Volterra-Type Operators on Zygmund Spaces
- PDF / 128,141 Bytes
- 11 Pages / 467.717 x 680.315 pts Page_size
- 51 Downloads / 188 Views
Research Article Volterra-Type Operators on Zygmund Spaces Songxiao Li and Stevo Stevi´c Received 26 November 2006; Accepted 4 March 2007 Recommended by Robert Gilbert
The boundedness and the compactness of the two integral operators Jg f (z) = z z f (ξ)g (ξ)dξ; Ig f (z) = 0 f (ξ)g(ξ)dξ, where g is an analytic function on the open unit 0 disk in the complex plane, on the Zygmund space are studied. Copyright © 2007 S. Li and S. Stevi´c. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Let D denote the unit disk in the complex plane C and ∂D its boundary. Denote by H(D) the class of all analytic functions on D. Let ᐆ denote the space of all f ∈ H(D) ∩ C(D) such that f ᐆ = sup
i(θ+h) f e + f ei(θ−h) − 2 f eiθ
h
< ∞,
(1.1)
where the supremum is taken over all eiθ ∈ ∂D and h > 0. By a Zygmund theorem (see [1, Theorem 5.3]) and the closed graph theorem, we have that f ∈ ᐆ if and only if sup 1 − |z|2 f (z) < ∞,
(1.2)
z∈D
moreover the following asymptotic relation holds: f ᐆ sup 1 − |z|2 f (z).
(1.3)
z∈D
Therefore, ᐆ is called Zygmund class. Since the quantities in (1.3) are semi norms (they do not distinguish between functions differing by a linear polynomial), it is natural to add them to the quantity | f (0)| + | f (0)| to obtain two equivalent norms on the Zygmund
2
Journal of Inequalities and Applications
class of functions. Zygmund class with such defined norm will be called Zygmud space. This norm will be again denoted by · ᐆ . By (1.3), we have f (z) − f (0) ≤ C f ᐆ ln
1 . 1 − |z|
(1.4)
Also, we have f (z) − f (0) − z f (0) z 1 z 1 |ζ |dt f (tζ)ζ dt dζ |dζ | = ≤ f ᐆ 1 − t | ζ | 0 0 0 0 |z| 1 = f ᐆ |z| + |z| − 1 ln 1 ≤ f ᐆ ln , ds 1−s 1 − |z| 0
(1.5)
for every z ∈ D. From this and since the quantity
sup x∈[0,1)
x + (x − 1)ln
1
(1.6)
1−x
is bounded, it follows that f ∞ ≤ C f ᐆ ,
(1.7)
for every f ∈ ᐆ, and for some positive constant C independent of f . We introduce the little Zygmund space ᐆ0 in the following natural way: f ∈ ᐆ0 ⇐⇒ lim 1 − |z| f (z) = 0.
(1.8)
|z|→1
It is easy to see that ᐆ0 is a closed subspace of ᐆ. Suppose that g : D → C is a holomorphic map, f ∈ H(D). The integral operator, called Volterra-type operator, Jg f (z) =
z
f dg =
0
1 0
f (tz)zg (tz)dt =
z 0
f (ξ)g (ξ)dξ,
z ∈ D,
(1.9)
was introduced by Pommerenke in [2]. Another natural integral operator is defined as follows: Ig f (z) =
z 0
f (ξ)g(ξ)dξ.
(1.10)
The importance of the operators Jg and Ig comes from the fact that Jg f + Ig f = Mg f − f (0)g(0),
(1.11)
where the multiplication operator Mg is defined by
Mg f (z) = g(z) f (z),
f ∈ H(D), z ∈ D.
(1.12)
S. Li and S. Stevi´c 3 In [2] Pommerenke showed that Jg is a bounde
Data Loading...
