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)