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Fan, S. Lu, D. Yang, Regularity in Morrey spaces of strong solutions to nondivergence elliptic equations with VMO coeﬃcients, Georgian Math. J. 5 (1998) 425–440. [18] D. Gilbarg, N. , SpringerVerlag, Berlin, 1983. [19] A. Hinz, H. Kalf, Subsolution estimates and Harnack’s inequality for Schrödinger operators, J. Reine Angew. Math. 404 (1990) 118–131. [20] F. John, L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 4 (1961) 415–426. [21] D. Kim, N. Krylov, Elliptic diﬀerential equations with coeﬃcients measurable with respect to one variable and VMO with respect to the others, SIAM J.

For this purpose, we need the following well-known lemma. 1. ) Let 1 < p < ∞. Then (i) A function u belongs to W 1,p (Rn+ ) if and only if its even extension u with respect to x1 belongs to W 1,p (Rn+ ). Moreover, there exists C = C(n) > 0 such that 124 G. Pan, L. Tang / Journal of Functional Analysis 270 (2016) 88–133 C −1 u C −1 Du Lp (Rn +) Lp (Rn +) ≤ u Lp (Rn +) ≤ Du ≤C u Lp (Rn +) , Lp (Rn +) ≤ C Du . 6) (ii) A function u belongs to W 1,p (Rn+ ) and vanishes on ∂Rn+ if and only if its odd extension u with respect to x belongs to W 1,p (Rn+ ).

Fan, S. Lu, D. Yang, Regularity in Morrey spaces of strong solutions to nondivergence elliptic equations with VMO coeﬃcients, Georgian Math. J. 5 (1998) 425–440. [18] D. Gilbarg, N. , SpringerVerlag, Berlin, 1983. [19] A. Hinz, H. Kalf, Subsolution estimates and Harnack’s inequality for Schrödinger operators, J. Reine Angew. Math. 404 (1990) 118–131. [20] F. John, L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 4 (1961) 415–426. [21] D. Kim, N. Krylov, Elliptic diﬀerential equations with coeﬃcients measurable with respect to one variable and VMO with respect to the others, SIAM J.