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A converse to the Andreotti-Grauert theorem

A. Sadullaev

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A. SADULLAEV
On separately subharmonic functions (Lelong’s problem)
Tome XX, no S2 (2011), p. 183-187.
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Annales de la Faculté des Sciences de Toulouse Vol. XX, n◦ Spécial, 2011
pp. 183–187
On separately subharmonic functions
(Lelong’s problem)
A. Sadullaev(1)
ABSTRACT. — The main result of the present paper is : every separately-
subharmonic function u(x, y), which is harmonic in y, can be represented
locally as a sum two functions, u = u∗ + U , where U is subharmonic and
u∗ is harmonic in y , subharmonic in x and harmonic in (x, y) outside of
some nowhere dense set S.
RÉSUMÉ. — Le résultat essentiel de ce papier est le suivant : toute fonc-
tion séparément sous-harmonique u(x, y) qui est harmonique en y peut
être reprétée localement comme la somme de deux fonctions u = u∗ +U ,
où U est sous-harmonique et u∗ est harmonique en y, sous-harmonique
en x et harmonique en (x, y) en dehors d’une ensemble nulle part dense
S.
1. Introduction
We will consider functions u(x, y) of two groups of variables x ∈ Rn,
y ∈ Rm. If u is separately harmonic, i.e., harmonic in x for fixed y and
harmonic in y for fixed x, then u will be harmonic in both variables (Lelong
[2],see also [1]). Lelong investigated also separately subharmonic functions,
and proved a series of special results in this area. Here originates the question
about subharmonicity of separately subharmonic functions.
(∗) * Partially supported by the fundamental research of Khorezm Mamun Academy,
Grant Vuzgorodok, 100174
(1) Mathematics department, National University of Uzbekistan, Vu Gorodok, 700174
Tashkent, Uzbekistan
sadullaev@mail.ru
– 183 –
On separately subharmonic functions (Lelong’s problem)
However, Wiegerinck, [3] (see also [4]) has shown that a separately
subharmonic function need not to be subharmonic in general. He con-
structed a separately subharmonic function u (x, y) in the bidisk U2 =
{|x| < 1} × {|y| < 1} ⊂ R2x × R2y ≈ C × C, which is not bounded above
near 0.
The problem of subharmonicity of a separately subharmonic
function u(x, y) that is in addition harmonic in y, is still open.
In the present paper we will study the class of these functions. Let us
begin by recalling the following well-known results:
1. If a separately subharmonic function is bounded above, then it is
subharmonic (Lelong [2], Avanissian [5]);
2. If u+ ∈ L1loc, then u is subharmonic (Arsove [6]);
3. If u+ ∈ Lploc, p > 0, then u is subharmonic(Riihentaus [7]);
4. There are also positive results under weak growth conditions (see [8],
[9]).
We note that the conditions in the above results are not separated in x and
y. The following results demand separate conditions:
5. Suppose that u(x, y) is defined on the product domain B = B1 ×
B2 ⊂ Rnx × Rmy . If u is subharmonic in x and harmonic in y, then
there are nowhere dense closed sets S1 ⊂ B1, S2 ⊂ B2 such that u
is subharmonic in G = (B1 ×B2) \ (S1 × S2) (Cegrell and Sadullaev
[10]);
6. If u(x, y) real analytic, subharmonic in x, and harmonic in y, then u
is subharmonic (Imomkulov [11]);
7. There exists a separately subharmonic function u(x, y), which is real
analytic in x, but which is not subharmonic (Cegrell and Sadullaev
[10]);
8. If u(x, y) is C2 and subharmonic in x, harmonic in y, then u is sub-
harmonic (Ko:lodziej and Thorbiörnson [12]).
2. Results
Let u(x, y) be a separately subharmonic function in the product domain
B = B1 × B2, which is harmonic in y. We will assume that u satisfies
– 184 –
A. Sadullaev
this condition in a slightly larger domain B̃ = B̃1 × B̃2 such that B̃ ⊃
B. Then u(x, y) is subharmonic in a domain
(
B̃1 × B̃2
)
\ (S1 × S2), where
S1 ⊂ B̃1, S2 ⊂ B̃2 are closed, nowhere dense sets. Moreover, for every fixed
y ∈ B̃2 the Laplacian ∆xu (x, y) defines a positive distribution as follows
F (ϕ) =
∫
u (x, y) ∆xϕ (x) dx ϕ ∈ C∞0 ,
thus for every test function ϕ (x) ∈ C∞0 (B1), suppϕ ⊂⊂ B1 , ϕ  0 we
have F (ϕ)  0. Hence, ∆xu (x, y) is a Borel measure, depending on the
parameter y.
Theorem 2.1. — For every test-function ϕ (x) ∈ C∞0 (B1) F (ϕ) is har-
monic in y for y ∈ B2\S2. Moreover, if suppϕ
⋂
S1 = ∅. then F (ϕ) is
harmonic in y for all y ∈ B2.
We say that the measure ∆xu (x, y) has the harmonic property with
respect to y in the domain G=(B1 ×B2) \ (S1 × S2).
Proof. — The result 5) above states that u (x, y) is subharmonic and
therefore u is locally bounded above in G=(B1 ×B2) \ (S1 × S2). Hence
the integral
F (ϕ) (y) =
∫
B1
ϕ (x) ∆xu (x, y) =
∫
B1
u (x, y) ∆xϕ (x)
is harmonic in B2\S2. If suppϕ
⋂
S1 = ∅, then this integral is harmonic in
all B2. 
Corollary 2.2 The measure FE (y) =
∫
E
∆xu (x, y) is harmonic in
B2 for any E ⊂⊂ B1\S1.
Corollary 2.3. — The total measure ‖∆xu (x, y)‖B1 =
∫
B1
∆xu (x, y)
is finite (=∞) for every fixed y ∈ B2 and is harmonic function in B2\S2.
Theorem 2.4. — The function FB1\S1 (y) =
∫
B1\S1 ∆xu (x, y) is bounded
and positive harmonic in B2.
Proof. — Let us take an increasing sequence of compacts Ej ⊂ Ej+1 ⊂⊂
B1\S1 such that
⋃
j
Ej = B1\S1. Then the functions FEj (y) =
∫
Ej
∆xu (x, y)
– 185 –
On separately subharmonic functions (Lelong’s problem)
are harmonic in B2 and form an increasing sequence in j. By Harnacks’ the-
orem either FEj (y)↗ +∞ or (FEj )j converges to a harmonic function. The
first possibility is ruled out, because Corollary 2.3 provides a bound on the
FEj (y) for every y ∈ B2\S2.
Thus lim
j→∞
FEj (y) =
∫
B1\S1 ∆xu (x, y) is harmonic in B2, which com-
pletes the proof. 
Now we consider the potential
U (x, y) =
∫
B1\S1
K (x− w) ∆wu (w, y) ,
where K is the Newtonian kernel,
K(w) =



1
2π
ln |w| , if n = 2
− 1
(n− 2)σn |w|n−2
, if n > 2.
The measure ∆xu (x, y) has the harmonic property in (B1\S1) × B2.
Moreover, for some constant C the total measure
∫
B1\S1 ∆xu (x, y)  C, y ∈
B2. It follows that the integral
∫
B1\S1 ϕ (w) ∆wu (w, y) is harmonic in y for
every continuous function ϕ ∈ C
(
B̄1
)
. Let Kj(w) ∈ C∞(Rn) approximate
K from above, Kj (w) ↓ K (w). Then for every fixed x ∈ B1 we have
∫
B1\S1
Kj (x− w) ∆wu (w, y) ↓
∫
B1\S1
K (x− w) ∆wu (w, y)
for j →∞, hence U (x, y) is harmonic in y for fixed x ∈ B1. Moreover, U is
subharmonic in x and bounded above in B1×B2. It follows by the theorem
of Lelong and Avanissian (1), that U is subharmonic in B1 ×B2.
Now we take the difference u∗ (x, y) = u (x, y) − U (x, y) . The function
u∗ (x, y) is separately subharmonic and is harmonic in y. Moreover, u∗ (x, y)
is harmonic in x outside S1. Thus we have
Theorem 2.5. — Every separately subharmonic function, which is har-
monic in y, can locally be represented as a sum of two functions:
u (x, y) = u∗ (x, y) + U (x, y) ,
where U is a subharmonic function and u∗ is separately subharmonic and
harmonic in y, such that the associated measure ∆xu
∗ (x, y) is supported on
S1 for every fixed y ∈ B2.
– 186 –
A. Sadullaev
Problem 2.6. — We finish our discussion by recalling an open prob-
lem on the definition of plurisubharmonic functions: in this definition one
demands two conditions.
a. The function u (z) is upper semicontinuous;
b. For each complex line l the restriction u |l is subharmonic.
The above results on separately subharmonic functions seem to indicate,
that the condition a. may be implied by b. But this is still open.
Bibliography
[1] Lelong (P.). — Les fonctions plurisousharmoniques, Ann. Sci. Ecol. Norm. Sup.
62, p. 301-328 (1945).
[2] Lelong (P.). — Fonctions plurisousharmoniques et fonctions analytiques de vari-
ables réelles, Ann. Inst. Fourier 11, p. 515-562 (1961).
[3] Wiegerinck (J.). — Separately subharmonic functions need not be subharmonic,
Proc. Amer. Math. Soc. 104, p. 770-771 (1988).
[4] Wiegerinck (J.) and Zeinstra (R.). — Separately subharmonic functions: when
are they subharmonic, Proc. Sympos. Pure Math. 52:1, p. 245-249 (1991).
[5] Avanissian (V.). — Fonctions plurisousharmoniques et fonctions doublement
sousharmoniques, Ann. Sci. École Norm. Sup. 78, p. 101-161 (1961).
[6] Arsove (M.G.). — On subharmonicity of doubly subharmonic functions, Proc.
Amer. Math. Soc. 17, p. 622-626 (1966).
[7] Riihentaus (J.). — On a theorem of Avanissian-Arsove, Exposition. Math. 7,
p. 69-72 (1989).
[8] Riihentaus (J.). — Subharmonic functions, generalizations and separately sub-
harmonic functions, arXiv:math / 0610259v5 (2008).
[9] Armitage (D.M.) and Gardiner (S.J.). — Conditions for separately subharmonic
functions to be subharmonic, Potential Anal. 2, p. 225-261 (1993).
[10] Cegrell (U.) and Sadullaev (A.). — Uzbek Math. J. 1, p. 78-83 (1993).
[11] Imomkulov. — Separately subharmonic functions, S.A. Dokl. UzSSR. 21, p. 8-10
(1990).
[12] Ko(lodziej (S.) and Thorbiörnson (J.). — Separately harmonic and subharmonic
functions, Potential Anal. 5, p. 463-466 (1996).
– 187 –


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