Functional analysis, Harmonic analysis
Boundedness of second-order Riesz transforms on weighted Hardy and $BMO$ spaces associated with Schrödinger operators
Comptes Rendus. Mathématique, Volume 359 (2021) no. 6, pp. 687-717.

Let $d\in \left\{3,4,5,...\right\}$ and a weight $w\in {A}_{\infty }^{\rho }$. We consider the second-order Riesz transform $T={\nabla }^{2}\phantom{\rule{0.166667em}{0ex}}{L}^{-1}$ associated with the Schrödinger operator $L=-\Delta +V$, where $V\in R{H}_{\sigma }$ with $\sigma >\frac{d}{2}$. We present three main results. First $T$ is bounded on the weighted Hardy space ${H}_{w,L}^{1}\left({ℝ}^{d}\right)$ associated with $L$ if $w$ enjoys a certain stable property. Secondly $T$ is bounded on the weighted $BMO$ space $BM{O}_{w,\rho }\left({ℝ}^{d}\right)$ associated with $L$ if $w$ also belongs to an appropriate doubling class. Thirdly $BM{O}_{w,\rho }\left({ℝ}^{d}\right)$ is the dual of ${H}_{w,L}^{1}\left({ℝ}^{d}\right)$ when $w\in {A}_{1}^{\rho }$.

Revised:
Accepted:
Published online:
DOI: https://doi.org/10.5802/crmath.213
Classification: 30H35,  42B20,  42B25,  42B30,  42B37
Trong Nguyen Ngoc 1; Truong Le Xuan 1; Do Tan Duc 1

1. University of Economics Ho Chi Minh City, Vietnam
@article{CRMATH_2021__359_6_687_0,
author = {Trong Nguyen Ngoc and Truong Le Xuan and Do Tan Duc},
title = {Boundedness of second-order {Riesz} transforms on weighted {Hardy} and $BMO$ spaces associated with {Schr\"odinger} operators},
journal = {Comptes Rendus. Math\'ematique},
pages = {687--717},
publisher = {Acad\'emie des sciences, Paris},
volume = {359},
number = {6},
year = {2021},
doi = {10.5802/crmath.213},
language = {en},
}
TY  - JOUR
AU  - Trong Nguyen Ngoc
AU  - Truong Le Xuan
AU  - Do Tan Duc
TI  - Boundedness of second-order Riesz transforms on weighted Hardy and $BMO$ spaces associated with Schrödinger operators
JO  - Comptes Rendus. Mathématique
PY  - 2021
DA  - 2021///
SP  - 687
EP  - 717
VL  - 359
IS  - 6
PB  - Académie des sciences, Paris
UR  - https://doi.org/10.5802/crmath.213
DO  - 10.5802/crmath.213
LA  - en
ID  - CRMATH_2021__359_6_687_0
ER  - 
%0 Journal Article
%A Trong Nguyen Ngoc
%A Truong Le Xuan
%A Do Tan Duc
%T Boundedness of second-order Riesz transforms on weighted Hardy and $BMO$ spaces associated with Schrödinger operators
%J Comptes Rendus. Mathématique
%D 2021
%P 687-717
%V 359
%N 6
%I Académie des sciences, Paris
%U https://doi.org/10.5802/crmath.213
%R 10.5802/crmath.213
%G en
%F CRMATH_2021__359_6_687_0
Trong Nguyen Ngoc; Truong Le Xuan; Do Tan Duc. Boundedness of second-order Riesz transforms on weighted Hardy and $BMO$ spaces associated with Schrödinger operators. Comptes Rendus. Mathématique, Volume 359 (2021) no. 6, pp. 687-717. doi : 10.5802/crmath.213. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.213/

[1] Bruno Bongioanni; Adrián Cabral; Eleonor Harboure Extrapolation for classes of weights related to a family of operators and applications, Potential Anal., Volume 38 (2013) no. 4, pp. 1207-1232 | Article | MR 3042701 | Zbl 1273.42017

[2] Bruno Bongioanni; Adrián Cabral; Eleonor Harboure Schrödinger type singular integrals: weighted estimates for $p=1$, Math. Nachr., Volume 289 (2016) no. 11-12, pp. 1341-1369 | Article | MR 3541813 | Zbl 1350.42021

[3] Bruno Bongioanni; Adrián Cabral; Eleonor Harboure Regularity of maximal functions associated to a critical radius function, Rev. Unión Mat. Argent., Volume 60 (2019) no. 2, pp. 539-566 | Article | MR 4049802 | Zbl 1428.42027

[4] Bruno Bongioanni; Eleonor Harboure; P. Quijano Weighted inequalities for Schrödinger type singular integrals, J. Fourier Anal. Appl., Volume 25 (2019) no. 3, pp. 595-632 | Article | MR 3953479 | Zbl 1416.42012

[5] Bruno Bongioanni; Eleonor Harboure; Oscar Salinas Riesz transforms related to Schrödinger operators acting on $BMO$ type spaces, J. Math. Anal. Appl., Volume 357 (2009) no. 1, pp. 115-131 | Article | MR 2526811 | Zbl 1180.42013

[6] Bruno Bongioanni; Eleonor Harboure; Oscar Salinas Classes of weights related to Schrödinger operators, J. Math. Anal. Appl., Volume 373 (2011) no. 2, pp. 563-579 | Article | MR 2720705 | Zbl 1203.42029

[7] Bruno Bongioanni; Eleonor Harboure; Oscar Salinas Weighted inequalities for commutators of Schrödinger-Riesz transforms, J. Math. Anal. Appl., Volume 392 (2012) no. 1, pp. 6-22 | Article | MR 2914933 | Zbl 1246.42018

[8] The Anh Bui; Ji Li; Fu Ken Ly $T1$ criteria for generalised Calderón-Zygmund type operators on Hardy and BMO spaces associated to Schrödinger operators and applications, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 18 (2018) no. 1, pp. 203-239 | MR 3783788 | Zbl 1394.35148

[9] Jacek Dziubański; Gustavo Garrigós; Teresa Martínez; José L. Torrea; Jacek Zienkiewicz $BMO$ spaces related to Schrödinger operators with potentials satisfying a reverse Hölder inequality, Math. Z., Volume 249 (2005) no. 2, pp. 329-356 | Article | Zbl 1136.35018

[10] Jacek Dziubański; Jacek Zienkiewicz ${H}^{p}$ spaces associated with Schrödinger operators with potentials from reverse Hölder classes, Colloq. Math., Volume 98 (2003) no. 1, pp. 5-38 | Article | Zbl 1083.42015

[11] David Gilbarg; Neil S. Trudinger Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften, 224, Springer, 1983 | MR 737190 | Zbl 0562.35001

[12] Kazuhiro Kurata; Satoko Sugano Estimates of the fundamental solution for magnetic Schrödinger operators and their applications, Tôhoku Math. J., Volume 52 (2000) no. 3, pp. 367-382 | MR 1772803 | Zbl 0967.35035

[13] Heping Liu; Lin Tang; Hua Zhu Weighted Hardy spaces and BMO spaces associated with Schrödinger operators, Math. Nachr., Volume 285 (2012) no. 17-18, pp. 2173-2207 | MR 3002608 | Zbl 1266.42060

[14] Fu Ken Ly Second order Riesz transforms associated to the Schrödinger operator for $p\le 1$, J. Math. Anal. Appl., Volume 410 (2014) no. 1, pp. 391-402 | MR 3109848 | Zbl 1319.42020

[15] Fu Ken Ly Classes of weights and second order Riesz transforms associated to Schrödinger operators, J. Math. Soc. Japan, Volume 68 (2016) no. 2, pp. 489-533 | MR 3488134 | Zbl 1348.35060

[16] Tao Ma; Pablo Raúl Stinga; José L. Torrea; Chao Zhang Regularity estimates in Hölder spaces for Schrödinger operators via a $T1$ theorem, Ann. Mat. Pura Appl., Volume 193 (2014) no. 2, pp. 561-589 | Zbl 1301.42046

[17] El Maati Ouhabaz Analysis of heat equations on domains, London Mathematical Society Monographs, 31, London Mathematical Society, 2005 | MR 2124040 | Zbl 1082.35003

[18] Guixia Pan; Lin Tang Solvability for Schrödinger equations with discontinuous coefficients, J. Funct. Anal., Volume 270 (2016) no. 1, pp. 88-133 | MR 3419757 | Zbl 1342.35426

[19] Cruz Prisuelos Arribas Weighted Hardy Spaces Associated with Elliptic Operators. Part III: Characterisations of ${H}_{L}^{p}\left(w\right)$ and the Weighted Hardy Space Associated with the Riesz Transform, J. Geom. Anal., Volume 29 (2019) no. 1, pp. 451-509 | Article | MR 3897023 | Zbl 07024076

[20] Zhongwei Shen ${L}^{p}$ estimates for Schrödinger operators with certain potentials, Ann. Inst. Fourier, Volume 45 (1995) no. 2, pp. 513-546 | Article | Numdam | MR 1343560 | Zbl 0818.35021

[21] Zhongwei Shen Estimates in ${L}^{p}$ for magnetic Schrödinger operators, Indiana Univ. Math. J., Volume 45 (1996) no. 3, pp. 817-841 | MR 1422108 | Zbl 0880.35034

[22] Nguyen Ngoc Trong; Le Xuan Truong Riesz transforms and Littlewood-Paley square function associated to Schrödinger operators on new weighted spaces, J. Aust. Math. Soc., Volume 105 (2018) no. 2, pp. 201-228 | Article | MR 3859181 | Zbl 1401.42017

[23] J. Zhong Harmonic analysis for some Schrödinger type operators (1993) (Ph. D. Thesis)

Cited by Sources: