Weighted integrals of holomorphic functions in the unit polydisc
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Let f be a measurable function defined on the unit polydisc U n in Cn and let ω j (z j ), j = 1,...,n, be admissible weights on the unit disk U, with distortion functions ψ j (z j ), p,q q q ᏸω,N (U n ) = { f | f ᏸ p,q < ∞}, where f ᏸ p,q = [0,1)n M p ( f ,r) nj=1 ω j (r j )dr j , and p,q Ꮽω,N (U n )
,N ω
=
,N ω
p,q ᏸω,N (U n ) ∩ H(U n ).
We prove the following result: if p, q ∈ [1, ∞) and for p,q p,q all j = 1,...,n, ψ j (z j )(∂ f /∂z j )(z) ∈ ᏸω,N , then f ∈ Ꮽω,N and there is a positive constant n C = C(p, q,ω j ,n) such that f Ꮽ p,q ≤ C | f (0)| + j =1 ψ j (∂ f /∂z j )ᏸ p,q . ,N ω
,N ω
1. Introduction Let U 1 = U be the unit disk in the complex plane, dm(z) = (1/π)dr dθ the normalized Lebesgue measure on U, U n the unit polydisc in complex vector space Cn and H(U n ) the space of all analytic functions on U n . For z,w ∈ Cn we write z · w = (z1 w1 ,...,zn wn ); eiθ is an abbreviation for (eiθ1 ,...,eiθn ); dt = dt1 · · · dtn ; dθ = dθ1 · · · dθn and r, θ are vectors in Cn . If we write 0 ≤ r < 1, where r = (r1 ,...,rn ) it means 0 ≤ r j < 1 for j = 1,...,n. For f ∈ H(U n ) and p ∈ (0, ∞) we usually write
1 M p ( f ,r) = (2π)n
[0,2π]n
f r · eiθ p dθ
1/ p
,
for 0 ≤ r < 1
(1.1)
for the integral means of f . Let ω(s), 0 ≤ s < 1, be a weight function which is positive and integrable on (0,1). We extend ω on U by setting ω(z) = ω(|z|). We may assume that our weights are normalized 1 so that 0 ω(s)ds = 1. p p Let ᏸω = ᏸω (U n ) denotes the class of all measurable functions defined on U n such that p f ᏸ p ω
=
Un
n
f (z) p ω j z j dm z j < ∞, j =1
where ω j (z j ), j = 1,...,n, are admissible weights on the unit disk U. Copyright © 2005 Hindawi Publishing Corporation Journal of Inequalities and Applications 2005:5 (2005) 583–591 DOI: 10.1155/JIA.2005.583
(1.2)
584
Weighted integrals of holomorphic functions p
p
The weighted Bergman space Ꮽω is the intersection of ᏸω and H(U n ). For ω j (z j ) = (1 − |z j |2 )α j , α j > −1, j = 1,...,n, we obtain the classical Bergman space Ꮽ p (dVα ), see [1, page 33]. p,q p,q Let ᏸω,N = ᏸω,N (U n ), p, q > 0, denotes the class of all measurable functions defined on U n such that
q f ᏸ p,q ,N ω
=
p,q
q
[0,1)n
M p ( f ,r)
n
ω j r j dr j < ∞,
(1.3)
j =1
p,q
p,q
p
and Ꮽω,N be the intersection of ᏸω,N and H(U n ). When p = q we denote Ꮽω,N by Ꮽω,N . In the case p = q, these two norms are equivalent on the space H(U n ), but the later one is more suitable for calculations than the first one. The result is contained in the following lemma. Lemma 1.1. The norms · Ꮽ p and · Ꮽ p are equivalent on the space H(U n ). ω
,N ω
Proof. By the polar coordinates it is easy to see that f Ꮽ p ≤ 2n f Ꮽ p for every f ∈ ,N ω ω H(U n ), moreover f ᏸ p ≤ 2n f ᏸ p for every f measurable on U n . ,N ω ω Now we prove that there is a positive constant C, which is independent of f , such that f Ꮽ p ≤ C f Ꮽ p , ,N ω
(1.4)
ω
for every f ∈ H(U n
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