Continuity of Hardy-Littlewood maximal function
Denote by \(Tf\) the uncentered Hardy-Littlewood maximal function \(Mf\) and the centered Hardy-Littlewood maximal function \(M_{c}f\) on \(\mathbb{R}^{n}\). In the paper under review, the authors obtain the necessary and sufficient condition for the continuity of \(Tf\) at a point. That is,\\ Theorem. Let \(f \in L^{1}_{\text{loc}}(\mathbb{R}^{n})\) and \(Tf(a) < \infty\). Then \(Tf(x)\) is continuous at the point \(a \in \mathbb{R}^{n}\) if and only if there exists \(\widetilde{f}\) such that \(\widetilde{f}(x) = f(x)\ \text{a.e.}\ x\in \mathbb{R}^{n}\), and \(\lim\sup_{x\to a}\vert\widetilde{f}(x)\vert\leq Tf(a)\). As one application, the authors obtain that \(Mf\) is continuous at \(a\in \mathbb{R}^{1}\) if \(f\in L^{1}_{\text{loc}}(\mathbb{R}^{1})\cap BV_{\text{loc}}(a)\), where \(BV_{\text{loc}}(a)\) is the space of functions of locally bounded variation. They also give a counterexample to show that this result is not true for \(n \geq 2\).
scientific article; zbMATH DE number 7347399