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