Some new classes of Hardy spaces (Q1123290)

The authors develop a Hardy space theory for certain function spaces, among them the spaces \[ B^ p=\{f\in L^ 1_{loc}({\mathbb{R}}):\quad \| f\| =\sup_{T\geq 1}[(2T)^{-1}\int^{T}_{-T}| f(t)|^ p dt]^{1/p}<\infty \}, \] 1\(<p<\infty\) (formerly considered by \textit{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 \(B^ p\) above. Also considered are duality relations; for example, an analogue to the Fefferman-Stein theorem [\textit{C. Fefferman} and \textit{E. M. Stein}, Acta Math. 129, 137-193 (1972; Zbl 0257.46078)] on the duality between the classical Hardy space \(H^ 1\) and BMO is proved.