Hardy type estimates for commutators of fractional integrals associated with Schrodinger operators

Volume 10, Issue 6, pp 3155--3167 http://dx.doi.org/10.22436/jnsa.010.06.29

Publication Date: June 25, 2017 Submission Date: April 28, 2017

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Authors

Yinhong Xia - School of Mathematics and Statistics, Huanghuai University, Zhumadian 463000, P. R. China. Min Chen - School of Mathematics and Statistics, Huanghuai University, Zhumadian 463000, P. R. China.

Abstract

We consider the Schrödinger operator \(L = -\Delta + V\) on \(\mathbb{R}^n\), where \(n \geq 3\) and the nonnegative potential \(V\) belongs to reverse Hölder class \(RH_{q1}\) for some \(q_1 > n/2\) . Let \(I_\alpha\) be the fractional integral associated with \(L\), and let \(b\) belong to a new Campanato space \(\Lambda_\beta^\theta(\rho)\). In this paper, we establish the boundedness of the commutators \([b, I_\alpha]\) from \(L^p(R^n)\) to \(L^q(R^n)\) whenever \(1/q=1/p-(\alpha+\beta)/n, 1<p<n/(\alpha+\beta)\). When \(\frac{n}{n+\beta}<p\leq 1,1/q=1/p-(\alpha+\beta)/n\), we show that \([b, I_\alpha]\) is bounded from \(H^p_ L(R^n)\) to \(L^q(R^n)\). Moreover, we also prove that \([b, I_\alpha]\) maps \(H_L^{\frac{n}{n+\beta}}(R^n)\) continuously into weak \(L^{\frac{n}{n-\alpha}}(R^n)\).

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ISRP Style

Yinhong Xia, Min Chen, Hardy type estimates for commutators of fractional integrals associated with Schrodinger operators, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 6, 3155--3167

AMA Style

Xia Yinhong, Chen Min, Hardy type estimates for commutators of fractional integrals associated with Schrodinger operators. J. Nonlinear Sci. Appl. (2017); 10(6):3155--3167

Chicago/Turabian Style

Xia, Yinhong, Chen, Min. "Hardy type estimates for commutators of fractional integrals associated with Schrodinger operators." Journal of Nonlinear Sciences and Applications, 10, no. 6 (2017): 3155--3167

Keywords

Schrödinger operator
commutator
Campanato space
fractional integral
Hardy space.

MSC

42B30
35J10

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