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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 ${ℝ}^{n}$, where $V$ is a positive potential in ${B}_{q}$ for some $q>n/2$: ${T}_{1}=V{L}^{-1}$, ${T}_{2}={V}^{1/2}{L}^{-1/2}$, ${T}_{3}=\nabla {L}^{-1/2}$.

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 $b\in BMO\left({ℝ}^{n}\right)$, the commutators $\left[b,{T}_{i}\right]$ $\left(i=1,2,3\right)$ are bounded on ${L}^{p}\left({ℝ}^{n}\right)$ 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 $f\in {H}^{1}\left({ℝ}^{n}\right)$, $\left[b,{T}_{i}\right]f$ may not be in ${L}^{1}\left({ℝ}^{n}\right)$. 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 $V\in {B}_{q}$, $q>n/2$, and showed that, if $f\in {H}_{L}^{1}\left({ℝ}^{n}\right)$, then ${T}_{3}f\in {L}^{1}\left({ℝ}^{n}\right)$.

In this paper, the authors discuss the ${H}_{L}^{1}$-boundedness of $\left[b,{T}_{i}\right]$ $\left(i=1,2,3\right)$. In particular, it is shown that for $b\in BMO\left({ℝ}^{n}\right)$, the commutator $\left[b,{T}_{3}\right]$ is not bounded from ${H}_{L}^{1}\left({ℝ}^{n}\right)$ to ${L}^{1}\left({ℝ}^{n}\right)$ and instead, $\left[b,{T}_{i}\right]$ $\left(i=1,2,3\right)$ are of $\left({H}_{L}^{1},{L}_{\text{weak}}^{1}\right)$-boundedness.

##### MSC:
 47B32 Operators in reproducing-kernel Hilbert spaces 47A75 Eigenvalue problems (linear operators) 42C40 Wavelets and other special systems 94A40 Channel models (including quantum)