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On weak convergence in \(H^ 1(\mathbf R^ d)\). (English) Zbl 0814.42011
Let \(H^ 1(\mathbb{R}^ n)\) be the Hardy space of functions on \(\mathbb{R}^ n\) whose Poisson maximal function lies in \(L^ 1\), \(\text{BMO}(\mathbb{R}^ n)\) be the dual space of functions of bounded mean oscillation, and let \(\text{VMO}(\mathbb{R}^ n)\) be the closure of the Schwartz space in \(\text{BMO}(\mathbb{R}^ n)\), the predual of \(H^ 1(\mathbb{R}^ n)\). Then the authors show the following: Suppose \(\{f_ j\}\) is a sequence of \(H^ 1(\mathbb{R}^ n)\) functions such that \(| f_ j|_{H^ 1}\leq 1\) for all \(j\) and \(f_ j(x)\to f(x)\) for a.e. \(x\in \mathbb{R}^ n\). Then \(f\in H^ 1(\mathbb{R}^ n)\), \(| f|_{H^ 1}\leq 1\), and \(\int f_ n\varphi dx\to \int f\varphi dx\) for all \(\varphi\in \text{VMO}(\mathbb{R}^ n)\).
They also note that this theorem is valid for martingale \(H^ 1\).
Reviewer: K.Yabuta (Nara)

42B30 \(H^p\)-spaces
30D55 \(H^p\)-classes (MSC2000)
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