Weak type estimates for commutators on Herz-type spaces (Q1417169)

Let \([b, T]\) be the commutator generated by the Calderón-Zygmund operator \(T\) and a \(\text{ BMO}(\mathbb R^n)\) function \(b\). In this paper, the author proves that if \(1<q<\infty\) and \(0<p\leq 1\), then \([b, T]\) is bounded from the Herz-type Hardy space \(H\dot K^{n(1-1/q),p}_q(\mathbb R^n)\) to the weak Herz space \(\dot K^{n(1-1/q),p,\infty}_q(\mathbb R^n)\). The author proves that the above result is also true for the maximal operator associated with the commutator generated by the Bochner-Riesz operator with any \(\text{BMO}(\mathbb R^n)\) function. Two counterexamples are given, respectively, to indicate that the above result fails if \(p>1\) or if \(n(1-1/q)\) is replaced by \(-n/q\). The weak Herz space with a different notation was also introduced by \textit{G. Hu, S. Lu} and \textit{D. Yang} [Adv. Math., Beijing 26, No. 5, 417--428 (1997; Zbl 0915.42012); J. Beijing Norm. Univ., Nat. Sci. 33, No. 1, 27-34 (1997; Zbl 0887.42016)].