Weighted Composition Operators between Mixed Norm Spaces and Spaces in the Unit Ball

Journal of Inequalities and Applications, Jan 2008

Let be an analytic self-map and let be a fixed analytic function on the open unit ball in . The boundedness and compactness of the weighted composition operator between mixed norm spaces and are studied.

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Weighted Composition Operators between Mixed Norm Spaces and Spaces in the Unit Ball

Journal of Inequalities and Applications Hindawi Publishing Corporation Weighted Composition Operators between Mixed Norm Spaces and Hα∞ Spaces in the Unit Ball Stevo Stevic´ 0 0 Stevo Stevic ́: Mathematical Institute of the Serbian Academy of Sciences and Arts , Knez Mihailova 36, 11000 Beograd , Serbia Let ϕ be an analytic self-map and let u be a fixed analytic function on the open unit ball B in Cn. The boundedness and compactness of the weighted composition operator uCϕ f = u · ( f ◦ ϕ) between mixed norm spaces and Hα∞ are studied. 1. Introduction Let B be the open unit ball in Cn, ∂B = S its boundary, D the unit disk in C, dV the normalized Lebesgue volume measure on B, dσ the normalized surface measure on S, and H(B) the class of all functions analytic on B. An analytic self-map ϕ : B→B induces the composition operator Cϕ on H(B), defined by Cϕ( f )(z) = f (ϕ(z)) for f ∈ H(B). It is interesting to provide a functional theoretic characterization of when ϕ induces a bounded or compact composition operator on various spaces. The book [1] contains a plenty of information on this topic. Let u be a fixed analytic function on the open unit ball. Define a linear operator uCϕ, called a weighted composition operator, by uCϕ f = u·( f ◦ ϕ), where f is an analytic function on B. We can regard this operator as a generalization of the multiplication operator Mu( f ) = u f and a composition operator. A positive continuous function φ on [0, 1) is called normal if there exist numbers s and t, 0 < s < t, such that φ(r)/(1 − r)s decreasingly converges to zero and φ(r)/(1 − r)t increasingly tends to ∞, as r→1− (see, e.g., [2]). For 0 < p < ∞, 0 < q < ∞, and a normal function φ, let H(p, q, φ) denote the space of all f ∈ H(B) such that 1 − r < ∞, where Mq( f , r) = ( S | f (rζ)|qdσ(ζ))1/q, 0 ≤ r < 1. For 1 ≤ p < ∞, H(p, q, φ), equipped with the norm · H(p,q,φ), is a Banach space. When 0 < p < 1, f H(p,q,φ) is a quasinorm on H(p, q, φ), and H(p, q, φ) is a Frechet space but not a Banach space. Note that if 0 < p = q < ∞, then H(p, p, φ) becomes a Bergman-type space, and if φ(r) = (1 − r)(γ+1)/p, γ > −1, then H(p, p, φ) is equivalent to the classical weighted Bergman space Aγp(B). For α ≥ 0, we define the weighted space Hα∞(B) = Hα∞ as the subspace of H(B) consisting of all f such that f Hα∞ = supz∈B(1 − |z|2)α| f (z)| < ∞. Note that for α = 0, Hα∞ becomes the space of all bounded analytic functions H∞(B). We also define a little version of Hα∞, denoted by Hα∞,0(B), as the subset of Hα∞ consisting of all f ∈ H(B) such that lim|z|→1−0(1 − |z|2)α| f (z)| = 0. It is easy to see that Hα∞,0 is a subspace of Hα∞. Note also that for α = 0, in view of the maximum modulus theorem, we obtain H0∞,0 = { } 0 . For the case of the unit disk, in [3], Ohno has characterized the boundedness and compactness of weighted composition operators between H∞ and the Bloch space and the little Bloch space 0. In [4], Li and Stevic´ extend the main results in [3] in the settings of the unit ball. In [5], A. K. Sharma and S. D. Sharma studied the boundedness and compactness of uCϕ : Hα∞(D)→Aγp(D) for the case of p ≥ 1. For related results in the setting of the unit ball, see, for example, [1, 6, 7] and the references therein. Here, we study the weighted composition operators between the mixed norm spaces H(p, q, φ) and Hα∞ (or Hα∞,0). As corollaries, we obtain the complete characterizations of the boundedness and compactness of composition operators between Bergman spaces and H∞. In this paper, positive constants are denoted by C; they may differ from one occurrence to the next. The notation a b means that there is a positive constant C such that a ≤ Cb. If both a b and b a hold, then one says that a b. 2. Auxiliary results In this section, we give some auxiliary results which will be used in proving the main results of the paper. They are incorporated in the lemmas which follow. Lemma 2.1. Assume that f ∈ H(p, q, φ)(B). Then there is a positive constant C independent of f such that f (z) ≤ C Proof. By the monotonicity of the integral means, the following asymptotic relations: 1 − r w ∈ B z, 3 1 − |z| /4), 1 − |z|, r ∈ and [8, Theorem 7.2.5], we have dr ≥ Mqp f , (1 + |z| /2 (1+|z|)/2 1 − r from which the result follows. Proof. It can be proved in a standard way (see, e.g., [9, Theorem 2]) that f (z) = 0. where fr (z) = f (rz), r ∈ (0, 1). Also since f ∈ H(p, q, φ), by the monotonicity of the integral means, we have fr ∈ H(p, q, φ), for every r ∈ (0, 1). From this and by inequality (2.1), we have that for each r ∈ (0, 1), f (z) ≤ fr (z) 1 − |z| n/qφ |z| + C f − fr H(p,q,φ). f − fr H(p,q,φ) < ε, r ∈ r0, 1 . The following criterion for compactness is followed by standard arguments. Lemma 2.4. The operator uCϕ : H(p, q, φ)→Hα∞(or Hα∞→H(p, q, φ)) is compact if and only if for any bounded sequence ( fk)k∈N in H(p, q, φ) (corresp. Hα∞), which converges to zero uniformly on compact subsets of B as k→∞, one has uCϕ fk Hα∞ →0 as k→∞ (corr (...truncated)


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Stevo Stević. Weighted Composition Operators between Mixed Norm Spaces and Spaces in the Unit Ball, Journal of Inequalities and Applications, 2008, pp. 028629, 2007,