On some properties of a differential operator on the polydisk

PROPERTIES OF DIFFERENTIAL OPERATOR ON THE POLYDISK 69
The Hardy spaces, denoted by H p (U n )(0 < p ≤ ∞), are defined by
H p (U n ) = {f ∈ H(U n ) :
sup M p (f, r) < ∞},
0≤r −1, j = 1, · · · , n, 0 < p < ∞, recall that the weighted Bergman
space A p ⃗α (U n ) consists <strong>of</strong> all holomorphic functions on the polydisk satisfying the
condition
∫
∏ n
‖f‖ p = |f(z)| p (1 − |z
A p i | 2 ) α i
dm 2n (z) < ∞.
⃗α
U n
i=1
Throughout this paper, constants are denoted by C, C α , or C(α), they are
positive and may differ from one occurrence to other. The notation A ≍ B
means that there is a positive constant C such that B/C ≤ A ≤ CB.
Let z = (z 1 , . . . , z n ) ∈ U n , f j (z) ∈ H(U n ), j = 1, · · · , n. It is easy to see that
if
f j (z) =
∑
a (j)
k 1 ,...,k n
z k 1
1 · · · zn kn
, j = 1, · · · n
k 1 ,··· ,k n≥0
is a usual Taylor expansion in U n <strong>of</strong> f j , then
S(f 1 , · · · , f n ) = f 1 + · · · f n =
∑ ( n∑ )
a (j)
k 1 ,...,k n
z k 1
1 · · · zn kn
.
We consider a very particular case when
where a k1 ,...,k n
k 1 ,...,k n≥0
j=1
a (j)
k 1 ,...,k n
= k j a k1 ,...,k n
, j = 1, · · ·, n,
is a certain sequence. We have
S(f 1 , · · · , f n ) =
∑
k 1 ,...,k n≥0
(k 1 + · · · k n )a k1 ,...,k n
z k 1
1 · · · z kn
Motivated by the above expression we define a <strong>operator</strong> in the polydisk as
follows
Rf =
∑
(k 1 + · · · + k n + 1)a k1 ,...,k n
z k 1
1 · · · z kn
or more general form
R s f =
∑
where
It is easy to see that
k 1 ,...,k n≥0
k 1 ,...,k n≥0
f(z) =
n .
(k 1 + · · · + k n + 1) s a k1 ,...,k n
z k 1
1 · · · zn kn
, s ∈ R,
∑
k 1 ,··· ,k n≥0
Rf = R 1 f = f +
a k1 ,...,k n
z k 1
1 · · · z kn
n ∈ H(U n ).
n∑
j=1
n
z j
∂f
∂z j
. (1.1)