Consider the following Riesz transforms , , and associated with the Schrödinger operator on , where is a positive potential in for some : , , .
Z.-H. Guo, P.-T. Li and L.-Z. Peng [J. Math. Anal. Appl. 341, No. 1, 421–432 (2008; Zbl 1140.47035)] have shown that for , the commutators are bounded on for . For the case of , E. Harboure, C. Segovia and J. L. Torrea [Ill. J. Math. 41, No. 4, 676–700 (1997; Zbl 0892.42009)] proved that even if we restrict , may not be in . However, J. Dziubański and J. Zienkiewicz [Rev. Mat. Iberoam. 15, No. 2, 279–296 (1999; Zbl 0959.47028)] studied the Hardy space associated with the Schrödinger operator for , , and showed that, if , then .
In this paper, the authors discuss the -boundedness of . In particular, it is shown that for , the commutator is not bounded from to and instead, are of -boundedness.