# American Institute of Mathematical Sciences

May  2017, 16(3): 899-914. doi: 10.3934/cpaa.2017043

## Weighted lorentz estimates for nondivergence linear elliptic equations with partially BMO coefficients

 Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China

* Corresponding author

Received  August 2016 Revised  December 2016 Published  February 2017

Fund Project: The first author is supported by the Fundamental Research Funds for the Central Universities of China grant 2016YJS154, and the second author is supported by NSF of China grant 11371050.

We prove weighted Lorentz estimates of the Hessian of strong solution for nondivergence linear elliptic equations $a_{ij}(x)D_{ij}u(x)=f(x)$. The leading coefficients are assumed to be measurable with respect to one variable and have small BMO semi-norms with respect to the other variables. Here, an approximation method, Lorentz boundedness of the Hardy-Littlewood maximal operators and an equivalent representation of Lorentz norm are employed.

Citation: Junjie Zhang, Shenzhou Zheng. Weighted lorentz estimates for nondivergence linear elliptic equations with partially BMO coefficients. Communications on Pure & Applied Analysis, 2017, 16 (3) : 899-914. doi: 10.3934/cpaa.2017043
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##### References:
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