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.
Highlights
Let ∆be the forward difference operator defined on sequences { } μ ∞ n n=0 by∆μn = μn − μn+1
Hausdorff matrix and Hausdorff operator have studied on various space of holomorphic functions, see, e.g
In [3], the author obtained that the Hausdorff operator μ is bounded on Hardy space H p (1 ≤ p < ∞), and in [4] we showed that this conclusion cannot be extended to the Bloch space directly
Summary
How to cite this paper: Hu, R. and Zhang, C.F. (2014) β-Hausdorff Operator on Lipschitz Space in the Unit Polydisk. ( ) Let U n be the unit polydisk in the complex vector space n , H U n be the space of all holomorphic ( ) functions on U n , and μi ,i = 1, , n be the Borel measures on (0,1) , μ. Dμ (t ) = Πnj=1dμ j tj ( ) In [2], the Lipschitz space Lipα U n (0 ≤ α < 1) is defined on U n by
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