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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)

##### MSC:
 42B30 $$H^p$$-spaces 30D55 $$H^p$$-classes (MSC2000)
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