Holomorphic Functions on the Mixed Norm Spaces on the ... - YSU

JOURNAL OF COMPUTATIONAL ANALYSIS AND APPLICATIONS,VOL.11,NO.2,239-251,2009,COPYRIGHT 2009 EUDOXUS PRESS, 239 LLC

Holomorphic Functions on the Mixed Norm Spaces on the Polydisc II Karen Avetisyan Faculty of Physics, Yerevan State University, Alex Manoogian st. 1, Yerevan, 375025, Armenia E-mail: [email protected]

Stevo Stevi´ c Mathematical Institute of the Serbian Academy of Science, Knez Mihailova 36/III, 11000 Beograd, Serbia E-mail: [email protected]; [email protected]

Abstract The paper continues the investigation of holomorphic mixed norm spaces Ap,q in the unit polydisc of n . We prove that a mixed norm ω  is equivalent to a “derivative norm” for all 0 < p ≤ ∞, 0 < q < ∞ and a large class of weights ω  . As an application, we prove that pluriharmonic conjugation is bounded in these mixed norm spaces. 2000 AMS Subject Classification: 32A37, 32A36. Key words and phrases: holomorphic function, polydisc, mixed norm space, weight function, pluriharmonic conjugate.

1

Introduction

Let U 1 = U be the unit disc in the complex plane, U n the unit polydisc in Cn , and H(U n ) the set of all holomorphic functions on U n . For the integral means of a function f given in U n , we write  1/p  1 Mp (f, r) = |f (r1 eiθ1 , . . . , rn eiθn )|p dθ , (2π)n [0,2π)n r = (r1 , . . . , rn ), 0 ≤ rj < 1, j ∈ {1, . . . , n}, 0 < p < ∞, θ = (θ1 , . . . , θn ), dθ = dθ1 · · · dθn and M∞ (f, r) =

sup θ∈[0,2π)n

|f (r1 eiθ1 , . . . , rn eiθn )|.

Let ω(x), 0 ≤ x < 1, be a weight function which is positive and integrable on (0,1). We extend ω on U by setting ω(z) = ω(|z|), and also on U n by ω  = (ω1 , . . . , ωn ).

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p,q n Let Lp,q ω  = Lω  (U ), 0 < p ≤ ∞, 0 < q < ∞, denote the mixed norm space, the class of all measurable functions defined on U n such that  n  f qp,q,ω = Mpq (f, r) ωj (rj )drj < ∞, (0,1)n

Ap,q ω 

j=1

n Ap,q ω  (U )

p,q = be the intersection of Lω and H(U n ). When p = q we and  p,p p come to weighted Bergman spaces Aω = Aω with general weights ω  . Mixed norm, weighted Bergman and closely related spaces have been studied, for example, in [1, 2, 3, 4, 6, 7, 8, 10, 11, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. Following [12], for a given weight ω on U , define the distortion function of ω by  1 1 ψ(r) = ψω (r) = ω(t)dt, 0 ≤ r < 1. ω(r) r We put ψ(z) = ψ(|z|) for z ∈ U . Also, a class of admissible weights, a large class of weight functions ω in U is defined in [12]. For a list of examples of admissible weights, see [12, pp. 660-663]. In [20, Theorem 1] the second author, among others, proved the following result.

Theorem A. Let f ∈ H(U n ) and ωj (zj ), j = 1, . . . , n are admissible weights on the unit disc U , with distortion functions ψj (zj ). If 0 < p, q < ∞, and p,q ∂f f ∈ Ap,q ω  , then for all j = 1, . . . , n, ψj (zj ) ∂zj (z) ∈ Lω  , and there is a positive constant C = C(p, q, ω  , n) such that  n    ∂f    . (1.1) f p,q,ω ≥ C|f (0)| + C ψj ∂zj  p,q, ω j=1 For 1 ≤ p, q < ∞ the reverse inequality holds as well. Remark 1. For all 0 < p, q < ∞ the equivalence between the left-hand and right-hand sides of (1.1) is established in [15, 20] for standard weights ωj (zj ) = (1 − |zj |)αj , αj > −1. See also [13] and [14]. In [9] the authors solved an open problem posed by S. Stevi´c ([13, 14]) regarding the reverse inequality in (1.1) for the case of the unit disk, by proving the following result: Theorem B. Assume 0 < p ≤ ∞, 0 < q < ∞, and that ω is a differentiable weight function on U satisfying the following condition  ω  (r) 1 ω(s)ds ≤ L < ∞, r ∈ (0, 1), (1.2) ω 2 (r) r for a positive constant L. Then   1 q q Mp (f, r)ω(r)dr  |f (0)| + 0

for all f ∈ H(U ).

0

1

Mpq (f  , r)(ψω (r))q ω(r)dr

(1.3)

AVETISYAN,STEVIC:HOLOMORPHIC FUNCTIONS

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We write a  b if the ratio a/b is bounded from above and below by two positive constants when the variable varies, and say that a and b are comparable. Note that condition (1.2) is weaker than that of admissible weights, see [9]. An interesting problem is to extend Theorem B to the polydisc case. This will be done by proving the next theorem. Theorem 1. Let f ∈ H(U n ), 0 < p ≤ ∞, 0 < q < ∞, and the weights ωj (zj ), j = 1, . . . , n, satisfy condition (1.2), with distortion functions ψj (zj ), j = ∂f if and only if ψj (zj ) ∂z (z) ∈ Lp,q for all j = 1, . . . , n. 1, . . . , n. Then f ∈ Ap,q ω  ω  j Moreover,  n    ∂f    . (1.4) f p,q,ω  |f (0)| + ψj ∂zj  p,q, ω j=1 Theorem 1 generalizes both Theorems A and B. In Section 2 we present several auxiliary results which will be used in the proofs of the main results of this paper. A proof of Theorem 1 is given in Section 3. In Section 4 we turn to pluriharmonic functions in U n , that is, the real parts of holomorphic functions. As an application of Theorem 1, we prove that the operator of pluriharmonic p,q n conjugation is bounded in mixed norm spaces Lω  (U ) for all 0 < p ≤ ∞, 0 < q < ∞.

2

Auxiliary results

In this section we collect and prove several auxiliary lemmas which we use in the proof of the main result. Throughout the paper, the letters C(p, q, α, β, . . . ), Cα etc. stand for positive constants depending only on the parameters indicated and which may vary from line to line. Lemma 1. ([9]) Let {Ak }∞ k=0 be a sequence of complex numbers, α, γ > 0. Then the quantities Q1 =

∞ 

e−kα |Ak |γ ,

Q2 = |A0 |γ +

k=0

∞ 

e−kα |Ak+1 − Ak |γ

k=0

are comparable. Lemma 2. ([9]) Given a function ϕ on [0, 1) define the sequence {rk }∞ k=0 ⊂ [0, 1) by ϕ(rk ) = ek , k ≥ 0. (a) If the function ϕ satisfies ϕ(0) = 1 and ϕ (r)ϕ(r) ≤ M < ∞, ϕ (r)2 0 0, c > 0, β ∈ R, γ ∈ R, 1−r (1 − r)α is a typical weight function satisfying (2.2), see [9]. Pluriharmonic conjugation makes it possible to extend Theorem 1 to pluriharmonic functions. The partial differential operators ∂z∂ j and ∂z∂ j are defined by   ∂ ∂ ∂ 1 ∂ 1 ∂ ∂ , , zj = xj + iyj . = −i = +i ∂zj 2 ∂xj ∂yj ∂z j 2 ∂xj ∂yj Theorem 4. Let u ∈ P h(U n ) and one of the following two conditions holds: (a) 1 ≤ p ≤ ∞, 0 < q < ∞, and the weights ωj (zj ), j = 1, . . . , n, satisfy condition (3.1), with distortion functions ψj (zj ), j = 1, . . . , n. (b) 0 < p ≤ ∞, 0 < q < ∞, and the weight functions ωj (zj ), j = 1, . . . , n, together with their corresponding functions ϕj = ϕωj defined by (3.2), satisfy (2.2). Then   n  n     ∂u   ∂u      up,q,ω  |u(0)| +  |u(0)| + . (4.7) ψj ∂zj  ψj ∂z j  p,q, ω p,q, ω j=1 j=1 Proof. Since the function u is real–valued, the second equivalence in (4.7) is obvious. Let now f ∈ H(U n ), f = u + iv, and v be the pluriharmonic conjugate of u normalized so that v(0) = 0. Then by Theorems 1-3 and Cauchy-Riemann equations   n  n     ∂u   ∂f      |u(0)| + = |f (0)| + C  f p,q,ω  up,q,ω , ψj ∂zj  ψj ∂zj  p,q, ω p,q, ω j=1 j=1 as desired. Remark 2. It is not difficult to see that Theorem B holds for the case of holomorphic functions on the unit ball B ⊂ Cn , where ∇f appears instead of f  in (1.3). Note that by the maximal theorem the inequality in Lemma 3 becomes Mp (f, ρ) − Mp (f, r) ≤ C(ρ − r) Mp (∇f, ρ), 0 < r < ρ < 1, f ∈ H(B), where = min{1, p}, p ∈ (0, ∞].

AVETISYAN,STEVIC:HOLOMORPHIC FUNCTIONS

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