On some properties of a differential operator on the polydisk

PROPERTIES OF DIFFERENTIAL OPERATOR ON THE POLYDISK 83
Corollary 3.9. Let u ∈ H(U n ), q ∈ (0, ∞), α j > 0, j = 1, · · · , n, n ∈ N. Then
∫
|u(z, · · · , z)| q (1 − |z| 2 ) P n
j=1 αj−1 dm 2 (z)
≤
U
∫
C
T
∫
Γ t(ξ)
Pro<strong>of</strong>. Fix ξ ∈ T , then (see [3])
∫
· · · |u(z 1 , · · · , z n )| q
Γ t(ξ)
∫ 1
0
n
∏
k=1
|ũ(ρξ)| q
1 − ρ
∫Γ dρ ≤ C |ũ(z)| q dm 2 (z)
.
t(ξ) (1 − |z| 2 ) 2
(1 − |z k | 2 ) α k−2 dm 2n (z)dm(ξ).
Choosing ũ = u(ρξ)(1 − ρ) α/q , u ∈ H(U), α, q ∈ (0, ∞), we have
∫ 1
∫
|u(ρξ)| q (1 − ρ) α−1 dρ ≤ C |u(z)| q (1 − |z| 2 ) α−2 dm 2 (z).
0
Γ t(ξ)
Using the last inequality, by each variable we get the following
≤
∫ 1
0
∫
C
· · ·
Γ t(ξ)
∫ 1
0
∫
· · ·
|u(ρ 1 ξ, · · · , ρ n ξ)| q
Γ t(ξ)
|u(z)| q
n
∏
k=1
n
∏
k=1
(1 − ρ k ) α k−1 dρ 1 · · · dρ n
(1 − |z k | 2 ) α k−2 dm 2n (z).
Integrating both sides by T and using i) <strong>of</strong> Theorem 3.6, we have
∫
|u(z, · · · , z)| q (1 − |z| 2 ) P n
j=1 αj−1 dm 2 (z)
≤
≤
U
∫
C
∫
C
T
T
∫ 1
∫
0
· · ·
Γ t(ξ)
∫ 1
0
∫
· · ·
|u(ρ 1 ξ, · · · , ρ n ξ)| q
Γ t(ξ)
|u(z 1 , · · · , z n )| q
This completes the pro<strong>of</strong> <strong>of</strong> Corollary 3.9.
n
∏
k=1
∏ n
(1 − ρ k ) α k−1 dρ 1 · · · dρ n dm(ξ)
k=1
(1 − |z k | 2 ) α k−2 dm 2n dm(ξ).
Remark 3.10. For n = 1, the assertion <strong>of</strong> Corollary 3.9 is contained in [3, 18].
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