On some properties of a differential operator on the polydisk

PROPERTIES OF DIFFERENTIAL OPERATOR ON THE POLYDISK 75
Theorem 2.7. Let 0 < p < ∞, α > −1, s ∈ N, f ∈ H(U n ). If γ > α+2 − 1 for
p
p ≤ 1 and γ > α+1 + 1 (1 − 1 ) for p > 1, v = sp + αn − γpn + n − 1, then
p n p
∫
|D γ f(z)| p (1 − |z| 2 ) α dm 2n (z)
U
∫ n 1 ∫
≤ C |R s f(w)| p (1 − |w| 2 ) v dm n (ξ)d|w| ,
0 T n
where w = |w|ξ.
Pro<strong>of</strong>. Let f ∈ H(U n ), p ≤ 1. Then (see [14])
M 1 (f, τ 2 ) ≤ C(1 − τ) n(1−1/p) M p (f, τ), τ ∈ (0, 1). (2.6)
Using (2.6) and the fact that M p (f, r) is increasing as a function <strong>of</strong> r (see [5]) we
easily get
( ∫ 1
) p
|f(w)|(1 − |w|)
∫T t dm n (ξ)d|w|
n
≤ C
0
∫ 1 ∫T n
0
|f(w)| p (1 − |w|) tp+(n+1)(p−1) dm n (ξ)d|w|, (2.7)
where t > n(1−p) − 1, f ∈ H(U n ), p ≤ 1. By the Cauchy formula,
p
∫
∣ |D γ f ρ (z 1 , · · · , z n )| p ≤ C
f ρ (ξ 1 , · · · , ξ n )dm n (ξ) ∣∣∣
p
∣
, ρ ∈ (0, 1).
T n (1 − ξz) γ+1
Using (1.2) or (2.4), (2.5) for large s and (2.7) we obtain
∫
∣ f ρ (ξ 1 , · · · , ξ n )dm n (ξ) ∣∣∣
p
∣ ∏ n
T n k=1 (1 − ξ kz k ) γ+1
( ∫ 1 ∫
)
|R s f ρ (w)|(1 − |w|) s−1 p
dm n (ξ)d|w|
≤ C
0 T |1 − wz| n γ+1
∫ 1
|R
≤ C
∫T s f ρ (w)| p (1 − |w|) (s−1)p (1 − |w|) (n+1)(p−1)
dm
|1 − wz| n (γ+1)p n (ξ)d|w|. (2.8)
0
Therefore, from (2.6)-(2.8) and using Fubini’s theorem we have
∫
|D γ f ρ (z)| p (1 − |z| 2 ) α dm 2n (z)
U n ∫ 1 ∫
|R
≤ C
∫U s f ρ (w)| p (1 − |w|) sp+n(p−1)−1 dm n (ξ)d|w|
(1 − |z| 2 ) α dm n 0 T |1 − wz| n (γ+1)p 2n (z)
∫ 1 ∫
(1 − |z|
≤ C
0
∫T 2 ) α dm 2n (z)
|R s f n U |1 − wz| n (γ+1)p ρ (w)| p (1 − |w|) sp+n(p−1)−1 dm n (ξ)d|w|.
Using the following estimate
∫
(1 − |z|) t dm 2n (z)
∏ n
k=1 |1 − z kw k | ≤ C
∏ t 1 n
k=1 (1 − |w k|) , t > −1, t t 1 > t + 2,
1−t−2
U n