Weighted mixed-norm $L_p$-estimates for elliptic and parabolic equations in non-divergence form with singular coefficients

Abstract

In this paper, we study non-divergence form elliptic and parabolic equations with singular coefficients. Weighted and mixed-norm $L_p$-estimates and solvability are established under suitable partially weighted BMO conditions on the coefficients. When the coefficients are constants, the operators are reduced to extensional operators which arise in the study of fractional heat equations and fractional Laplace equations. Our results are new even in this setting and in the unmixed norm case. For the proof, we explore and utilize the special structures of the equations to show both interior and boundary Lipschitz estimates for solutions and for higher-order derivatives of solutions to homogeneous equations. We then employ the perturbation method by using the Fefferman–Stein sharp function theorem, the Hardy–Littlewood maximal function theorem, as well as a weighted Hardy’s inequality.