| Title:
|
Some estimates for commutators of Riesz transform associated with Schrödinger type operators (English) |
| Author:
|
Liu, Yu |
| Author:
|
Zhang, Jing |
| Author:
|
Sheng, Jie-Lai |
| Author:
|
Wang, Li-Juan |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
66 |
| Issue:
|
1 |
| Year:
|
2016 |
| Pages:
|
169-191 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $\mathcal {L}_1=-\Delta +V$ be a Schrödinger operator and let $\mathcal {L}_2=(-\Delta )^2+V^2$ be a Schrödinger type operator on ${\mathbb {R}^n}$ $(n \geq 5)$, where $V \neq 0$ is a nonnegative potential belonging to certain reverse Hölder class $B_s$ for $s\ge {n}/{2}$. The Hardy type space $H^1_{\mathcal {L}_2}$ is defined in terms of the maximal function with respect to the semigroup $\{{\rm e}^{-t \mathcal {L}_2}\}$ and it is identical to the Hardy space $H^1_{\mathcal {L}_1}$ established by Dziubański and Zienkiewicz. In this article, we prove the $L^p$-boundedness of the commutator $\mathcal {R}_b=b\mathcal {R}f-\mathcal {R}(bf)$ generated by the Riesz transform $\mathcal {R}=\nabla ^2\mathcal {L}_2^{-{1}/{2}}$, where $b\in {\rm BMO}_\theta (\rho )$, which is larger than the space ${\rm BMO}(\mathbb {R}^n)$. Moreover, we prove that $\mathcal {R}_b$ is bounded from the Hardy space $H_{\mathcal {L}_2}^1(\mathbb {R}^n)$ into weak $L_{\rm weak}^1(\mathbb {R}^n)$. (English) |
| Keyword:
|
commutator |
| Keyword:
|
Hardy space |
| Keyword:
|
reverse Hölder inequality |
| Keyword:
|
Riesz transform |
| Keyword:
|
Schrödinger operator |
| Keyword:
|
Schrödinger type operator |
| MSC:
|
35J10 |
| MSC:
|
42B20 |
| MSC:
|
42B30 |
| MSC:
|
42B35 |
| idZBL:
|
Zbl 06587882 |
| idMR:
|
MR3483231 |
| DOI:
|
10.1007/s10587-016-0248-z |
| . |
| Date available:
|
2016-04-07T15:05:04Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144882 |
| . |
| Reference:
|
[1] Bongioanni, B., Harboure, E., Salinas, O.: Commutators of Riesz transforms related to Schrödinger operators.J. Fourier Anal. Appl. 17 (2011), 115-134. Zbl 1213.42075, MR 2765594, 10.1007/s00041-010-9133-6 |
| Reference:
|
[2] Bramanti, M., Cerutti, C. M.: Commutators of singular integrals on homogeneous spaces.Boll. Unione Mat. Ital. 10 (1996), 843-883. MR 1430157 |
| Reference:
|
[3] Cao, J., Liu, Y., Yang, D.: Hardy spaces $H^1_{\mathcal L}({\mathbb R}^n)$ associated to Schrödinger type operators $(-\Delta)^2+V^2$.Houston J. Math. 36 (2010), 1067-1095. MR 2753734 |
| Reference:
|
[4] Coifman, R. R., Rochberg, R., Weiss, G.: Factorization theorem for Hardy spaces in several variables.Ann. Math. 103 (1976), 611-635. MR 0412721, 10.2307/1970954 |
| Reference:
|
[5] Duong, X. T., Yan, L.: Commutators of BMO functions and singular integral operators with non-smooth kernels.Bull. Aust. Math. Soc. 67 (2003), 187-200. Zbl 1023.42010, MR 1972709, 10.1017/S0004972700033669 |
| Reference:
|
[6] Dziubański, J., Zienkiewicz, J.: Hardy space $H^1$ associated to Schrödinger operators with potentials satisfying reverse Hölder inequality.Rev. Mat. Iberoam 15 (1999), 279-296. MR 1715409, 10.4171/RMI/257 |
| Reference:
|
[7] Guo, Z., Li, P., Peng, L.: $L^p$ boundedness of commutators of Riesz transform associated to Schrödinger operator.J. Math. Anal. Appl. 341 (2008), 421-432. MR 2394095, 10.1016/j.jmaa.2007.05.024 |
| Reference:
|
[8] Janson, S.: Mean oscillation and commutators of singular integral operators.Ark. Math. 16 (1978), 263-270. Zbl 0404.42013, MR 0524754, 10.1007/BF02386000 |
| Reference:
|
[9] Li, H. Q.: $L^p$ estimates of Schrödinger operators on nilpotent groups.J. Funct. Anal. 161 (1999), French 152-218. MR 1670222, 10.1006/jfan.1998.3347 |
| Reference:
|
[10] Li, P., Peng, L.: Endpoint estimates for commutators of Riesz transforms associated with Schrödinger operators.Bull. Aust. Math. Soc. 82 (2010), 367-389. Zbl 1210.47048, MR 2737950, 10.1017/S0004972710000390 |
| Reference:
|
[11] Liu, Y.: $L^p$ estimates for Schrödinger type operators on the Heisenberg group.J. Korean Math. Soc. 47 (2010), 425-443. Zbl 1187.22008, MR 2605991, 10.4134/JKMS.2010.47.2.425 |
| Reference:
|
[12] Liu, Y., Dong, J.: Some estimates of higher order Riesz transform related to Schrödinger type operators.Potential Anal. 32 (2010), 41-55. Zbl 1197.42008, MR 2575385, 10.1007/s11118-009-9143-7 |
| Reference:
|
[13] Liu, Y., Huang, J. Z., Dong, J. F.: Commutators of Calderón-Zygmund operators related to admissible functions on spaces of homogeneous type and applications to Schrödinger operators.Sci. China. Math. 56 (2013), 1895-1913. Zbl 1278.42031, MR 3090862, 10.1007/s11425-012-4551-3 |
| Reference:
|
[14] Liu, Y., Huang, J., Xie, D.: Some estimates of Schrödinger type operators on the Heisenberg group.Arch. Math. 94 (2010), 255-264. Zbl 1189.22006, MR 2602452, 10.1007/s00013-009-0098-0 |
| Reference:
|
[15] Liu, Y., Sheng, J.: Some estimates for commutators of Riesz transforms associated with Schrödinger operators.J. Math. Anal. Appl. 419 (2014), 298-328. MR 3217150, 10.1016/j.jmaa.2014.04.053 |
| Reference:
|
[16] Liu, Y., Wang, L., Dong, J.: Commutators of higher order Riesz transform associated with Schrödinger operators.J. Funct. Spaces Appl. 2013 (2013), Article ID 842375, 15 pages. Zbl 1279.47052, MR 3053277 |
| Reference:
|
[17] Shen, Z.: $L^{p}$ estimates for Schrödinger operators with certain potentials.Ann. Inst. Fourier (Grenoble) 45 (1995), 513-546. MR 1343560, 10.5802/aif.1463 |
| Reference:
|
[18] Sugano, S.: $L^p$ estimates for some Schrödinger operators and a Calderón-Zygmund operator of Schrödinger type.Tokyo J. Math. 30 (2007), 179-197. MR 2328062, 10.3836/tjm/1184963655 |
| Reference:
|
[19] Zhong, J.: Harmonic Analysis for some Schrödinger Type Operators.Ph.D. Thesis, Princeton University (1993). MR 2689454 |
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