherical (respectively, solid) monotonicity, near as

well as weak subharmonicity. Several results of Gunning [9, K and L] are extendable via

regularity to complex spaces. In particular, plurisubharmonicity amounts (on a normal

space) essentially to regularized weak plurisubharmonicity, and similarly for subharmonic-

ity (on a general space). A generalized Hartogs’ lemma and constancy criteria for certain

matrix-valued mappings are given.

RESUMEN

En esta nota son estudiadas intŕınsicamente las funciones subarmonicas y plurisubarmon-

icas sobre un espacio complejo. Para aplicaciones, subarmonicidad es caracterizada mas

eficientemente en términos de propiedades que necesitan ser verificadas solamente local-

mente en un subconjunto anaĺıtico delgado; estas aplicaciones incluyen la desigualdad

del valor-submedio, la monotonicidad esférica (respectivamente, sólida), bien como subar-

monicidad debil. Varios resultados de Gunning [9, K and L] son extendibles v́ıa regulari-

dad a espacios complejos. En particular, plurisubarmonicidad (sobre un espacio normal)

1Supports by the ”Globale Methoden in der komplexen Geometrie” Grant of the German research society DFG and

the Faculty Improvement Grant of Minnesota State University, Mankato, are gratefully acknowledged.



236 Chia-chi Tung CUBO
12, 2 (2010)

importa esencialmente para plurisubarmonicidad débil regularizada y similarmente para

subarmoniciada (sobre un espacio general). Son dados un lema de Hartogs generalizado y

un criterio de constancia para ciertas aplicaciones matriz-valuada.

Key words and phrases: Subharmonicity, seminear subharmonicity, Jensen function, weak sub-

harmonicity, weak plurisubharmonicity

2000 Math. Subj. Class.: Primary: 31C05; Secondary: 31C10, 32C15

1 Introduction

It is well-known that subharmonic functions are widely used in potential theory and partial differential

equations, and their complex analogue, the plurisubharmonic functions, have played a significant

role in the development of higher dimensional complex analysis. In this work such functions are

studied on a general complex space from an intrinsic viewpoint, the main purpose here being to show

that they admit intrinsic distributional representations despite the presence of singularities. In what

follows let Y denote a (reduced) complex space of pure dimension m > 0. This means that Y is

a Hausdorff space which admits a countable basis of open sets and an open covering {Uj} together

with homeomorphisms αj : Uj → Vj where Vj is a pure m-dimensional analytic subset in some open

subset of Cnj such that each mapping αj ◦ α
−1
k : αk(Uj ∩ Vk) → αj (Uj ∩ Vk) is biholomorphic. A

Riemann covering of an m-dimensional complex space Y at a point a ∈ Y is a holomorphic map

of a neighborhood of a into Cm with discrete fibers. A mappping φ : Y → [−∞,∞) is called

subharmonic (φ ∈ SH (Y )) if: (i) φ is upper semicontinuous Y (φ ∈ Cusc(Y )); (ii) for any compact

set K ⊂ Y and any function h : K → R which is continuous in K and (locally) semiharmonic

([24]) in int (K), and h(z) ≥ φ(z) for every z ∈ ∂K, it follows that h(a) ≥ φ(a) for every a ∈ K

([17, p. 1][10, p. 16]). Subharmonicity is both a global and a local property (Lemma 3.2), the link

between the two being rested with a maximum principle for almost everywhere solidly submedian

functions (Proposition 3.1). In fact, via local Riemann coverings characterizations of subharmonicity

can be given, as in the Euclidean case, in terms of the local property of being solidly, respectively,

spherically, submedian. With the continuity assumption subharmonicity is also equivalent, by virtue of

a maximum principle for seminearly subharmonic functions (Proposition 3.3), to near subharmonicity,

spherical (respectively, solid) monotonicity. For applications it gains in usefulness if such a property

needs only be verified locally off a thin analytic subset (Theorems 3.1 and 3.3). Especially, in the

definition of subharmonicity, it suffices to require the condition (ii) to hold for closed pseudoballs K

lying in Y off some thin analytic subset, instead of arbitrary compact subsets of Y (by Lemma 3.1

and Theorem 3.1; cf. [1, p. 36]).

A locally integrable (or an upper semicontinuous) function ψ : Y → [−∞,∞) is said to be

regularized at a ∈ Y (respectively, regularized in Y ) if

νp
∆

(a) ψ(a) = lim
ε→0

〈ψ ⌋∆〉a,ε (1.1)

for some standard domain ∆ at a (respectively, the said condition holds for each a ∈ Y ). An upper

semicontinuous function ψ : Y → [−∞,∞) is said to be strictly regularized in Y, if ψ satisfies not

only the condition (1.1) for any standard domain ∆ at every point of Y, but also the condition



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On Semisubmedian Functions and Weak Plurisubharmonicity 237

νp
∆

(a) ψ(a) = lim
ε→0

[ψ⌋∆]a,ε, (1.2)

for each pseudoball ∆ at every point a of Y. Here νp(a) denotes the multiplicity of a light holo-

morphic map p at a ([21, p. 22]; for a topological treatment see Radò and Reichelderfer [18]), and

〈ψ ⌋∆〉a,ε (respectively, [ψ⌋∆]a,ε) the solid (respectively, spherical) mean-value function of ψ ((2.3)-

(2.4)). Every plurisubharmonic function is strictly regularized (Propositions 3.2 and 4.2).

A mapping φ : Y → [−∞,∞) is plurisubharmonic (φ ∈ PSH (Y ) in the sense of Lelong and Oka)

iff φ ∈ Cusc(Y ) and the pull-back of φ by any holomorphic map from the unit disc ∆ ⊂ C into Y is

subharmonic in ∆. This definition turns out, owing to a notable result of Fornæss and Narasimhan

([5, p. 64]), to be equivalent to one in an apparently stronger sense ([15, p. 356]): a mapping φ :

Y → [−∞,∞) is plurisubharmonic if φ has a plurisubharmonic extension into the ambient space of

a local embedding of Y at every point. Consequently plurisubharmonicity of a mapping is a local

property. In contrast to ”subharmonicity”, such a mapping is necessarily radially submedian ((4.2))

and, if not identically equal to −∞ on any open subset, a Jensen function (Proposition 4.2). On

a normal complex space the (possibly weaker) property of being semiradially submedian suffices to

characterize plurisubharmonicity (Theorem 4.1).

A locally integrable function with nonnegative local distributional Laplacian is called weakly

subharmonic. With C2-differentiability such a function can be regarded as possessing (off a thin

analytic subset) nonnegative local spherical mean-radial derivative. By a similar token the requirement

of semidefinite positivity of the distributional Levi form of locally integrable functions gives rise to

a class PSHw(Y ) of weakly plurisubharmonic functions. Denote by SH {6≡−∞}(Y ) the set of all

subharmonic functions in Y not identically equal to −∞ on any open subset of Y. It turns out

that a weakly subharmonic function φ belongs to SH {6≡−∞}(Y ) if and only if φ is regularized in

Y (Theorem 3.4). Owing to the strict regularity of plurisubharmonic functions, several results of

Gunning [9, K and L] are extendable to complex spaces. Some of these are indicated in §4 and

§5. Especially, given φ ∈ PSHw(Y ), there exists a unique locally integrable, regularized function

ψ : Y → [−∞,∞) such that ψ = φ almost everywhere and ψ ∈ SM rad(Y ) ∩ PSH {6≡−∞}(Yreg); if

further Y is normal, then a function φ : Y → [−∞,∞) is weakly plurisubharmonic and regularized in

Y if and only if φ ∈ PSH {6≡−∞}(Y ) (Theorem 4.2 and Corollary 4.1). As applications, a generalized

Hartogs’ lemma, conditions for the constancy of certain matrix-valued mappings (including extensions

of two results of Bochner and Montgomery [2, p. 155]), and the maximum modulus principle for weakly

real-analytic, subharmonic mappings, are given. Also, some lemmas and examples of recurring use on

subharmonicity and plurisubharmonicity are gathered in an Appendix.

The author is indebted to the referee for suggestions which led to the improvement of this paper.

2 Preliminaries

Denote by ‖z‖ the Euclidean norm of z = (z1, · · · ,zm) ∈ C
m, where each component zj = xj +iyj. Let

the space Cm be oriented so that the Eulidean Kähler form υm := ((i/2π) ∂∂̄ ‖z‖2)m is positive. In

what follows let X,Y denote (reduced) complex spaces of pure dimension m > 0, and let p : Y → Cm

be a holomorphic map. Set a′ := p(a), p[a] := p−a′, and ra := ‖p
[a]‖ for each a ∈ Y. If U ⊆ Y is an

open set, a ∈ U and r > 0, set U[a](r) := {z ∈ U | ra(z) < r}, and U[a][r] := {z ∈ U | ra(z) ≤ r}.



238 Chia-chi Tung CUBO
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Denote by B[a′](r) the open ball in C
m with center a′ and radius r, omitting the subscript if

a′ = 0. Let dυ (respectively, dσr) be the Euclidean volume element of C
m (respectively, the sphere

S(r) = ∂B(r)) and | B(r) | (respectively, | S |) the volume of B(r) (respectively, | S(1) |).

A complex space X together with a holomorphic map p : X → Ω, where Ω is an open, connected

subset of Cm, is called a semi-Riemann domain (of dimension m > 0) if there exists a thin analytic

set Σ in Ω such that Σp := p
−1(Σ) is thin in X, and the restriction p : X0 := X \ Σp → Ω0 := Ω \ Σ

has discrete fibers; the map p = (p1, · · · ,pm) : X → Ω is called a Riemann semicovering. If Σ = ∅,

then (X,p) is called a Riemann domain and the map p a Riemann covering. If p : X → Ω is, in

addition, a local homeomorphism, then (X,p) is said to be unramified. Every proper modification

of an affine algebraic variety is a semi-Riemann domain (as such it is parabolic in the sense of [22,

pp. 73–74]). Riemann domains, in particular, analytic coverings of Cm, play a fundamental role in

complex analysis; in fact, every pure m-dimensional complex space is locally an analytic covering of

a domain in Cm (hence a Riemann domain).

Let (X,p) be a semi-Riemann domain; denote by X∗ the largest open subset of X on which

p is locally biholomorphic. For each open subset D ⊆ X, set D0 := D ∩ X0 and D∗ := D ∩ X∗.

Suppose that p : Ũ → Ω, is a holomorphic map defined on an open neighborhood Ũ of a point

a ∈ Y such that: i) p−1(a′) ∩ Ũ = {a}; ii) for a sufficiently small ball U′ := B[a′](r) in C
m, Ua :=

p−1(U′) ∩ Ũ = p−1(U′) ∩ Ũ is connected and the mapping p⌋Ua : Ua → U
′ is an analytic covering

(biholomorphic if a ∈ D∗); iii) every branch V ka , k = 1, · · · ,sa, of Ua contains a; and iv)

sa = deg (p⌋Ua) = ν
y
p (a) (2.1)

([21, Proposition 1.3]). Such an Ua is called, for convenience, a pseudoball (of radius r) at a ([24,

p. 557]). A pseudopolydisc ∆[a](R) over a polydisc (of polyradius R) in C
m is similarly defined. An

open neighborhood ∆ of a point a in a complex space Y is called a standard domain at a if and only

if there exists a Riemann covering p : ∆ → Ω exhibiting ∆ as either a pseudoball (of radius R) or a

pseudopolydisc (of polyradius R) at a. If p : Y → Ω is a Riemann semicovering and a ∈ Y 0, define

d(a) (respectively, dpd(a)) := the supremum of R > 0 (respectively, R = (R1, · · · ,Rm) > (0, · · · , 0))

such that a pseudoball (respectively, polydisc) ∆[a](R) exists.

The notions of Ck-differential forms, the exterior differentiation d, the operators ∂, ∂̄ and dc :=

(1/4πi)(∂ − ∂̄), are well-defined on a complex space Y despite the presence of singularities ([23,

Chapter 4]). If G ⊆ Y is open subset, denote by dG the (maximal) boundary manifold of Greg in

Yreg, the manifold of simple points of Y, oriented to the exterior of Greg ([23, p. 218]). If p : G → C
m

is a holomorphic map and a ∈ G, the form

υp : = dd
c r2a =

i

2π
∂∂̄ r2a (2.2)

is nonnegative ([23, §4]) and independent of a. The Poincaré form

σa :=
2 dcra

r2m−1a
∧ υm−1p

is d-closed. Also set dυ̃ := p∗dυ, dσ[a],r := (p
[a])∗dσr, where dυ (respectively, dσr ) denotes the

Euclidean volume element of Cm (respectively, S(r)). If p : Y → Cm is a Riemann semicovering and

φ ∈ Cusc(U), where U = U[a](r0) ⊂ Y, the associated spherical mean-value function is defined by



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On Semisubmedian Functions and Weak Plurisubharmonicity 239

[φ⌋U]a,r :=

∫

dU[a](r)

φσa, 0 < r < r0, (2.3)

([24, (3.2)]). Similarly, if φ ∈ Cusc(∆), where ∆ = ∆[a](R0) is a standard domain in Y, the associated

solid mean-value function is defined by

〈φ⌋∆〉a,R :=
1

vol (∆(a′,R))

∫

∆a(R)

φ(z) dṽ(z), 0 < R < R0, (2.4)

where vol (∆(a′,R)) denotes the Euclidean volume of the induced domain ∆(a′,R) := p(∆a(R)).

Let j
dG

: dG → Y denote the inclusion mapping and dσ = dσ
dG

the (Lebesgue) surface measure on

Y ∗ ∩ dG induced by the local patches p
U

:= p : U → B[a′](r) on an unramified neighborhood U of a

point a ∈ dG.

3 Subharmonicity

An upper semicontinuous map φ : Y → [−∞,∞) is called (locally) semispherically submedian in Y

(φ ∈ SM sm.sph(Y )), if there exists a thin analytic subset A of Y such that: (i) at every z ∈ Y \A,

there is a pseudoball ∆ of radius r0 for which the inequality

νp
∆

(z) φ(z) ≤ [φ⌋∆]z,r, 0 < r < r0, (3.1)

holds; (ii) if φ admits an exceptional peak point, that is, a point z∗ ∈ A with φ(z∗) = supY φ,

then φ(z∗) ≤ lim infn→∞ φ(an) for all sequences {an} in ∆\A converging to z∗. Similarly, the set

SM sm.sol(Y ) of (locally) semisolidly submedian maps is defined by requiring (in place of (3.1)) the

inequality

νp
∆

(z) φ(z) ≤ 〈φ⌋∆〉z,R, 0 < R < R0, (3.2)

to hold for some pseudoball ∆ = ∆z(R0). If in the preceding the set A can be taken to be the empty

set, then φ is called spherically submedian (respectively, solidly submedian) in Y.

An upper semicontinuous map φ : Y → [−∞,∞) is said to be weakly Jensen if there exists a thin

analytic subset A of Y such that at each z ∈ Y0(φ) := Y \A, the inequality (3.2) holds with respect

to some standard domain ∆, where, in the case of a pseudoball (respectively, pseudopolydisc), R0

denotes the radius (respectively, the polyradius) of ∆. (In view of Proposition 4.1 and Theorem 3.1)

the following extension of [24, Proposition 3.2] generalizes the maximum principle for subharmonic

(and plurisubharmonic) functions:

Proposition 3.1. (Maximum principle for weakly Jensen functions) Let D be a domain in Y and

φ : D → [−∞,∞) a weakly Jensen function with finite supremum M. Assume that (i) the peak set

P := {z ∈ D |φ(z) = M} is nonempty; (ii) if no peak point of φ lies in D0 := D0(φ), then every

sequence {an} in D0 converging to a point z0 ∈ P satisfies the inequality φ(z0) ≤ lim inf n→∞ φ(an).

Then φ = constant.



240 Chia-chi Tung CUBO
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Proof. Let F := {z ∈ D |φ(z) < M}. Assume at first that φ admits a peak point in D0. Choose a

standard domain ∆ ⊆ D0 at such a point a for which the inequality (3.2) holds. Then the proof of

[24, Proposition 3.2] carries over here and yields the desired conclusion. Now consider the case where

φ has no peak point in D0. Let z0 be any peak point of φ, and {an} a sequence in D0 converging

to z0. Choose a sequence of standard domains ∆[an](Rn) (relative to a Riemann covering pn ) with

norm ‖Rn‖ → 0 as n → ∞ such that the solid submean-value property (3.2) holds. Suppose that

F ∩B[z0](Rk) 6= ∅ for every standard domain B[z0](Rk). Then φ(z) < M for all z in a neighborhood

of each point in F ∩ B[z0](Rk). Since ∆[an](Rn) ∩ B[z0](Rk) 6= ∅ for sufficiently large n and an ∈ F,

it follows from [23, Proposition 5.2.2] and [24, (2.4)] that

vol (∆(a′n,Rn)) νpn (an) φ(an) ≤

∫

∆an (Rn)

φ(z) dṽ(z)

<

∫

∆an (Rn)

M dṽ(z) = M vol (∆(a′n,Rn)) deg(pn ⌋∆).

This inequality and the relation [24, (2.5)] imply that

φ(z0) ≤ lim inf
n→∞

φ(an) < M,

hence a contradiction. Therefore F ∩ B[z0](Rk) = ∅ for some B[z0](Rk). Thus the set P is open and

nonvoid. Then by the connectedness of D one must have F = ∅. Consequently φ(z) ≡ M in D.

In the following, if V k is a branch of a standard domain ∆ at point a ∈ Y, and φ ∈ L1(∆),

denote by φ̃k a locally integrable function on ∆
′ such that p∗φ̃k = φ on V

k. Since each branch V k

contains a, the function φ̃k can be chosen to be continuous in ∆
′, provided so is φ in ∆.

Lemma 3.1. Let (X,p) be a Riemann domain. Assume that φ : X → [−∞,∞) is upper semicon-

tinuous, and for each pseudoball U ⊂ X and every continuous function h : U → R with h ≥ φ⌋∂U

such that h is semiharmonic in U, one has h ≥ φ⌋U. Then νp(a) φ(a) ≤ [φ]a,r for all a ∈ X and

all r ∈ (0,d(a)), (in particular, φ is spherically submedian).

Proof. Let U be a pseudoball at a ∈ X, and r ∈ (0,d(a)). There exists, for each branch V k, k =

1, · · · , l, of U , a decreasing sequence of continuous functions φ̃nk on U
′
a′ [r] converging pointwise to

φ̃k. The Poisson integral hn := Pa′,r(φ̃
n
k ) is continuous and harmonic in U

′
a′ (r), and hn = φ̃

n
k ≥ φ̃k

on ∂U′a′ (r). By the subharmonicity of φ̃k and the formula [25, (4.16)], one has

φ̃k(z
′) ≤ hn(z

′) =

∫

∂U′[a′](r)

Pa′,r(z
′,ζ′) φ̃nk (ζ

′) dσ[a′],r,

for each z′ ∈ U′a′ (r). Hence by the monotone convergence theorem,

φ̃k(z
′) ≤ lim

n→∞
hn(z

′) =

∫

∂U′[a′](r)

Pa′,r(z
′,ζ′) φ̃k(ζ

′) dσ[a′],R.

In particular, by the definition (2.3),



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On Semisubmedian Functions and Weak Plurisubharmonicity 241

φ̃k(a
′) ≤ [φ̃k]a′,r =

∫

∂U′[a′](r)

1

r | S |

r2

‖a′ − ζ′‖2m
φ̃k(ζ

′) dσ[a′],r.

Consequently φ satisfies the spherical submean-value inequality (3.1) for all r ∈ (0,d(a)).

Observe that for each ψ ∈ L1(B[a′][r0]), the Fubini type formula holds:

∫

B[a′](r)

ψ dυ[a′] =

∫ r

0

(

∫

S[a′](t)

ψ dσ[a′],t
)

dt, 0 < r < r0. (3.3)

Lemma 3.2. Let Y, be a pure m-dimensional complex space and φ : Y → [−∞,∞). Then φ ∈

SH (Y ) if and only if φ is locally subharmonic in Y.

Proof. Using the formula (3.3), the following preliminary assertion can be proved (in the same way as

in [24, Lemma 3.1]): For every pseudoball ∆ = ∆[a](R0) ⊂ Y and for each φ ∈ L
1(∆), the inequality

”[φ⌋∆]a,R ≥ (constant) C for all R ∈ (0,R0)” implies that ”〈φ⌋∆〉a,R ≥ C for all R ∈ (0,R0)”.

Suppose that φ is locally subharmonic in Y. Let K ⊂ Y be a compact subset and h : K → R a

continuous function which is semiharmonic in int (K) and h(z) ≥ φ(z) for all z ∈ ∂K. It suffices to

show that the difference ψ := φ − h satisfies the maximum principle on each connected component

G of int (K). Suppose now that supG ψ is attained at a point of G. By assumption, at each point

a ∈ G there is an open neighborhood Q such that φ ∈ SH (Q). Then by Lemma 3.1 and the

preliminary assertion, φ is solidly submedian in a neighborhood U of a ∈ Q. Thus by virtue of

the Gauss mean-value formula for semiharmonic functions one has νp(a) ψ(a) ≤ 〈ψ ⌋U〉a,R for all

R ∈ (0,R0). Therefore, by the maximum principle for solidly submedian functions (Proposition 3.1),

ψ = (constant) C in G, so that for every sequence {an} ⊂ G converging to a point a ∈ ∂G, one

must have

0 ≥ (φ − h)(a) ≥ lim sup
n→∞

(φ − h)(an) = C.

Consequently h(z) ≥ φ(z) for all z ∈ K, as desired.

Theorem 3.1. (1) SM sm.sph(Y ) = SM sm.sol(Y ) = SH (Y ). (2) If φ ∈ SH (Y ), then φ is solidly

submedian (relative to any pseudoball) at every point of Y.

Proof. The preliminary assertion in Lemma 3.2 shows that SM sm.sph(Y ) is a subset of SH sm.sol(Y ).

As in the proof of Lemma 3.2, the inclusion ” SM sm.sol(Y ) ⊆ SH (Y )” is a consequence of Proposition

3.1. Finally the assertion (2) follows from Lemma 3.1 and the preliminary assertion in Lemma 3.2.

Remark 3.1. The second assertion of the above theorem gives a generalization of Lemma 15.2 of [7]

to complex spaces. Also, it will be seen later (Theorem 3.4) that if φ is only weakly subharmonic,

then the solid submean-value inequality holds almost everywhere in Y.

Corollary 3.1. If φ : Y → C is semiharmonic, then there exists ψ ∈ C0(Y ) ∩ SH (Y ) such that

ψ = φ almost everywhere in Y.



242 Chia-chi Tung CUBO
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Proof. Since φ is semiharmonic in any given pseudoball U ⊂ Y, there exists, by [24, Theorems 4.2], a

continuous function ψ
U

in U such that ψ
U

= φ almost everywhere and ψ
U

has the solid mean-value

property. Hence Theorem 3.1 implies that ψ
U

is subharmonic in U. The conclusion follows then from

Lemma 3.2.

An upper semicontinuous function φ : Y → [−∞,∞) is said to be spherically (respectively,

solidly) monotone at z ∈ Y if there exists a pseudoball U at z of radius r0 such that

[φ⌋U]z,r ≤ [φ⌋U]z,s, if 0 < r ≤ s < r0, (3.4)

(respectively,

〈φ⌋U〉z,r ≤ 〈φ⌋U〉z,s, if 0 < r ≤ s < r0). (3.5)

Let p
U

: U → Ω be an unramified Riemann covering at a point a ∈ Yreg. The radial derivative

of φ ∈ C1(U) at z ∈ dUa(r)\p
−1(a′) (for sufficiently small r > 0) is defined by

j∗a,r(d
cφ ∧ υm−1p

U
) (z) =

r2m−1

2
(Rp

U
,a φ)(z) σa, (3.6)

where ja,r : dUa(r) → X, denotes the inclusion (for more detail see [24, p. 567]). The spherical mean

radial derivative

[Rp
U

,z(φ)]z,r :=

∫

dU[z](r)

Rp
U

,z(φ) σz, z ∈ U,

exists for sufficiently small r > 0. It follows from the identity (3.6) (and [24, p. 571]) that

d

dρ
[φ⌋U]z,ρ

]

ρ=r
= [Rp

U
,z(φ)]z,r, ∀z ∈ U, (3.7)

for sufficiently small r > 0. For a real-valued φ ∈ C2(Yreg) the spherical monotonicity at a point

a ∈ Yreg amounts to the condition that φ possesses (locally) nonnegative spherical mean radial

derivative at a, that is, there is an unramified Riemann covering p
U

at a such that [Rp
U

,a(φ)]a,r ≥ 0

for sufficiently small r > 0. Let △p
U

denote the pull-back (under p
U
) of the Euclidean Laplace

operator on Cm. By differentiating under the integral sign and using the divergence theorem, it is

easy to show that

d

dρ
[φ⌋B]a,ρ

]

ρ=r
=

1

| S |r2m−1

∫

B[a](r)

△pU φdυ[a],r.

Consequently one has, for each z ∈ U,

[Rp
U

,z(φ)]z,r =
r

2m
〈△p

U
φ〉z,r, (3.8)

for sufficiently small r > 0.



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On Semisubmedian Functions and Weak Plurisubharmonicity 243

Lemma 3.3. If p
U

: U → Ω is an unramified Riemann covering and φ ∈ C2(U) is real-valued and

solidly submedian, then △p
U
φ ≥ 0.

Proof. Suppose that △p
U
φ < 0 in U[a](r0) for some a ∈ U and r0 > 0. By the relations (3.7)-(3.8)

one has
d

dρ
[φ⌋U]a,ρ

∣

∣

ρ=r
< 0, ∀r ∈ (0,r0).

Hence it follows from the relation (1.2) that

νp
U

(a) φ(a) >
1

| S |ρ2m−1

∫

dX[a](ρ)

φdσ[a],ρ, ∀ρ ∈ (0,r0). (3.9)

Thus, multiplying (3.9) by| S |ρ2m−1 and integratingover (0,r), 0 < r < r0, yields

|B(r)|νp
U

(a) φ(a) >

∫

X[a](r)

φdυ̃,

which implies that νp
U

(a) φ(a) > 〈φ⌊U〉a,r, a contradiction. Therefore the Lemma is proved.

An upper semicontinuous map φ : Y → [−∞,∞) is called nearly subharmonic at z ∈ Y if there

exists a pseudoball U at z of radius r0 such that

〈φ⌋U〉z,r ≤ [φ⌋U]z,r ∀r ∈ (0,r0); (3.10)

φ is said to be seminearly subharmonic in Y (φ ∈ SM sm.near(Y ) if there exists at each a ∈ Y an

open neighborhood Q and a thin analytic subset AQ of Q such that φ is nearly subharmonic at

every point of Q\AQ. The class Msm.sph(Y ) (respectively, Msm.sol(Y )) of functions semispherically

monotone (respectively, semisolidly monotone) in Y is similarly defined. Note that by Lemma 3.1

and Theorem 3.1-(2), if φ ∈ SH (Y ), then φ is spherically monotone and nearly subharmonic relative

to every pseudoball at any point of Y.

Lemma 3.4. If φ is spherically monotone at a ∈ Y (with U and r0 as in (3.4) for z = a), and φ

is not identically equal to −∞ in U, then the function ”r 7→ [φ⌋U]a,r” is continuous on (0,r0).

Proof. Since

∫

dU[a](r)

φdσ[a],r =

sa
∑

k=1

∫

dV k
[a]

(r)

φdσ[a],r =

sa
∑

k=1

∫

S[a′](r)

φ̂k dσ[a′],r, ∀r ∈ (0,r0),

it suffices to prove that the function ”r 7→ [φ⌋V k]a,r” is continuous on (0,r0). Let ψ denote the

induced function of φ⌋V k on U′. There exists a monotonically decreasing sequence {ψ(µ)} of C∞

subharmonic functions (with increasing domains) converging pointwise on U′ to ψ. Let ε > 0 and

t ∈ (0,r0) be given. Suppose that there exists a decreasing sequence {tn} converging to t such that

[ψ ⌋U′]a′,t + ε ≤ [ψ ⌋U
′]a′,tn for each n. Then, for a sufficiently large N,

[ψ ⌋U′]a′,t + ε ≤ [ψ
(µ) ⌋U′]a′,tn →

n→∞
[ψ(µ)⌋U′]a′,t,



244 Chia-chi Tung CUBO
12, 2 (2010)

where the last limit relation follows from the Stokes Theorem. Applying the monotone convergence

theorem to the sequence {ψ(N ) − ψ(µ)} for a sufficiently large N, one obtains

[ψ ⌋U′]a′,t + ε ≤ lim
µ→∞

[ψ(µ)⌋U′]a′,t = [ψ ⌋U
′]a′,t,

a contradiction. The conclusion now follows from the spherical monotonicity of φ.

If U is a pseudoball at a0 ∈ Y of radius r0, and φ ∈ L
1(U), applying the formula (3.3) to the

restriction of φ to each branch of U, one obtains, for every r ∈ (0,r0), the relation

|B(r)| 〈φ⌋U〉a0,r =

∫ r

0

[φ⌋U]a0,t |S|t
2m−1 dt. (3.11)

Theorem 3.2. SH (Y ) ⊆ Msph(Y ) ⊆ Msol(Y ) = SM near(Y ).

Proof. It will be first shown that, for a given φ ∈ Cusc(Y ), ”subharmonicity” implies ”spherical

monotonicity”. Let U be a pseudoball at a ∈ Y of radius r0 such that φ is subharmonic in U.

Let Wε and the smooth approximations φ̃k,ε of φ̃k (extended to be 0 outside Wε), k = 1, · · · , l, be

defined as in the proof of Theorem 4.2 of [24]. It will be shown that each φ̃k,ε is solidly submedian

in Wε. Let mB(r) :=
1

|B(r)|
χB(r) be the measure defined by a uniform unit mass distribution over the

ball B(r). Take z′ ∈ U′. There exists rz′ > 0 such that

〈φ̃k⌋U
′〉z′,r =

1

|B(r)|

∫

B[z′](r)

φ̃k(y) dυ(y)

= (m
B(r) ∗ φ̃k) (z

′), 0 < r < rz′.

Thus for every z ∈ V j\{a},

φ̃k,ε(z
′) = (hε ∗ φ̃k) (z

′) ≤ hε ∗ (mB(r) ∗ φ̃k) (z
′), ∀r ∈ (0,rz′ ).

Hence (following an argument of [14, p. 205]) the inequality

φ̃k,ε(z
′) ≤ (m

B(r) ∗ φ̃k,ε) (z
′) = 〈φ̃k,ε⌋U

′〉z′,r

holds at each z′ ∈ Wε\{a
′} for sufficiently small r > 0. Thus by Lemma 3.3, △p

U
φ̃k,ε is nonnegative

in Wε\{a
′}, and therefore by the formulas (3.7)-(3.8), the function ”r 7→ [φ̃k,ε]z′,r” is increasing on

(0,r0 − ε). Moreover, for a fixed r ∈ (0,r0),

[φ̃k⌋U
′]z′,r = lim

ε→0
[φ̃k,ε⌋Wε]z′,r, ∀z

′ ∈ U′.

It follows from this that the function ”r 7→ [φ⌋U]z,r” is increasing on (0,r0) for every z ∈ U. Thus

φ is spherically monotone at every point of Y. By approximating the last integral in the formula

(3.11) by its Riemann sums, it follows easily that the ”spherical monotonicity” of φ at a point z ∈ Y

implies its ”solid monotonicity” at the same point.

To prove the assertion that φ is solidly monotone at a point z ∈ Y if and only if it is nearly

subharmonic at the same point, it may be assumed without loss of generality that φ is not identically



CUBO
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On Semisubmedian Functions and Weak Plurisubharmonicity 245

equal to −∞ in U. Then by Lemma 3.4, differentiation of the formula (3.11) yields for each z ∈ U

and r ∈ (0,r0),

r

2m

d

dρ
〈φ⌋U〉z,ρ

]

ρ=r
= [φ⌋U]z,r −

1

|B(r)|

∫ r

0

[φ⌋U]z,t t
2m−1 dt.

Therefore the desired equivalence follows.

To prove that ”spherical monotonicity” at a point implies ”near subharmonicity” at the same

point, suppose that (with Q and U as above) the inequality (3.4) holds for each z ∈ Q. Then the

relation (3.11) implies that

|B(r)| 〈φ⌋U〉z,r ≤ [φ⌋U]z,r
| S |r2m

2m
, ∀r ∈ (0,r0),

so that the inequality (3.13) holds, thus proving the desired claim.

An upper semicontinuous function ψ : Y → [−∞,∞) is called: (1) strongly regularized in Y,

if ψ satisfies both the conditions (1.1) and (1.2) for all pseudoballs ∆ at every point of Y ; (2) a

Jensen function (ψ ∈ J (Y )), if at each z ∈ Y the inequality (3.2) holds for ψ with respect to every

standard domain ∆ = ∆z (R0), where, in the case of a pseudoball (respectively, pseudopolydisc),

R0 = dpd(a) (respectively, R0 = d(a)). Observe that the expression [24, (3.9)] (where the proof,

with slight modifications, remains valid in the case of a pseudopolydisc) shows that every continuous

function ψ : Y → R is strictly regularized in Y. This property shall be extended to Jensen (hence

plurisubharmonic) functions. If ψ : Y → [−∞,∞), define

ψ(∗)(a) := lim sup
z→a

ψ(z), ∀a ∈ Y.

Since every subharmonic function is weakly Jensen, it follows from the maximum principle (Proposition

3.1) that ψ = ψ(∗) for all ψ ∈ SH (Y ) (compare [9, Theorem J2(c)]).

Proposition 3.2. (1) If ψ ∈ SH (Y ), then ψ is strongly regularized in Y. In particular, if ψ ∈

SH (Y ) is not identically zero, then the set {z ∈ Y |ψ(z) 6= 0} is of positive measure at some point

of Y. (2) If ψ ∈ J (Y ), then ψ is strictly regularized in Y.

Proof. Consider first the case ψ ∈ SH (Y ). For any pseudoball ∆ at a point a ∈ Y and a fixed small

positive r,

lim sup
ε→0

〈ψ⌋∆〉a,ε ≤ lim sup
ε→0

〈 sup
z∈∆a(r)\{a}

ψ(z)〉a,ε

= lim
ε→0

〈1⌋∆〉a,ε sup
z∈∆a(r)\{a}

ψ(z)

= νp
∆

(a) sup
z∈∆a(r)\{a}

ψ(z).

Here (on the above right-hand side) the relation (3.9) of [24] is invoked. Thus

lim sup
ε→0

〈ψ⌋∆〉a,ε ≤ νp
∆

(a) ψ(a). (3.12)



246 Chia-chi Tung CUBO
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It follows then from the solid submean-value property of ψ that the limit relation (1.1) holds. The

relation (1.2) can be similarly proved.

If ψ ∈ J (Y ), then the inequality (3.12) remains valid for any standard domain ∆ at each point

a ∈ Y, hence the limit relation (1.1) follows from the solid submean-value property (3.2)

A maximum principle for continuous nearly subharmonic functions in Cm was proved in [6,

Lemma 3.1, p. 5]. This result is generalized and strengthened below for later applications:

Proposition 3.3. (Maximum principle for seminearly subharmonic functions) Assume φ is a real-

valued, continuous, seminearly subharmonic function in a domain D ⊆ Y. If φ is bounded above and

admits a peak point in D, then φ = constant in D.

Proof. Let P := {z ∈ D |φ(z) = M} be the peak set of φ. There exists at every a ∈ D an open

neighborhood Q and a thin analytic subset AQ of Q such that φ is nearly subharmonic at every

point of Q\AQ. Let D̂ := ∪(Q\AQ). Consider first the case C0 ∩ P 6= ∅ for some component C0 of

D̂. Let C0 be any such component and U a pseudoball of radius r0 at a point a0 ∈ C0 ∩ P such

that the inequality

〈φ⌋U〉z,r ≤ [φ⌋U]z,r ∀r ∈ (0,r0), (3.13)

holds for each z ∈ U. Observe that the mapping r 7→ [φ⌋U]a0,r is continuous in r. Suppose there

exists t1 ∈ (0,r0) such that [φ⌋U]a0,t1 < M −
1
N

(for N > 1
|M|

). Following an idea of [6, p. 5

(last paragraph)], it will be shown that this assumption leads to a contradiction to the inequality

(3.13). Let ρ∗ be the infimum of all t ∈ (0, t1) such that the spherical means [φ⌋U]a0,t are bounded

above by [φ⌋U]a0,t1. Then ρ
∗ > 0, for, otherwise, there is a sequence {tµ} converging to 0 with

[φ⌋U]a0,tµ ≤ [φ⌋U]a0,t1 for each µ, such that, by virtue of [24, Proposition 3.1],

deg (p
U
) φ(a0) ≤ [φ⌋U]a0,t1 < M −

1

N
. (3.14)

Hence a contradiction results. The Fubini type formula (3.3) implies that

|B(ρ∗)| 〈φ⌋U〉a0,ρ∗ =
l

∑

j=1

∫

V
j

[a0]
(ρ∗)

φdυ[a0]

=

∫ ρ∗

0

(

l
∑

j=1

∫

S[a′
0
](t)

φ̃j dσ[a′0],t
)

dt

=

∫ ρ∗

0

[φ⌋U]a0,t |S|t
2m−1 dt > [φ⌋U]a0,t1

|S|(ρ∗)2m

2m
.

This leads to a contradiction 〈φ⌋U〉a0,ρ∗ > [φ⌋U]a0,ρ∗. Therefore one must have [φ⌋U]a0,r ≥ M for

every r ∈ (0,r0). Then the formula (3.11) yields that 〈φ⌋U〉a0,r ≥ M for such r. Since φ ≤ M,

this implies that φ = M in a neighborhood of a0. Thus C0 ⊆ P. Consequently, by the continuity of

φ, φ = M also in D.



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On Semisubmedian Functions and Weak Plurisubharmonicity 247

Now suppose that C0 ∩ P = ∅ for all components C0 of D̂. Let z∗ be any point of P. Then

z∗ ∈ AQ for every Q (chosen as above). There exists a sequence {an} in Q\(AQ ∪ P) (for a fixed

Q) converging to z∗. Choose a sequence of pseudoballs U[an](rn) such that φ is nearly submedian

at each point of Un = U[an](rn). Then (as shown in the equation (3.14)),

deg (p
U
) φ(an) ≤ [φ⌋Un]an,tn < M −

1

N
,

(where tn depends on an). Thus by the continuity of φ, one has φ(z∗) < M, a contradiction! Hence

the preceding argument showing that φ = M in a neighborhood of a0 implies the same for an.

Consequently the sequence {an} lies in P, again a contradiction! Therefore C0 ∩ P 6= ∅ for some

component C0 of D̂, and thereby the proposition is proved.

Theorem 3.3. For a continuous function φ : Y → R the following conditions are equivalent: (i)

φ ∈ Msm.sph(Y ); (ii) φ ∈ Msm.sol(Y ); (iii) φ ∈ SM sm.near(Y ); (iv) φ ∈ SH (Y ).

Proof. In view of Theorem 3.2, it suffices to prove that every element φ ∈ SM sm.near(Y ) is sub-

harmonic in Y. Let K ⊂ Y be a compact set and h : K → R a continuous function such that

h is semiharmonic in int (K) and h(z) ≥ φ(z) for every z ∈ ∂K. Suppose that φ is seminearly

subharmonic in Y. Then at each point a in a connected component G of int (K), there is an open

neighborhood Q and a thin analytic subset AQ of Q such that φ is nearly subharmonic at each

point of Q\AQ. Thus given z ∈ Q\AQ, there is a pseudoball U = Uz(r0) in which the inequality

(3.13) holds, and, consequently,

〈φ − h⌋U〉z,r ≤ [φ − h⌋U]z,r, ∀r ∈ (0,r0).

Thus by Proposition 3.3, the function φ − h satisfies the maximum principle on G. It follows that

φ − h ≤ 0 in int (K). This proves the subharmonicity of φ.

A locally Lipschitz function is semiharmonic if it has almost everywhere vanishing local radial

derivative (this will be shown in a later work). With C2-differentiability, ”subharmonicity” betokens

similarly the nonnegativity (off a thin analytic subset) of the local spherical mean radial derivative:

Proposition 3.4. (1) Let φ ∈ C0(Y ) ∩ C2(Yreg) be real-valued. Assume that for some thin analytic

subset A of Y there exists, at each a ∈ Y \A, an unramified Riemann covering p
U

: U → Ω satisfying

one of the following: (a) φ has (locally) nonnegative spherical mean radial derivative at each point of

U; (b) △p
U
φ ≥ 0 in U. Then φ is subharmonic in Y. (2) If φ ∈ SH(Y ) ∩ C2(Yreg), then both the

conditions (a) and (b) are valid for any unramified Riemann covering (on an open set in Yreg).

Proof. Let p
U

: U → Ω be an unramified Riemann covering at a ∈ Yreg. Clearly for any φ ∈ C
2(U)

the relation (3.8) implies that the conditions (a) and (b) are equivalent; moreover, by virtue of the

formula (3.7) each of these implies that φ is spherically monotone in U. Thus the first assertion follows

from Theorem 3.3. Since each subharmonic function is solidly submedian, the second assertion is a

consequence of the preceding and Lemma 3.3.

Definition 3.1. A locally integrable function φ : Y → [−∞,∞) is called weakly subharmonic in

Y (φ ∈ SHw(Y )) if there exists at each a ∈ Y a standard domain ∆ = ∆a(R) such that for all

nonnegative u ∈ C20 (∆),



248 Chia-chi Tung CUBO
12, 2 (2010)

∫

∆a(r)

φddcu ∧ υm−1p
U

≥ 0, ∀r ∈ (0,R). (3.15)

Observe that, if p : D → Ω, is a Riemann semicovering on an open set D ⊆ Y and if φ ∈

SHw(D), then the inequality (3.15) holds with D in place of ∆a(r) for all nonnegative u ∈ C
2
0 (D).

Theorem 3.4. (1) There exists a bijection between SHw(Y ) and SH {6≡−∞}(Y ) taking each φ ∈

SHw(Y ) to a unique element φ̂ ∈ SH {6≡−∞}(Y ) such that φ̂ = φ almost everywhere in Y. (2)

A function φ : Y → [−∞,∞) is weakly subharmonic and regularized in Y if and only if φ ∈

SH {6≡−∞}(Y ).

Proof. Suppose that φ ∈ SHw(Y ). Let a ∈ Y and φ̃k,ε be defined as in the proof of Theorem 3.2

(for U = ∆ as in the above definition). The weak subharmonicity of φ implies that the induced

function φ̃k ∈ SHw(∆
′), hence the function φ̃k,ε is weakly subharmonic in Wε for every ε > 0

(by the proof of the assertion ”(1) implies (2)” in [24, Theorem 4.2, p. 564]). Suppose that the form

ddcφ̃k,ε ∧ υ
m−1
[a′]

is negative at some z′ ∈ Wε. Then dd
cφ̃k,ε ∧ υ

m−1
[a′]

< 0 in a neighborhood Q ⊆ Wε
of z′, hence for all u ∈ C20 (Q) with u ≥ 0 and u = 1 in a neighborhood of z

′, one has

∫

∆′
φ̃k,ε dd

cu ∧ υm−1
[a′]

=

∫

∆′
uddcφ̃k,ε ∧ υ

m−1
[a′]

< 0,

a contradiction. Thus △id φ̃k,ε ≥ 0 in Wε, so that (by the proof of Theorems 3.2) the function φ̃k,ε
is spherically monotone, and for every z′ ∈ Wε\{a

′},

φ̃k,ε(z
′) ≤ 〈φ̃k,ε〉z′,r (3.16)

for sufficiently small r > 0. It follows from the argument of [13, p. 20] that the function ε 7→ φ̃k,ε(z
′)

is nondecreasing for fixed z′ ∈ Wε. Define

ψ
∆
(z) := min

k
lim
ε→0

{φ̃k,ε(z
′) |V k ∋ z}, ∀z ∈ ∆. (3.17)

Then ψ
∆

is upper semicontinuous in ∆. Moreover ψ
∆
(z) = φ(z) for almost every z ∈ ∆. Also, the

formula (3.17) implies that for each z ∈ ∆,

ψ
∆
(z) ≤ min

k
{φ̃k,ε(z

′) |V k ∋ z}

≤ min
k

1

|B(r)|

∫

B[z′](r)

φ̃k,ε(y) dv(y),

for sufficiently small positive ε and r, where the last inequality follows from (3.16). Letting ε → 0

this inequality implies that

νp(z) ψ∆ (z) ≤ 〈ψ∆〉z,r,



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On Semisubmedian Functions and Weak Plurisubharmonicity 249

for sufficiently small r > 0. Therefore by Theorem 3.1, ψ
∆
∈ SH {6≡−∞}(∆). It follows from Proposi-

tion 3.2 that setting ψ := ψ
∆

on each ∆ (as above) defines a subharmonic function ψ : Y → [−∞,∞)

which coincids with φ almost everywhere in Y.

Conversely, given φ̂ ∈ SH {6≡−∞}(Y ), it will first be shown that φ̂ is locally integrable in Y. For

this purpose a preliminary observation is needed: if Uz(r) ⊂ Y, 0 < r < d(z), is a pseudoball at z

and φ̂(z) > −∞, then φ̂ is integrable over Uz(r). This is an immediate consequence of the local solid

submean-value inequality for φ̂. Let I be the set of all points of Y where φ̂ is locally integrable.

Clearly I is an open set and since locally φ̂ 6≡ −∞, it follows from the preliminary observation that

I is nonempty. On the other hand, if b ∈ Y \I, the preliminary observation implies that φ̂ ≡ −∞

almost everywhere in some pseudoball Ub(R); for, otherwise, φ̂(z) > −∞ for some point z ∈ Ub(r)

for each (sufficiently small) r > 0; hence there is a sequence {zn} ⊂ Y converging to b such that

φ̂(zn) > −∞ for every n. If 0 < R < d(b)/2 and n is sufficiently large, one has ‖zn − b‖ < R and

φ̂ is locally integrable over Uzn (R); but since b ∈ Uzn (R), it follows that b ∈ I, a contradiction.

Thus Ub(R) ⊂ Y \I, so that Y \I is open. Applied to each component Yµ of Y, this implies that

Yµ\I = ∅, hence φ̂ is locally integrable in Y.

It remains to show that φ̂ satisfies the condition (3.15). Let ∆ be a standard domain at a ∈ Y

and denote by φ̃k the function on ∆′ induced by φ̂⌋V k. Then φ̃k is subharmonic, hence solidly

submedian, in ∆′. It follows as in the proof of Theorem 3.2 that the smooth approximations φ̃kε of

φ̃k are solidly submedian in Wε. Consequently by Lemma 3.3, △id φ̃
k
ε ≥ 0 in W

′
ε (for a given open

set ∆′′ ⋐ ∆′). Then by the L1-convergence of φ̃kε to φ̃k, for each nonnegative g ∈ C
2
0 (∆

′),

∫

∆′
φ̃k ddcg ∧ υm−1

[a′]
= lim

ε→0

∫

∆′
g ddcφ̃kε ∧ υ

m−1
[a′]

≥ 0. (3.18)

Let u ∈ C20 (∆) with u ≥ 0. Since the function ũ
k induced by u⌋V k has compact support in ∆′,

there exists a sequence fn ∈ C
2
0 (∆

′) such that fn → △id ũ
k in L1(∆′). By solving the Dirichlet

problem

ddc (w υm−1
[a′]

) =
1

4m
fn υ

m
[a′] in ∆

′, w = 0 on ∂Dn,

where Dn is an open neighborhood of Spt(fn) with smooth boundary, one obtains a sequence {wn}

in C20 (∆
′) such that △id wn tends to △id ũ

k in L1(∆′). Without loss of generality it may be assumed

that each wn is nonnegative. Then by the inequality (3.18),

∫

V k
φ̂ddcu ∧ υm−1p =

∫

∆′\{a′}

φ̃k ddcũk ∧ υm−1
[a′]

≥ 0.

The proof of the assertion (1) is thereby completed. The second assertion follows from the preceding

proof and the regularity of a subharmonic function.

4 Plurisubharmonicity

Let π̃ : Cm\{0} → Pm−1(C) denote the natural projection. The induced map π : S2m−1 → Pm−1(C)

defines a fiber bundle with fiber π−1([a]) = la := {ζa |ζ ∈ C, ‖ζ‖ = 1} over [a] ∈ P
m−1(C), where



250 Chia-chi Tung CUBO
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‖a‖ = 1. Let j(r) : S
2m−1 → Cm be the dilation: j(r)(z) = rz. The Fubini-Study form ω̈ on P

m−1(C)

pulls back under π̃ to the projective form ω := ddc log ‖z‖2 on Cm\{0}. Thus

π∗(ω̈m−1) = j∗(r)(π̃
∗(ω̈m−1)) = j∗(r)(ω

m−1). (4.1)

An upper semicontinuous function φ : Y → [−∞,∞) is said to be (locally) radially submedian

(φ ∈ SM rad(Y )) if there exists at every point a ∈ Y a pseudoball U of radius r0 such that for all

a ∈ S2m−1,

νp
U

(a) φ(a) ≤
1

2π

2π
∫

0

∑

k

φ̃k(a
′ + reiθa) dθ, 0 < r < r0, (4.2)

(where the sum ranges over all branches of U) for some r0 = r0(a, a) ≤ r0. An upper semicontinuous

function φ : Y → [−∞,∞) is said to be (locally) semiradially submedian (φ ∈ SM sm.rad(Y )) if φ

belongs, locally at every a ∈ Y, to SM rad(Q\AQ) for some thin analytic subset AQ in an open

neighborhood Q of a. A peak point of φ⌋Q belonging to AQ is called a local exceptional peak point.

Proposition 4.1. PSH (Y ) ⊆ SM rad(Y ) ⊆ SH (Y ).

Proof. Let U be a pseudoball at a ∈ Y. The first inclusion relation follows from the fact that given

φ ∈ PSH (Y ), each induced function φ̃k is subharmonic at a
′
0 when restricted to the complex line

La = {z ∈ C
m |z = a′0 + ζa, ζ ∈ C}, for each a ∈ S

2m−1. To prove the second inclusion relation,

suppose that φ ∈ SM rad(Y ). The expression (4.1) allows one to evaluate the spherical means of each

φ̃k : U
′ → R by integrating along fibers of the bundle map π (as in [20, p. 202]). The detail is given

below for completeness.

Without loss of generality assume that a′0 = 0. Write z = e
iθ a, a ∈ S2m−1, 0 ≤ θ ≤ 2π. Let

ia : La →֒ C
m, ja : la →֒ La, Ja : la →֒ S

2m−1 be the inclusion mappings. Then

J
∗
a
j
∗
(r)(d

c ln ‖z‖2) = j∗
a
i
∗
a
(dc ln ‖z‖2) = j∗

a
(dc ln ‖ζ‖2) =

1

2π
dθ.

Therefore for small r > 0 one has

νp(a0) φ(a0) =

∫

[a]∈Pm−1(C)

νp(a0) φ(a0) ω̈
m−1

≤

∫

[a]∈Pm−1(C)

[

∫ 2π

0

l
∑

j=1

φ̃j (a
′
0 + r e

iθ
a)
dθ

2π

]

ω̈m−1

=
l

∑

j=1

∫

S2m−1

φ̃j (a
′
0 + rz) j

∗
r (σ0) =

l
∑

j=1

∫

dV
j

[a0]
(r)

φσa0

= [φ⌋U]a0,r

Thus φ is spherically submedan in Y Therefore, by Theorem 3.1, φ ∈ SH (Y ).

Theorem 4.1. (1) If φ ∈ SM sm.rad(Y ) and is continuous at every local exceptional peak point (if

any), then φ is subharmonic. (2) If Y is a normal space, then SM sm.rad(Y ) = PSH (Y ).



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On Semisubmedian Functions and Weak Plurisubharmonicity 251

Proof. By Proposition 4.1 and Theorem 3.1, a function φ ∈ SM sm.rad(Y ) that is continuous at every

local exceptional peak point is semisolidly submedian, hence subharmonic, in Y. To prove the second

assertion, note that in the definition of the set SM sm.rad(Y ), the local analytic set AQ at a point

z can be chosen (without loss of generality) so that Q\AQ ⊆ Qreg. The desired conclusion follows

then from the plurisubharmonicity test in Cm ([26, p. 73]) and the extension theorem of Grauert and

Remmert for plurisubharmonic functions on a normal complex space ([8, Satz 3, p. 181]).

A function φ : Y → [0,∞) is called logarithmically subharmonic (respectively, logarithmically

plurisubharmonic), if log φ ∈ SH (Y ) (respectively, log φ ∈ PSH (Y ); for C2-functions in Cm, another

definition of logarithmic plurisubharmonicity is given in [3, p. 200]). If f = (f
1
, · · · ,f

N
) : Y → CN

and c = (c1, · · · ,cN ), where each cj is an arbitrary positive constant, set

⌊f⌋c := ‖f1‖
c1 + · · · + ‖fN ‖

cN .

If f is a holomorphic map, then the function ⌊f⌋c is logarithmically plurisubharmonic (see Example

6.3 of the Appendix); furthermore, log ⌊f⌋c is a Jensen function, provided f has thin zero set. This

is in fact a special case of the next proposition (which generalizes the classical Jensen’s inequality):

Proposition 4.2. PSH (Y ){6≡−∞} ⊆ J (Y ).

Proof. Let φ ∈ PSH {6≡−∞}(Y ). By Theorems 4.1 and 3.4, φ is subharmonic and locally integrable in

Y. The submean-value inequality relative to pseudoballs follows from Theorem 3.1. Suppose now that

U = ∆a(R) ⋐ Y is a pseudopolydisc of polyradius R. Each induced function φ̃k is plurisubharmonic

in p(U)\{a′}, hence also in p(U) (by [8, Satz 3, p. 181]), so that it is subharmonic in each variable

zj, 1 ≤ j ≤ m. Repeated integration yields the inequality

φ(a) ≤ vol(∆(a′,R))−1
∫

∆(a′,R)

φ̃k(z
′) dv(z′).

From this and the identity (2.1) the the submean-value inequality (3.2) follows.

Definition 4.1. A locally integrable function ψ : Y → [−∞,∞) is said to be weakly plurisubharmonic

in Y (ψ ∈ PSHw(Y )) if at each point a ∈ Y there exists a standard domain ∆ such that the Levi

form of ψ, L(ψ,λ), is positive semidefinite in ∆, that is, for some thin analytic subset A of ∆ (with

∆\A ⊆ ∆∗) and for all nonnegative u ∈ C20 (∆),

〈L(ψ,λ),u〉∆ :=

∫

∆\A

ψ
∑

µ,ν

∂2u

∂pµ ∂p̄ν
λµλ̄ν dṽ ≥ 0, ∀λ ∈ C

m. (4.3)

Similarly, a locally integrable function ψ : Y → [−∞,∞) is said to be weakly pluriharmonic in

Y (ψ ∈ PHw(Y )) if at each a ∈ Y there exists a standard domain ∆ in which the above Levi form

L(ψ,λ) ≡ 0.

Theorem 4.2. (Compare [13, Theorem 3, p. 21] and [9, Theorem K15]) (1) PSH {6≡−∞}(Y ) ⊆

PSHw(Y ). (2) Assume that φ ∈ PSHw(Y ). Then there exists a unique locally integrable, regu-

larized function ψ : Y → [−∞,∞) such that ψ = φ almost everywhere and ψ ∈ SM rad(Y ) ∩

PSH {6≡−∞}(Yreg).



252 Chia-chi Tung CUBO
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Proof. Let φ ∈ PSH {6≡−∞}(Y ). By Theorems 4.1 and 3.4, φ is locally integrable in Y. Consider

first the special case of a C2-functions φ : Y → [−∞,∞). Suppose that Y is nonsingular. For any

standard domain ∆ ⊂ Y and α ∈ C2(∆), denote by α̃ the function on ∆′ induced by α and set

L(α,λ)(z) :=
∑

µ,ν

∂2α̃

∂zµ ∂z̄ν
(z′) λµλ̄ν, z

′ ∈ ∆′, λ ∈ Cm.

Then φ ∈ PSH (U) if and only if L(φ,λ)(z) ≥ 0 for every z ∈ ∆ and every λ ∈ Cm; but that is

clearly equivalent to the requirement that there exists a thin analytic subset A of ∆ such that for

all nonnegative u ∈ C20 (∆),

∫

∆\A

uL(φ,λ) υmp ≥ 0.

By [9, Lemma K14], the latter condition is in turn equivalent to the requirement (4.3) (with D = ∆).

If Y is singular, then by using local desingularizations of Y, it is easily seen that a C2-function φ :

Y → R belongs to PSH (Y ) precisely when it satisfies the condition (4.3) of weak plurisubharmonicity.

Thus the C2 case of the assertion (1) is established.

For the general case, observe that each induced function φ̃k : ∆
′ → R is plurisubharmonic in

U′ := ∆′, hence so are the smooth approximations φ̃k,ε in Wε. Let u ∈ C
2
0 (∆) and denote by ũk

the function induced by u⌋V k in ∆′. Since the support of ũk lies in Wε whenever ε is sufficiently

small, and for such values ε it follows from the first part of the proof that

∫

∆′

φ̃k,ε L(ũk,λ) dv(z) ≥ 0, λ ∈ C
m.

As ε → 0, the function φ̃k,ε converges to φ̃k in L
1-norm on the support of ũk. Consequently φ

satisfies the condition (4.3). Thus the assertion (1) is established in the general case.

Now suppose that φ ∈ PSHw(Y ). Let ∆ be a pseudoball at a ∈ Y relative to which the

condition (4.3) holds. Define ψ∆ : ∆ → [−∞,∞) by

νp
∆

(z) ψ∆(z) := lim
r→0

∑

{〈φ⌋V k〉z,r |V
k ∋ z}, ∀z ∈ ∆.

Then ψ∆ is regularized in ∆ and ψ∆ = φ almost everywhere; moreover, the Levi form L(ψ∆,λ) ≥ 0

in ∆. Consider the smooth approximations ψ̃k,ε in Wε of the induced function ψ̃k of ψk := ψ∆⌋V
k.

The second half of the proof of [9, Theorem K15] shows that ψ̃k is plurisubharmonic in ∆
′\{a′},

hence also in ∆′ (by Grauert and Remmert’s extension theorem [8, Satz 3, p. 181]). It follows from

Proposition 3.2 that the function ψ given by ψ := ψ∆ on each ∆ is well-defined. Also, ψ is locally

integrable, radially submedian and regularized in Y. Since ψ∆ is plurisubharmonic in each open set

∆\{a}, so is the function ψ in Yreg.

Remark 4.1. The above theorem and its proof (of the first assertion) imply that, if D ⊆ Y is a

normal, open subset and φ ∈ PSH {6≡−∞}(D), then for any Riemann semicovering p : D → Ω, the

inequality (4.3) remains valid: 〈L(φ,λ),u〉D ≥ 0 for all nonnegative u ∈ C
2
0 (D) and every λ ∈ C

m.



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On Semisubmedian Functions and Weak Plurisubharmonicity 253

Corollary 4.1. Let Y be a normal space. A function φ : Y → [−∞,∞) is weakly plurisubharmonic

and regularized in Y if and only if φ ∈ PSH {6≡−∞}(Y ).

Proof. Suppose that φ ∈ PSHw(Y ) is regularized in Y. Then by Theorem 4.2, φ is plurisubharmonic

in Yreg. The Grauert and Remmert’s extension theorem ([8, Satz 3, p. 181]) asserts that φ⌋Yreg admits

a plurisubharmonic extension ψ to Y, which, by subharmonicity, must coincide with φ (Proposition

3.2). Conversely, every element of PSH {6≡−∞}(Y ) is weakly plurisubharmonic and regularized in Y

by Theorem 4.2-(1) and Proposition 3.2.

Remark 4.2. (Compare [9, Corollary K18]) Let Y be a normal space. A function φ : Y → [−∞,∞)

is pluriharmonic if and only if φ is weakly pluriharmonic and regularized in Y.

5 Applications

As a consequence of Theorem 3.1, subharmonicity and plurisubharmonicity are closed under a ba-

sic sequential limiting process (Lemma 6.1). By allowing to decrease the values of a subharmonic

(respectively, plurisubharmonic) function on a null set, the failure of the preservation of subhar-

monicity (respectively, plurisubharmonicity) under more general limit operations can be remedied.

A mapping φ : Y → [−∞,∞) is called presubharmonic (φ ∈ SH[Y ]) (respectively, preplurisub-

harmonic, (φ ∈ PSH[Y ])) if φ admits a subharmonic (respectively, plurisubharmonic) majorant

φ̂ : Y → [−∞,∞) such that φ = φ̂ almost everywhere.

Lemma 5.1. (Compare [9, Theorem L8]) A mapping φ : Y → [−∞,∞) is presubharmonic (respec-

tively, preplurisubharmonic) if and only if the upper envelop

φ[∗](z) := max (φ(z),φ(∗)(z)), ∀z ∈ Y, (5.1)

is subharmonic (respectively, plurisubharmonic), with φ[∗] = φ almost everywhere, in Y ; hence φ[∗]

is the least such majorant of φ.

Proof. The sufficiency part of the lemma is trivial. Suppose that φ admits a subharmonic majorant

φ̂ which agrees with φ almost everywhere. Then the same argument as in the proof of [9, Theorem

L8] shows that φ(z) ≤ φ[∗](z) ≤ φ̂[∗](z) = φ̂(z) for all z ∈ Y and hence φ(z) = φ[∗](z) almost

everywhere. On the other hand, if φ[∗](a) < φ̂(a) at some a ∈ Y, then by the definition (5.1), there

are constants ε > 0, δ > 0, such that supz∈∆a(ε) φ(z) = φ̂(a) − ε. But since φ̂ is subharmonic and

φ̂ = φ almost everywhere in Y, it follows from the regularity of φ̂ (Proposition 3.2) that, for some

standard domain ∆ at a,

νp
∆

(a) φ̂(a) = lim
r→0

〈φ⌋∆〉a,r ≤ νp
∆

(a) (φ̂(a) − ε),

a contradiction, thus proving the desired claim. The remaining case of preplurisubharmonicity is

entirely similar.

Thanks to Corollary 4.1 and Lemma 5.1, the preservation of preplurisubharmonicity on a complex

space can be assured in reference to: (1) the pointwise limit of a monotonically decreasing sequence in



254 Chia-chi Tung CUBO
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PSH[Y ]; (2) the supremum and lim sup of a sequence in PSH[Y ] that is locally uniformly bounded

from above (as in [9, Theorems L9]); and, for a normal space Y, (3) the supremum of a family

{φt}t∈T ⊂ PSH[Y ] that is locally uniformly bounded from above (as in [9, Theorems L10]), and

(4) the operation of forming ψt0 := lim supt→t0 φt, for {φt}t∈T ⊂ PSH[Y ] that is locally uniformly

bounded from above (where t0 ∈ T and T is an open (or closed) subset of R ∪{∞} or C
m). Similar

assertions hold on a general space for presubharmonic functions. An application of such preservation

properties is given by the following generalization of the Hartogs’ lemma:

Corollary 5.1. (Hartogs’ lemma for presubharmonic functions) Assume that {φ
t
}

t>0
is a family in

SH[Y ] that is locally uniformly bounded from above. Then (1) the function G := lim supt→∞ φt is

presubharmonic in Y ; (2) if for some g ∈ C0(Y ), G ≤ g in Y, then for every compact set K ⊂ Y

and every positive ε, there is a positive number N such that supK φt < g + ε for all t ≥ N.

Proof. Let Ft := sups≥t φs. Then G = limt→∞ Ft. By the preceding remark, both Ft and G are

presubharmonic in Y, (alternatively, this can be proved using Theorem 3.1 and Lemma 6.2). Since

{Ft} is a decreasing sequence, so is also the sequence {F
[∗]
t }, hence the limit function limt→∞ F

[∗]
t is

subharmonic. It follows from Lemma 5.1 that G[∗] is equal to limt→∞ F
[∗]
t outside a set of measure

zero, and hence by Proposition 3.2, everywhere in Y. The second assertion follows by showing that

the nested family Et := {z ∈ K |F
[∗]
t (z) − g(z) ≥ ε}, for t > 0, has an empty intersection (by the

same argument as in [12, p. 93]).

In a similar vein it is of interest to see if the theorem of Lelong-Norguet-Bremermann [13, p. 54] on

the representation of plurisubharmonic functions remains valid for a P -convex domain D in a general

complex space, where P = PSH (D), that is, whether each ψ ∈ PSH (D) admits an representation

ψ =
(

lim sup
k→∞

1

k
log ||fk||

)[∗]
,

for some sequence of holomorphic functions fk in D.

Let p : Y → Ω be a Riemann semicovering and G ⋐ Y a weak Stokes domain ([24, p. 568]).

The (real) energy of a function τ ∈ C1(G) is defined by

E
G
(τ) :=

∫

G

dτ ∧ dcτ̄ ∧ υm−1p

([25, § 3]). If τ ∈ C1,1(G) ([24, p. 562]) is real-valued, then its energy is given by

E
G
(τ) =

∫

dG

τ dcτ ∧ υm−1p −

∫

G

τ ddcτ ∧ υm−1p . (5.2)

Suppose now that p : Y → Ω is an unramified Riemann covering. By the expression [24, (5.7)] for

the normal derivatives one has

∫

dG

τ,dcτ ∧ υm−1p = (−1)
m(m−1)

2
1

2‖S‖

∫

dG

τ ∂ντ dσdG. (5.3)



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On Semisubmedian Functions and Weak Plurisubharmonicity 255

Also, since the energy E
G
(τ) is never negative [24, (3.7)], the expression (5.2) (applied to the pseu-

doannular region G = Ua(r)\Ua(s), where 0 < s < r < r0) and the identity (5.6) of [24] imply that

the spherical mean [τ Rp,a(τ)]a,r of a subharmonic C
2-function τ : Y → [0,∞) is nonnegative and

increasing in r (for small r > 0).

If φ = (φ
1
, · · · ,φ

N
) : Y → CN, set τρ := ‖φ‖

2ρ for each positive constant ρ. A mapping

φ : Y → CN is called subharmonic, respectively, semiharmonic, if so are the real and imaginary

parts of each component of φ (regarded as an R-valued function); φ is called weakly real-analytic if

each component φ
j

is continuous in Y and real-analytic in Yreg. Kellog ([11, Theorem IV, p. 213])

showed that a real-valued harmonic function on a closed, regular plane region is nonconstant unless

it has vanishing normal derivative at every boundary point. The relations (5.2) and (5.3) lead to a

generalization of this result to subharmonic mappings:

Proposition 5.1. (1) Assume that G ⋐ Y is a Stokes domain and φ : G → CN, with components

φj ∈ C
1,1(G) ∩ C2(G), is subharmonic in G. If for some constant ρ ≥ 1

2
,

∫

dG

τρ d
cτρ ∧ υ

m−1
p = 0, (5.4)

then φ = constant in G. (2) Assume that Y is a connected complex space and φ : Y → CN a weakly

real-analytic subharmonic map. Then φ is a constant if and only if there exist a point a ∈ Yreg
and ρ ≥ 1

2
such that the function [τρ Rp,a(τρ)]a,r is monotonically decreasing in r (for small r > 0)

relative to an unramified Riemann covering p at a.

Proof. By Example 6.1 of the Appendix and Lemma 3.2, the function τρ = ‖φ‖
2ρ is subharmonic in

G for all ρ ≥ 1
2
. (1) Observe that

ddc‖φj‖
2 = φ̄jdd

cφj + φjdd
cφ̄j + dφj ∧ d

cφ̄j − d
cφj ∧ dφ̄j (5.5)

almost everywhere in G. Writing φj = uj + ivj (in terms of its real and imaginary parts), one has

‖φ‖2 =
∑N

j=1(‖uj‖
2 + ‖vj‖

2). Thus the mapping

φ̃ := (|u1|, |v1|, · · · , |uN |, |vN |) : G → R
2N

is continuous in G and subharmonic in G with ‖φ̃‖2 = ‖φ‖2. Therefore it can be assumed without

loss of generality that each φj is real-valued. Then the expression (5.5) implies that

ddcτ1 ∧ υ
m−1
p ≥ 2

N
∑

j=1

(duj ∧ d
cuj + dvj ∧ d

cvj ) ∧ υ
m−1
p ≥ 0 (5.6)

almost everywhere in G. To prove the assertion (1), observe that by the condition (5.4) and Proposi-

tion 3.4, the energy (5.2) (for τρ) is nonpositive. Since the integral EG (τρ) is always nonnegative, the

norm of the gradient, ‖ ▽ τρ‖, must vanish locally almost everywhere in G. Consequently τρ, hence

also τ1, is constant in G. Hence the relation (5.6) implies that φ = constant in G. For the assertion

(2), assume that relative to an unramified Riemann covering p at a point a ∈ Yreg, the function

[τρ Rp,a(τρ)]a,r is decreasing in r for some ρ ≥
1
2
. Then by the expression (5.2), the energy E

D
(τρ)



256 Chia-chi Tung CUBO
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for the pseudoannular region G := Ua(r)\Ua(s), 0 < s < r < r0, is nonpositive. It follows that

‖ ▽ τρ‖
2 = 0 in G. Therefore τρ is constant in every component of G\Z, where Z is the zero set of

f = ‖φ‖2. Since each component φj of φ is real analytic in Y off a thin analytic subset, the same is

true for the function f; hence (choosing a to be a point where f is real analytic and r0 sufficiently

small) the closure of every component of G\Z must contain a ([16, Lemma 1, p. 96]). This implies

that τρ is constant in G; in particular, so is τ1. It follows as above from the relation (5.6) that φ =

constant in G. Hence by the identity theorem for real-analytic functions, φ = constant in Y .

Corollary 5.2. Let Y be a connected complex space and φ = (φ
1
, · · · ,φ

N
) : Y → CN a weakly

real-analytic and subharmonic map. If ‖φ‖ attains a local maximum at some point of Y, then φ =

constant.

Proof. Observe that the function τ
1

= ‖φ‖2 : Y → R is solidly submedian in Y. Assume that ‖φ‖

attains a local maximum at a point z0 ∈ Y. Hence Proposition 3.1 (applied to a Riemann covering

p
U

of a small open, connected neighborhood U of z0) implies that τ1 is constant in U. Thus τ1
has vanishing mean spherical radial derivative at some point of U, and by Proposition 5.1-(2), φ =

constant in U. Since each φj is real analytic in Yreg, φj = constant in Yreg by the identity theorem

for real-analytic functions, whence the conclusion follows.

By means of the solid submean-value property of subharmonic functions on a general space, two

results of Bochner and Montgomery on matrix-valued mappings [2, Theorems 10 and 11, p. 155] can

be extended:

Proposition 5.2. Assume that Y is a connected complex space. (1) If φ = (φ
1
, · · · ,φ

kN
) : Y → RkN

is continuous, subharmonic such that the image of each map g
j

:= (φ
(j−1)N +1

, · · · ,φ
jN

) : Y → RN, 1 ≤

j ≤ k, is contained in the unit sphere in RN, then φ = constant. (2) Every continuous mapping

φ : Y → U(n) (a unitary group) with semiharmonic components is a constant.

6 Appendix

Lemma 6.1. If {φn} is a monotonically decreasing sequence of subharmonic (respectively, plurisub-

harmonic) functions in a complex space Y, then the limit function lim φn is subharmonic (respec-

tively, plurisubharmonic) in Y.

Lemma 6.2. Assume that (i) Y is locally irreducible; (ii) ψ : Y → [−∞,∞) is locally integrable

and locally bounded from above; and (iii) ψ is solidly submedian with respect to every pseudoball in

Y. Then ψ is presubharmonic in Y.

Proof. Since ψ is locally bounded from above, it can be proved in the same way as in [9, Lemma

L6-(c)] that ψ[∗] is upper semicontinuous in Y. Let ∆ be a pseudoball at at point z ∈ Y. Then

either ψ[∗](z) = ψ(z), hence the solid submean-value inequality holds at z, or there exists a sequence

{ζn} in ∆\{z} tendng to z such that ψ
[∗](z) = limn→∞ ψ(ζn); in the latter case one has

ψ[∗](z) ≤ lim inf
n→∞

〈ψ ⌋∆〉ζn ,r ≤ lim inf
n→∞

〈ψ[∗] ⌋∆〉ζn,r = 〈ψ
[∗] ⌋∆〉z,r,



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On Semisubmedian Functions and Weak Plurisubharmonicity 257

for sufficiently small r > 0. Thus ψ[∗] is subharmonic in Y by Theorem 3.1. Also,

ψ[∗](z) ≤ lim inf
n→∞

〈ψ ⌋∆〉ζn,r = 〈ψ ⌋∆〉z,r.

It follows then from the regularity of ψ[∗] that

lim
r→0

〈ψ[∗] ⌋∆〉z,r = lim
r→0

〈ψ ⌋∆〉z,r.

Consequently ψ[∗] = ψ almost everywhere in Y.

Lemma 6.3. (Compare [17, 3.13]) Let g : I → R be an increasing function on a finite (or infinite)

interval I ⊆ [−∞,−∞) such that g⌋I ∩ R is convex. Assume that φ ∈ SH (Y ) (respectively, φ ∈

PSH (Y )) with φ(Y ) ⊂ I. Then g(φ) ∈ SH (Y ) (respectively, g(φ) ∈ PSH (Y )).

Proof. Suppose that φ ∈ SH (Y ). Let ∆ be a pseudoball at a point a ∈ Y of radius r0. By Theorem

3.1, one has φ(a) ≤ 〈φ⌋∆〉a,r for every r ∈ (0,r0). Note that, since g(φ) = g
+(φ) − g−(φ), the

function g(φ) is locally integrable on Y. Using the supporting line argument of [19, p. 115], it can be

shown that the Jensen’s inequality for convex functions holds for the pair (g,φ):

g(〈φ⌋∆〉a,r) ≤ 〈g ◦ φ⌋∆〉a,r, ∀r ∈ (0,r0).

Therefore, by Theorem 3.1, the composition g ◦ φ is subharmonic in Y. The corresponding assertion

for φ ∈ PSH (Y ) follows then from the definition of plurisubharmonicity.

Example 6.1. Assume that φ = (φ
1
, · · · ,φ

N
) : Y → CN is a subharmonic map with components

φ
j
∈ C2(Y \A) for some thin analytic subset A of Y. Then for all constants c, ρ with c > 1, cρ ≥ 1,

the function (⌊φ⌋ĉ)ρ ∈ SH (Y ) (where ĉ = (c, · · · ,c)).

Proof. It is a consequence of [17, 3.23, p. 19], Proposition 3.4 and Lemma 3.2 that the function

g := (⌊φ⌋ĉ)1/c is subharmonic in Y, hence so is gcρ = (⌊φ⌋ĉ)ρ by Lemma 6.3, provided cρ ≥ 1.

Lemma 6.4. (Compare [17, 2.13]) Assume that φ : Y → [0,∞) is upper semicontinuous and φ 6≡ 0.

Then φ is logarithmically subharmonic in Y if and only if φeh ∈ SH(D) for every subdomain

D ⋐ Y and every continuous function h : D → R which is semiharmonic in D. In particular, if φ

is logarithmically subharmonic in Y, then φ ∈ SH (Y ). The converse is false.

Proof. By virtue of Proposition 3.1 and Lemmas 3.2 and 6.3, the desired conclusion follows from the

same argument as in [17, 2.13].

Example 6.2. If φk : Y → [0,∞) is upper semicontinuous and logarithmically subharmonic for

1 ≤ k ≤ l then so are
∑l

k=1 φk and max1≤k≤l φk.

Example 6.3. (Compare [4, p. 119]) Let Fj = (fj1, · · · ,fjN ) : Y → C
N be a holomorphic map for

each 1 ≤ j ≤ s. Then for all constants ρ > 0 and c(j) = (cj1, · · · ,cjN ) with cjk > 0, the functions

(⌊Fj⌋
c(j) )ρ and max

1≤j≤s
⌊Fj⌋

c(j) are logarithmically plurisubharmonic in Y.



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12, 2 (2010)

Remark 6.1. If φ : Y → [0,∞) is logarithmically plurisubharmonic, it can be shown as in Lemma

6.4 that, for every subdomain D ⋐ Y and every continuous function h : D → R which is semi-

harmonic in D, one has φeh ∈ PSH (D) (in particular, φ ∈ PSH (Y )). In the converse direction,

using Theorem 4.1 the following Lemma can be proved (as in [12, Theorem 2.6.1]):

Lemma 6.5. (Compare also [3, Proposition VI.4.9] and, respectively, [17, 3.12]) Let (X,p) be a

normal semi-Riemann domain. Assume that φ : X → [0,∞) is upper semicontinuous and locally φ 6≡

0. Then φ is logarithmically plurisubharmonic (respectively, logarithmically subharmonic) in X if and

only if φ |e
P

ckpk | ∈ PSH (X) (respectively, φ |e
P

ckpk | ∈ SH (X)) for all constants (c1, · · · ,cm) ∈ C
m.

Received: January 2009. Revised: May 2009.

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