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Some new classes of Hardy spaces. (English) Zbl 0677.30030
The authors develop a Hardy space theory for certain function spaces, among them the spaces $$ B\sp p=\{f\in L\sp 1\sb{loc}({\bbfR}):\quad \Vert f\Vert =\sup\sb{T\ge 1}[(2T)\sp{-1}\int\sp{T}\sb{-T}\vert f(t)\vert\sp p dt]\sp{1/p}<\infty \}, $$ 1$<p<\infty$ (formerly considered by {\it A. Beurling} [Ann. Inst. Fourier 14, No.2, 1-32 (1964; Zbl 0133.075)]), and the harmonic extension of their elements to the upper half-plane. Their results include a Burkholder-Gundy-Silverstein maximal function characterization of spaces related to the spaces $B\sp p$ above. Also considered are duality relations; for example, an analogue to the Fefferman-Stein theorem [{\it C. Fefferman} and {\it E. M. Stein}, Acta Math. 129, 137-193 (1972; Zbl 0257.46078)] on the duality between the classical Hardy space $H\sp 1$ and BMO is proved.
Reviewer: R.Mortini

30H05Bounded analytic functions
30D55H (sup p)-classes (MSC2000)
46J15Banach algebras of differentiable or analytic functions, $H^p$-spaces
Full Text: DOI
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