Continuity properties for the maximal operator associated with the commutator of the Bochner-Riesz operator
In this paper, the authors discuss the maximal commutator \(T_\lambda^*\) between a BMO function \(b\) and the Bochner-Riesz operator \(T_\lambda^r\). The \(L^p\)-boundedness of the maximal commutator \(T_\lambda^*\) was obtained by \textit{G. Hu} and \textit{S. Lu} [ Tôhoku Math. J., II. Ser. 48, No.2, 259-266 (1996; Zbl 0889.42011)] for \(1<p<\infty\) and \(\lambda>(n-1)| 1/p-1/2| \). In this paper, the author establishes the weak type \((H^1, L^1)\) estimate for \(T_\lambda^*\) for \(\lambda>(n-1)/2\), and the \((H_b^p, L^p)\) type estimate for \(0<p\leq 1\) and \(\lambda>n/p-(n+1)/2\). Here \(H^1\) is the usual Hardy space, and \(H_b^p\) is defined similarly to the usual Hardy space \(H^p\), except that the atoms satisfy a weighted vanishing moment condition with `weight' \(b\).
scientific article; zbMATH DE number 2057937