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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 $L=-\triangle+V$ on $\Bbb R^n$, where $V$ is a positive potential in $B_q$ for some $q>n/2$: $T_1=V L^{-1}$, $T_2=V^{1/2}L^{-1/2}$, $T_3=\nabla L^{-1/2}$. {\it Z.-H.\thinspace Guo, P.-T.\thinspace Li} and {\it L.-Z.\thinspace Peng} [J. Math. Anal. Appl. 341, No. 1, 421--432 (2008; Zbl 1140.47035)] have shown that for $b \in BMO(\Bbb R^n)$, the commutators $[b, T_i]$ $(i=1,2,3)$ are bounded on $L^p(\Bbb R^n)$ for $p>1$. For the case of $p=1$, {\it E. Harboure, C. Segovia} and {\it J. L.\thinspace Torrea} [Ill. J. Math. 41, No. 4, 676--700 (1997; Zbl 0892.42009)] proved that even if we restrict $f \in H^1(\Bbb R^n)$, $[b,T_i]f$ may not be in $L^1(\Bbb R^n)$. However, {\it J. Dziubański} and {\it 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 $V \in B_q$, $q>n/2$, and showed that, if $f \in H_L^1(\Bbb R^n)$, then $T_3 f \in L^1(\Bbb R^n)$. 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 $b \in BMO(\Bbb R^n)$, the commutator $[b,T_3]$ is not bounded from $H_L^1(\Bbb R^n) $ to $L^1(\Bbb R^n)$ and instead, $[b, T_i]$ $(i=1,2,3)$ are of $(H_L^1, L_{\text{weak}}^1)$-boundedness.

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