Commutators of singular integrals on \(H_b^p(R^n)\) at critical index (Q2747341)

Let \(T\) be a \(\delta\)-Calderón-Zygmund singular integral operator and \(b\) be a function in \(\text{BMO}(\mathbb{R}^n)\). Let \([b,T]= bT- Tb\) be the commutator. \textit{C. Pérez} [J. Funct. Anal. 128, No. 1, 163-185 (1995; Zbl 0831.42010)] provd that \([b,T]\) is a bounded operator from \(H^p_b(\mathbb{R}^n)\) to weak \(L^p(\mathbb{R}^n)\), where \(n/(n+\delta)< p\leq 1\), \(0<\delta\leq 1\).NEWLINENEWLINENEWLINEIs is conjectured that \([b,T]\) is a bounded operator from \(H^p_b(\mathbb{R}^n)\) to weak \(L^p(\mathbb{R}^n)\) for the critical value \(p= n/(n+\delta)\) \((0<\delta< 1)\). When \(b\) is a bounded function, the conjecture is true. In this paper, the author gives a counterexample in the case of \(n=1\).