Boundedness of high order commutators of Marcinkiewicz integrals associated with Schrödinger operators

Volume 10, Issue 8, pp 4295--4306 http://dx.doi.org/10.22436/jnsa.010.08.24

Publication Date: August 22, 2017 Submission Date: July 01, 2017

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Authors

Hui Wang - Teachers College, Nanyang Institute of Technology, Nanyang, 473000, P. R. China. Bijun Ren - Department of Information Engineering, Henan Institute of Finance and Banking, Zhengzhou, 451464, P. R. China.

Abstract

Suppose \(L=-\Delta+V\) is a Schrödinger operator on \(\mathbb{R}^n\), where \(n\geq 3\) and the nonnegative potential \(V\) belongs to reverse Hölder class \(RH_{n}.\) Let \(b\) belong to a new Campanato space \(\Lambda_\beta^\theta(\rho),\) and let \(\mu_j^L\) be the Marcinkiewicz integrals associated with \(L.\) In this paper, we establish the boundedness of the \(m\)-order commutators \([b^m, \mu_j^L]\) from \(L^p(\mathbb{R}^n)\) to \(L^q(\mathbb{R}^n),\) where \(1/q=1/p-m\beta/n\) and \(1<p<n/(m\beta).\) As an application, we obtain the boundedness of \([b^m, \mu_j^L]\) on the generalized Morrey spaces related to certain nonnegative potentials.

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ISRP Style

Hui Wang, Bijun Ren, Boundedness of high order commutators of Marcinkiewicz integrals associated with Schrödinger operators, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 8, 4295--4306

AMA Style

Wang Hui, Ren Bijun, Boundedness of high order commutators of Marcinkiewicz integrals associated with Schrödinger operators. J. Nonlinear Sci. Appl. (2017); 10(8):4295--4306

Chicago/Turabian Style

Wang, Hui, Ren, Bijun. "Boundedness of high order commutators of Marcinkiewicz integrals associated with Schrödinger operators." Journal of Nonlinear Sciences and Applications, 10, no. 8 (2017): 4295--4306

Keywords

Schrödinger operator
Marcinkiewicz integral
commutator
Campanato space
Morrey space.

MSC

42B25
35J10

References

[1] B. Bongioanni, E. Harboure, O. Salinas, Commutators of Riesz transforms related to Schrödinger operators, J. Fourier Anal. Appl., 17 (2011), 115–134.

[2] D. Chen, F. Jin, The boundedness of Marcinkiewicz integrals associated with Schrödinger operator on Morrey spaces, J. Funct. Spaces, 2014 (2014), 11 pages.

[3] D. Chen, D. Zou, The boundedness of Marcinkiewicz integral associated with Schrödinger operator and its commutator, J. Funct. Spaces, 2014 (2014), 10 pages.

[4] J. Dziubański, J. Zienkiewicz , Hardy space \(H^1\) associated to Schrödinger operator with potential satisfying reverse Hölder inequality , Rev. Mat. Iberoamericana, 15 (1999), 279–296.

[5] W. Gao, L. Tang , Boundedness for Marcinkiewicz integrals associated with Schrödinger operators, Proc. Indian Acad. Sci. Math. Sci., 124 (2014), 193–203.

[6] Y. Liu, J. Sheng, Some estimates for commutators of Riesz transforms associated with Schrödinger operators, J. Math. Anal. Appl., 419 (2014), 298–328.

[7] T. Mizuhara , Boundedness of some classical operators on generalized Morrey spaces, Harmonic analysis, ICM-90 Satell. Conf. Proc., Springer, Japan, (1991), 183–189.

[8] C. B. Jr. Morrey , On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc., 43 (1938), 126–166.

[9] B. Ren, H. Wang, Boundedness of higher order Riesz transforms associated with Schrödinger type operator on generalized Morrey spaces, J. Nonlinear Sci. Appl., 10 (2017), 2757–2766.

[10] Z. Shen, \(L^p\) estimates for Schrödinger operators with certain potentials, Ann. Inst. Fourier, 45 (1995), 513–546.

[11] L. Tang, J. F. Dong, Boundedness for some Schrödinger type operator on Morrey spaces related to certain nonnegative potentials, J. Math. Anal. Appl., 355 (2009), 101–109.