Boundedness of Hausdorff operators on the power weighted Hardy spaces

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...

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Bibliographic Details
Published in:Applied Mathematics-A Journal of Chinese Universities 2017-09, Vol.32 (4), p.462-476
Main Authors: Chen, Jie-cheng, He, Shao-yong, Zhu, Xiang-rong
Format: Article
Language:English
Subjects:
Applications of Mathematics
Mathematics
Mathematics and Statistics
Operators
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Summary: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.
ISSN:1005-1031
1993-0445
DOI:10.1007/s11766-017-3523-3