Some new classes of Hardy spaces
Yong-Zhuo Chen1989, Journal of Functional Analysis
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Abstract
Let BP= {f:))fj) =sup,,, (1/2T)~~,lflP)*'P< co), 1 <p< co. Then BP is the dual of a function algebra Aq on R (Beurling). In this paper, we study the harmonic extensions offin BP and in A", and the corresponding Hardy spaces H,,, HA*. It is shown that a parallel theory for L", L' and BMO, H' can be developed for the above pairs. In particular we prove that for 1 < q < 2, (HAM)* is isomorphic to the Banach space where mTf= (1/2T) I',! We also prove Burkholder, Gundy, and Silverstein's maximal function characterization for the new Hardy space HA', 1 < 4 < 2. 0 1989 Academic Press, Inc.
Key takeaways
- In Section 3, we prove some elementary theorems of harmonic extensions for functions in BP and A p, and define the corresponding Hardy spaces.
- The case AP follows from Theorem 3.4 and 3.5.
- We will prove the theorem in two parts.
- For 2 bp < co, BP/H,, z CMOp,.
- It follows from the open mapping theorem that BP/HBp is isomorphic to CMOP,.
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