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Endpoint estimates for commutators of Riesz transforms associated with Schrödinger operators. (English) Zbl 1210.47048

Consider the following Riesz transforms T 1 , T 2 , and T 3 associated with the Schrödinger operator L=-+V on n , where V is a positive potential in B q for some q>n/2: T 1 =VL -1 , T 2 =V 1/2 L -1/2 , T 3 =L -1/2 .

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 bBMO( n ), the commutators [b,T i ] (i=1,2,3) are bounded on L p ( n ) for p>1. For the case of p=1, 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 fH 1 ( n ), [b,T i ]f may not be in L 1 ( n ). However, J. Dziubański and J. Zienkiewicz [Rev. Mat. Iberoam. 15, No. 2, 279–296 (1999; Zbl 0959.47028)] studied the Hardy space H L 1 associated with the Schrödinger operator L for VB q , q>n/2, and showed that, if fH L 1 ( n ), then T 3 fL 1 ( n ).

In this paper, the authors discuss the H L 1 -boundedness of [b,T i ] (i=1,2,3). In particular, it is shown that for bBMO( n ), the commutator [b,T 3 ] is not bounded from H L 1 ( n ) to L 1 ( n ) and instead, [b,T i ] (i=1,2,3) are of (H L 1 ,L weak 1 )-boundedness.

MSC:
47B32Operators in reproducing-kernel Hilbert spaces
47A75Eigenvalue problems (linear operators)
42C40Wavelets and other special systems
94A40Channel models (including quantum)