Sabanci University,19 DEC 2008
Definition of the complex Monge-Ampere Operator for Arbitrary Plurisubharmonic
Functions
A.Sadullaev
Definition of the Monge-Ampere operator  dd cu  and solution of the corresponding
n
Dirichlet problem  dd cu  =  , u  Psh   , u |   has a long time history and there were
n
proved enough many fundamental results.
Bremermann (1959) noted that if u  C 2  
Psh   is maximal, i.e.
 
inf  u  v   inf  u  v  v  Psh G , v |G  u |G ,then
G
G
proved that if
the operator
 dd u  =0. Later
n
c
(1977) Kerzman
 dd u  = 0 then u is maximal. Chern, Levine and Nirenberg (1969) proved that
 dd u  is bounded in the mean for the class of uniformly bounded
n
c
n
c
plurisubharmonic functions of class C 2 .We note that this proposition easily follows also from
the next integral formula [1,2]:
If G  {  z   0} is a strictly pseudoconvex domain,   C 2  G  ,   min   z  and u is a
G
C 2 plurisubharmonic function in G, then for each r and k ,   r  0,1  k  n
r
 dt    dd  

c
 z  t
nk
  dd cu    M  m 
k
  dd  

c
n  k 1
 z  r
  dd cu  ,
k 1
where M  max u  z  , m  min u  z  .
G
G
After these results it became clear, that if u  Psh   Lloc   and u j  u  C 
approximation, then { dd cu j  } is a compact family of Borel measures, and one is possible to
n
define  dd cu  as a limit of the sequence { dd cu j  } . In this way Bedford and Taylor (1976)
n
n
gave first definition of operator Monge-Ampere for u  Psh   Lloc  
relation:
  dd u 
c
k
    u  dd cu 
But the definition
 dd u 
c
n
k 1
by recurrent
 dd c ,   D nk ,nk 
for arbitrary u  Psh   is still a strong problem and it
prevented to solve the Dirichlet problem  dd cu  =  , u  Psh   , u |   in general.
n
First in 1975 Shiffman and Taylor show that there is a u  Psh  C n  suсh that
  dd u 
c
j
n
  ,where u j  u . Kiselman noted that the function
u  z     ln z1 
1/ n
z
2
2

 ...  zn  1
2
is plurisubharmonic near the origin and it has unbounded mass near z1  0 .
Second the example of U.Cegrell:


n
1
2
2
2
u  z   ln z1  ...  ln zn ; u j  z   ln( z1...zn  )  u  z  , dd cu j
j
n
1
1
2
2
v j  z   ln( z1  )  ...  ln( zn  )  u  z  , dd cu j
n !4n  0 ,
j
j

0,

where  0  the Dirac measure, shows that it is not possible to well- define the  dd cu  for
n
arbitrary plurisubharmonic function. However for some class of unbounded plurisubharmonic
function one is possible to well-define  dd cu  .
n
1. Kiselman proposed the following variant to define
1

   w  : Im w   
2

measure
 dd F 
c
n 1
n 1
 dd u  : We consider the domain
n
c
and the auxiliary function F  z, w   u  z   Re w . Then the

supportes on the graph Re w  u  z  and its projection on
n
z
for bounded
 dd u  . This mass we denote as K(u). But for unbounded function the
projection is not well-defined: For u  z   ln z we have  dd u  =0.
Operator  dd u  in this definition does not take into account the mass supported on singular
set S  u  z    .
Another method to define  dd u  is due to Z.Blocki: For bounded plurisubharmonic function u
c
function is equal
n
2
c
n
c
n
c
n
1
it is clear that e 2u dd c u  2eu dd c eu  dd c e 2u .Therefore for unbounded functions we can define
2
p
 dd cu  ,1  p  n , outside S as e2 pu B p (u) = (e2u dd cu) p  (2eu dd ceu  12 dd ce2u ) p .
2. In the paper [3] author proved that for class

L  u  z   Psh  n  :   ln z  u  z     ln z , z  r , where r ,  ,   cons tan ts ,


the  dd c u  and u  dd cu  , 0  k  n, n  2, are well-defined. It follows that if a
k
k
plurisubharmonic function u  z  is bounded in some sphere S  0, r  , then these operators are
well-defined inside B  0, r  . Similarly results were also proved by Demailly [4].
3. U .Cegrell considered a few class of plurisubharmonic functions, where  dd cu  can be
n
well-defined. Let   n be a hyperconvex domain, i.e. there exists
v  Psh   , v |  0 and lim v  z   0. Then we put
z 

E0     u  Psh    L    , u |  0, lim u  z   0 and
z 

  dd u  < 
c

n
F     u  Psh    : u j  u , u j  E0    and sup   dd cu j   



n
E     u  Psh    : the sequense u j exists locally .
For u  E    F   it is possible to well-define
 dd u 
c
j
n

 dd u 
c
n
using compactness of
.
In this way Z.Blocki introduced class D, where operator Monge-Ampere can be well-defined.
Def. We will say that operator
 dd u 
c
n
is well-defined for u  Psh   if for each open set
U   there exist a Borel measure  such that for each u j  Psh U  C  U  , u j  u is
true
 dd u 
c
 . Then we put  dd cu    and denote the set of all such u by D.
n
n
j
Z.Blocki proves a series of properties of D [5,6]. In partiqular, u  D      dd cu j  is
n
weakly bounded for each sequence u j  u .Moreover, class D is characterized by boundedness of
currents: If
uj
n p2

du j  d cu j   dd cu j   dd c z
p

2 n  p 1
, p  0,1,..., n  2, is bounded for some sequence
u j  u , it follows that these currents are bounded for each sequence u j  u .
For hyperconvex domain   C n Cegrells’class E    is equal to D    and for them
well-defined also all  dd cu  ,1  p  n.
p
4. Let u  z   Psh   be an arbitrary plurisubharmonic function in a domain  
n
.We put
v  eu and ua  ln  v  a   ln  eu  a  , a  0. Then ua  u in a  0 and v  Psh   L   .
So the operators
 dd v 
c
p
are correctly defined.
We have
1
1
dd cua   v  a  dd cv   v  a  dv  d cv  ,


p
p 1

p

1
(dd c ua ) p   v  a   dd c v   p  v  a   dd c v   dv  d c v  


p
p 1
p
a
 p 1
dd c v   1,a  2,a
 v  a  v  dd c v   pdv  d cv   dd cv   
p 
(v  a )
The 1,a , 2,a are positive currents. In fact it is clear that for 2,a and for 1,a we take the
standard approximation u j  u and put v j  e j .We have
u
v
e
j
 a
 p 1u j
p 1
1,a, j  v j (dd c v j ) p  pdv j  d cv j   dd cv j 
 dd u
 du j  d cu j   pe
p
c
j
It is clear that 1,a , j
 p 1u j
p 1

du j  d cu j   dd cu j  du j  d cu j 
p 1
e
 p1u j
 dd u 
c
j
p
 0.
1,a in j   . It follows that 1,a  0 .
We put, formally, 1 p  lim 1,a , 2 p  lim 2,a .The 1p and 2p are well-defined outside of
a 0
j 
S  {u  z   } and here  2p =0. The current 1p  2P fully characterize  dd cu  : the 1p -n
regular part, and the  2p --singular part of
 dd u  ( supp 
c
n
p
2
 S ).
Problems:
1. Is  2p a bounded current, supported on S?
2. Is it true that 2p  B p and 2n  B n  K for arbitrary plurisubharmonic function u  z  ?
3. Is a Psh function u  z  maximal  1n  2n =0 ? The  part is true.
4. If u j  u is arbitrary approximation of u by bounded plurisubharmonic functions u j , such
that  dd cu j 
p
,
Then is the   ( 1p  2P ) positive? It means,that 1p  2P minimal current among the other
currents, corresponding to the approximations u j  u . Note that (4)  (3).
Examples.
2
2
2
2
2
1. u  ln z  ln z1  ...  zn
. Then v  z1  ...  zn ,


dd c v  dz1  d z1  ...  dzn  d zn and dv  d c v  z1 dz1  d z1  ...  zn dzn  d zn
2
It follows that for
p  n  1 the  
p
1
 
p
1
Moreover,



p
v dd c v  pdv  d c v  dd cv

 dd c z
2  p 1
z

2 n p

2
p 1
is borel tipe current and 1p, a  1p .
 0 in r  0. We note that 2p, a 
B o, r
But 1n, a  0 and 2n, a  const


z
adV
a
2

n 1
 
, so that
n
2, a
a
dd cv p  0 in a  0 .
p
v  a 
 1 in a  0
B 0, r
2. u  ln z1  z2 , v  z1  z2 .
2
2
2
2
Then dd c v  dz1  d z1  dz2  d z2 , dv  d c v  z1 dz1  d z1  z2 dz2  d z2
2
2
z2 dz1  d z1  z1 dz2  d z2
2
We have 
p
1, a

11  dd c z
 0 for p  3 and 
1
1, a
 
2
z
1
1
1

2 n 1
dV

z1  z2
2


2 n 1

z
1


2 n2

z
adV
2
1
 z2

2 2
.Therefore,
.
2
It is not difficult to see that 21, a  dd c z
22, a  dd c z
2
 z2  a
2

3
adV
2
 z2

2 2
 0 in a  0 and
  z1  0, z 2  0 (Dirac measure supported on the plane
z1  0, z2  0  C n ).
3. u  ln f z  , where f  OC n . Then v  f f , dv  d c v  f df  d f , dd c v  df  d f .
2
2
It is clear that for p  2 the 1p.a  2p, a  0 .We have
11, a  0 and 21, a 
f
a
2
a

2
df  d f   1f ,10 -- the Dirac (1,1) current supported on the
analytic set {f=0}.
REFERENCES
1. A.Sadullaev, Оператор (ddcu)n и емкости конденсаторов, ДАН СССР, 1980, т251,№1,
С.44-57 (translated :Operator
 dd u 
c
n
and capacity of condencer)
2. A.Sadullaev, Плюрисубгармонические функции (коллективная монография)(Psh
function,monography), Современные проблемы математики. Изд-во ВИНИТИ,
Springer,1985,т.8, 65-114
3. A.Sadullaev, Об одном классе Psh функций,Известия АНУзССР,1982,N 2, 12-15
4. J.-P. Demailly, Mesures de Monge-Ampere et measures Plurisousharmoniques,
Math.Z.,1987,V.194,519-564
5. Z.Blocki, On the definition of the Monge-Ampere operator in 2 ,
Math.Ann,2004,V.328,415-423
6. Z.Blocki, The domain of definition of the complex Monge-Ampere operator,preprint2004