On the weak\(*\) convergence in \(H_{\mathcal{F}}^1(\mathbb{R}^n)\)
The main result of this paper consists of proving the \(\mu\)-almost everywhere convergence of a sequence of functions in the Hardy space \(H^1_{\mathcal F}\) defined by \textit{Y. Ding} and \textit{C.-C. Lin} [Tohoku Math. J. (2) 57, No. 2, 147--170 (2005; Zbl 1085.42014)] as the predual of \(\mathrm{BMO}_{\mathcal F}\), that is, the space of bounded mean oscillations associated to a family of bounded convex sets \({\mathcal F}=\{S(x,t) \subset \mathbb{R}^N: x\in \mathbb{R}^N, \;t>0\} \) and a Borel measure \(\mu\) satisfying a doubling property with respect the parameter \(t>0\). For this purpose, the predual \(\mathrm{VMO}_{\mathcal F}(\mathbb{R}^N)\) of \(H^1_{\mathcal F}\) is previously identified, which is the closure of \(C_c(\mathbb{R}^N)\cap \mathrm{Lip}(\mathbb{R}^N)\) with respect the seminorm \(\| \cdot \|_{\mathrm{BMO}_{\mathcal F}}\).
scientific article; zbMATH DE number 6756246