Boundedness of Hausdorff operators on the power weighted Hardy spaces

• Published in: Applied Mathematics-A Journal of Chinese Universities Vol. 32; no. 4; pp. 462 - 476
• Main Authors: Chen, Jie-cheng, He, Shao-yong, Zhu, Xiang-rong
• Format: Journal Article
• Language: English
• Published: Hangzhou Editorial Committee of Applied Mathematics - A Journal of Chinese Universities 01-09-2017; Springer Nature B.V
• Subjects: Applications of Mathematics; Mathematics; Mathematics and Statistics; Operators; power weight; 42B35; 47B38; Hausdorff operator; Hardy space
• ISSN: 1005-1031; 1993-0445
• DOI: 10.1007/s11766-017-3523-3

In this paper, we consider the two-dimensional Hausdorff operators on the power weighted Hardy space
H
|
X
|
α
1
(
R
2
)
(
−
1
≤
α
≤
0
)
, defined by
H
Φ
,
A
f
(
x
)
=
∫
R
2
Φ
(
u
)
f
(
A
(
u
)
x
)
d
u
,
, where Φ ∈ L
loc
1
(
R
2
),
A
(
u
) = (
a
ij
(
u
))
i,j
=1
2
is a 2 × 2 matrix, and each
a
i,j
is a measurable function. We obtain that
H
Φ,
A
is bounded from
H
|
X
|
α
1
(
R
2
)
(
−
1
≤
α
≤
0
)
to itself, if
∫
R
2
|
Φ
(
u
)
|
|
det
A
−
1
(
u
)
|
‖
A
(
u
)
‖
−
α
ln
(
1
+
‖
A
−
1
(
u
)
‖
2
|
det
A
−
1
(
u
)
|
)
d
u
<
∞
.
. This result improves some known theorems, and in some sense it is sharp.