On some properties of a differential operator on the polydisk

PROPERTIES OF DIFFERENTIAL OPERATOR ON THE POLYDISK 79
For any functions f(z) and g(z) measurable in the unit disk
∫
∫
|f(z)||g(z)|
dm 2 (z) ≤ C A p (f)(ξ)C p ′(g)(ξ)dm(ξ), 1 < p ≤ ∞, 1 1 − |z|
p + 1 = 1.(3.1)
p ′
U
T
The above result can be easily by iteration extended to polydisk.
Using the polydisk version <strong>of</strong> (3.1), by Cauchy formula in polydisk, Littlewood-
Paley inequality (see [1]) and 2.1 we have for any holomorphic function f in the
polydisk,
∫
|R s f ρ (ϕτ 2 )| = C∣
f(τξ)R s 1
T n 1 − τρξϕ dm n(ξ) ∣
∫
≤ C |D α R s 1
U 1 − wϕρ |f(w)|(1 − |w|)α−1 dm 2n (w)
n
≤
≤
∫
C
∫
C
T n
T n
sup |D α R s 1
w∈Γ t(ξ) 1 − wϕρ (1 − |w|)α dm n (ξ) × ‖C 1 (f)‖ L ∞
sup |R s 1
w∈Γ t(ξ) 1 − wϕρ |dm n(ξ) × ‖C 1 (f)‖ L ∞.
Here α > 0, ϕ, ξ ∈ T n , ρ ∈ [0, 1] n , ϕτ 2 = (ϕ 1 τ1 2 , · · ·, ϕ n τn), 2 τ j ∈ (1/2, 1), j =
1, · · · , n. We used in the last inequality a maximal theorem for D s <strong>operator</strong>s, see
[7]. Therefore
( ∑ n∏
−1
|R s f(ρϕ)| × (1 − ρ k ) k) −α ≤ C‖C1 (f)‖ L ∞
α j ≥0
P αj =s
k=1
and
( ∑
sup M ∞ (D −1 R s f, ρ)| ×
ρ∈[0,1] n
α j ≥0
P αj =s
n∏
(1 − ρ k ) −α k
k=1
) −1
where
≤ C‖C 1 (f × (1 − |z|))‖ L ∞, (3.2)
C 1 (f × (1 − |z|))(ξ 1 , · · · , ξ n ) = sup
ξ 1 ∈I
∫
1
· · · sup
|I| S(I) ξ n∈I
∫
1
|f(z)|dm 2n (z).
|I| S(I)
Remark 3.3. Estimates (3.2) extends known one dimensional inequality which
can be found in [10]. We just give one application <strong>of</strong> Lemma 2.1. Various other
generalizations can be obtained with the help <strong>of</strong> Lemma 2.1. Estimate (3.2) is
just an example.
We see that the R s <strong>operator</strong> are connected with the quasinorms on subframe
via integral representations (2.4) and (2.5). So it is natural to find new estimates
for quasinorms on subframe. In the polydisk we consider the following three
expressions
∫ 1 ∫
|f(rξ 1 , · · · , rξ n )| p (1 − r) α rdrdm n (ξ), 0 < p < ∞, α > −1;
0 T n