﻿ <i>β</i>-Hausdorff Operator on Lipschitz Space in the Unit Polydisk

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

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 byLemma 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

1. Garabedian, H.L. (1939) Hausdorff Matrices. The American Mathematical Monthly, 46, 390-410. http://dx.doi.org/10.2307/2303033
2. Zhou, Z.H. (2003) Composition Operators on the Lipschitz Space in Polydiscs. Science in China, 46, 33-38. http://dx.doi.org/10.1360/03ys9004
3. Chang, D.C., Gilbert, R. and Stević, S. (2006) Hausdorff Operator on the Unit Polydisk in. Complex Variables and Elliptic Equations, 4, 329-345. http://dx.doi.org/10.1080/17476930600610528
4. Hu R., Xie L. and Hu, P.Y. (2012) Hausdorff-Type Operator on Bloch Space in the Unit Polydisk in. Acta Mathematica Scientia, 32, 521-529.
5. Timoney, R.M. (1980) Blochfunctions in Several Complex Variables. Bull London Math, 319, 1-22.
6. Zhou, Z.H. and Wei, Z.Q. (2005) Weighted Composition Operators on the Bloch Space in Polydiscs. Journal of Mathematics, 25, 435-440
7. Galanopoulos, P. and Siskakis, A. (2001) Hausdorff Matrices and Composition Operators. Illinois Hournal of Math, 45, 757-773.
8. Liflyand, E. and Móricz, F. (2000) The Hausdorff Operator Is Bounded on the Real Hardy Space. Proceedings of the American Mathematical Society, 128, 1391-1396. http://dx.doi.org/10.1090/S0002-9939-99-05159-X

NOTES

*This work is Supported by the Sichuan Provincial Natural Science Foundation (13ZB0101,13ZB0102).

#Corresponding author.