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Interpolation of weighted and vector-valued Hardy spaces

Authors: Serguei V. Kisliakov and Quan Hua Xu
Journal: Trans. Amer. Math. Soc. 343 (1994), 1-34
MSC: Primary 46E15; Secondary 42B30, 46E40, 46M35
MathSciNet review: 1236225
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Abstract: Real and complex interpolation methods, when applied to the couple $ ({H^{{p_0}}}({E_0};{w_0}),{H^{{p_1}}}({E_1};{w_1}))$, give what is expected if $ {E_0}$ and $ {E_1}$ are quasi-Banach lattices of measurable functions satisfying certain mild conditions and if $ \log (w_0^{1/{p_0}}w_1^{ - 1/{p_1}}) \in {\text{BMO}}\,({w_0},{w_1}$ being weights on the unit circle). The last condition is in fact necessary. (It is expected, of course, that the resulting spaces coincide with the subspaces of analytic functions in the corresponding interpolation spaces for the couple $ ({L^{{p_0}}}({E_0};{w_0}),{L^{{p_1}}}({E_1};{w_1})).)$

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Keywords: Interpolation, Hardy space, Banach lattice, $ {A_p}$-weight, Hilbert transform, BMO
Article copyright: © Copyright 1994 American Mathematical Society

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