Advances in Pure Mathematics
Vol.4 No.5(2014), Article ID:45618,5 pages DOI:10.4236/apm.2014.45022
β-Hausdorff Operator on Lipschitz Space in the Unit Polydisk*
Rong Hu, Chaofeng Zhang#
School of Mathematics and Finance-Economics, Sichuan University of Arts and Science, Dazhou, China
Email: #zbeyondiee@163.com
Copyright © 2014 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativecommons.org/licenses/by/4.0/
Received 27 February 2014; revised 27 March 2014; accepted 2 April 2014
ABSTRACT
In this paper, we define β-Hausdorff operator on the unit polydisk and study the boundedness of the operator on Lipschitz space. Firstly, we translate the problem of coefficient into integral of weighted composition operator, then give the sufficient conditions of boundedness, and also obtain an upper bound for the operator norm on Lipschitz space.
Keywords:Unit Polydisk, Lipschitz Space, β-Hausdorff Operator, Weighted Composition Operator
1. Introduction
Let be the forward difference operator defined on sequences
by
. Let operator
be
Define the -Hausdorff matrix
as the lower triangular matrix
with entries
For, it is the Hausdorff matrix
, see [1] .
When is the moment sequence of a measure i.e.
, the matrix arising from a Borel measure
is denoted by
, a simple calculation then gives
Let be the unit polydisk in the complex vector space
,
be the space of all holomorphic functions on
, and
be the Borel measures on
,
,
.
In [2] , the Lipschitz space is defined on
by
where. It is easy to prove that
is a Banach space under the norm
.
Let, suppose
, and
be the
- Hausdorff matrices arising by Borel measures
. The
-Hausdorff operator
is defined as follows:
. For
, we obtain the classical Hausdorff operator
, see [3] .
Hausdorff matrix and Hausdorff operator have studied on various space of holomorphic functions, see, e.g., [3] -[9] . In [3] , the author obtained that the Hausdorff operator is bounded on Hardy space
, and in [4] we showed that this conclusion cannot be extended to the Bloch space directly. Then we try to study on the Lipschitz space, found that when the measure is common Lebesgue measure
, the Hausdorff operator
is unbounded on Lipschitz space
, see the remark. In this paper, we study the operator which is got by amending the Hausdorff operator and called it
-Hausdorff operator. The results of this paper can be deemed as a continuation of the results in [3] on Lipschitz space.
2. Main Results
The main results in this paper is the following:
Theorem 1 Let be finite Borel measures on (0,1) and
be corresponding
-Hausdorff matrices,
be
-Hausdorff operator. For
,
is bounded on
if
In this case, the operator norm satisfies
for some constant.
In order to prove the main results, we need some auxiliary result.
Lemma 1 [2] Let,then
.
For each, we note the functions
given by
Lemma 2 Let
be finite Borel measures on
and
be corresponding
-Hausdorff matrices. Suppose
Then(a) The power series in (2) represents a holomorphic functions on
;
(b) can be written in terms of weighted composition operators as follows:
. For each
.
Proof (a) Let. Since
the sequence of Taylor coefficients of
is bounded by a constant
, then
Hence the coefficients of the series (2) are bounded and consequently is defined and analytic on
.
(b) By the Schwarz lemma we have for each
. Hence applying (3) we have
On the other hand,
Hence
is finite and analytic on.
Now we proof, in order to avoid tedious calculations, we may assume that
, For a fixed
we have
It easy to see that
Hence,
Denote as follows
where is defined in (4).
Now we obtain estimates for the norms of the weighted composition operator .
Lemma 3 Suppose, then
is bounded on
. Further more, there is a constant
such that. For each
.
Proof Let, in which
,
and the function
is defined in (4).
and. Hence we obtain that
Now we proof the main results.
The Proof of Theorem 1 For each, by (5) we can obtain
Then by (1) and (6),
from which the result follows.
Remark When the Borel measure is the common Lebesgue measure
, the Hausdorff operator arising from measure
is denoted as
.
is bounded on Hardy space
, see [3] .
However, it is unbounded on Lipschitz space. For example, fix, and let
, it is easy to see that
, then
From this it follows that
References
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. Acta Mathematica Scientia, 32, 521-529.
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NOTES
*This work is Supported by the Sichuan Provincial Natural Science Foundation (13ZB0101,13ZB0102).
#Corresponding author.