On some properties of a differential operator on the polydisk

PROPERTIES OF DIFFERENTIAL OPERATOR ON THE POLYDISK 77
3. Embedding theorems for <strong>some</strong> analytic spaces connected with
R s <strong>operator</strong>
In this section we state <strong>some</strong> new embedding theorems for various quasinorms
where the R s <strong>operator</strong> is participating, note that practically all results are well
known or obvious in the unit disk. Pro<strong>of</strong>s <strong>of</strong> our results are heavily based on
definitions and preliminaries from previous sections.
Theorem 3.1. i) Let 0 < q < ∞, α ∈ [0, ∞), s ∈ N, 1 < p < ∞, f ∈ H(U n ).
Then
( ∫ 1
q/p
(1 − |z|)
∫T α |f(z)| p |z| d|z|) sp dm n (ξ)
n
≤
0
( ∫ 1
q/p
C |R
∫T s f(uξ)| p (1 − u) du) sp+α dm n (ξ).
n
0
ii) Let 0 < q < ∞, α ∈ [0, ∞), s ∈ N, 1 < p < ∞, γ ∈ (−1/p, 1/p ′ ), 1/p ′ + 1/p =
1, f ∈ H(U n ). Then
≤
( ∫ 1
q/p
(1 − r)
∫T γ |f(rξ)| p r dr) p dm n (ξ)
n
0
( ∫ 1
q/p
C |R
∫T s f(rξ)| p (1 − r) dr) sp+γ dm n (ξ) .
n
Pro<strong>of</strong>. Using (1.2) we have
( ∫ 1
M = (1 − |z|) α |f(z)| p |z| sp d|z|
≤
= C
0
( ∫ 1 ( ∫ 1
C
0
∫ 1 ∫ 1
0
where ψ(|z|) ∈ L p′ (d|z|), 1/p ′ + 1/p = 1.
Changing the variables we have
∫ 1
0
0
0
0
) 1/p
) p(1 1/p
|R s f(ρz)|(1 − ρ) dρ) s−1 − |z|) α |z| sp d|z|
|R s f(ρz)|(1 − ρ) s−1 (1 − |z|) α/p ψ(|z|)|z| s d|z|dρ ,
|R s f(ρz)|(1 − ρ) s−1 dρ ≤
∫ |z|
Using the last inequality and Hölder inequality, we get
M ≤ C
≤
C
∫ 1 ∫ |z|
0
∫ 1
0
( ∫ 1
0
0
|R s s−1 du
f(uξ)|(1 − u)
|z| . s
|R s f(uξ)|(1 − u) s−1 (1 − |z|) α/p ψ(|z|)d|z|du
|R s f(vξ)|(1 − v) s
1 − v
∫ 1
v
) p ′
(1 − |z|) α/p ψ(|z|)d|z|dv
( ∫ 1 1
) 1/p ′( ∫ 1
1/p
≤ C
ψ(|z|)d|z| dv |R s f(uξ)| p (1 − u) du) α+ps .
0 1 − v v
0
From the last inequality and Lemma 2.4 we easily get the first part <strong>of</strong> the theorem.