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.

中文翻译：

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具有奇异系数的非发散形式的椭圆和抛物线方程的加权混合范数 $L_p$-估计

在本文中，我们研究了具有奇异系数的非发散形式椭圆和抛物线方程。加权和混合范数 $L_p$ 估计和可解性是在适当的部分加权 BMO 条件下对系数建立的。当系数为常数时，算子被简化为在分数热方程和分数拉普拉斯方程研究中出现的扩展算子。即使在这种情况下和未混合的规范情况下，我们的结果也是新的。为了证明，我们探索并利用方程的特殊结构来显示解的内部和边界 Lipschitz 估计以及齐次方程解的高阶导数。然后，我们通过使用 Fefferman-Stein 锐函数定理、Hardy-Littlewood 极大函数定理采用微扰方法，