Global weighted W2,p estimates for nondivergence elliptic equations with small BMO coefficients

Recently, Byun and Lee [On weighted W2,p estimates for elliptic equations with BMO coefficients in nondivergence form, Internat. J. Math.26 (2015), Article ID: 1550001, 28pp.] proved that $$|f|^2\in L_w^{\frac{p}{2}}(\Omega) \Rightarrow |D^2 u|^2\in L_w^{\frac{p}{2}}(\Omega)\quad \mbox{for any}\ p>2$$ for the solutions of $$\left\{\begin{array}{@{}l@{\quad}l@{}} a_{ij}(x)u_{x_{i}x_{j}}=f&\mbox{in}\ \Omega,\\[4pt] u(x)=0&\mbox{on}\ \partial\Omega \end{array} \right.$$ with small BMO coefficients. In this paper we shall extend the result in the above-mentioned paper to the following result under the same assumptions $$f \in L_w^p(\Omega) \Rightarrow |D^2 u| \in L_w^p(\Omega) \quad \mbox{for any}\ p>1.$$ As a corollary we obtain Lp-type regularity for such equations.