标题： 水平集的大尺度性质对于具有对数BMO矩阵权的退化$p$-Laplacian方程 显示英文标题

标题： A large scaling property of level sets for degenerate $p$-Laplacian equations with logarithmic BMO matrix weights

摘要： 在这项研究中，我们处理了具有退化矩阵权的$p$- 拉普拉斯方程解的广义正则性性质。 在之前的一些有趣工作中 [A. Kh. Balci, L. Diening, R. Giova, A. Passarelli di Napoli, SIAM J. Math. Anal. 54 (2022), 2373-2412] 和 [A. Kh. Balci, S.-S. Byun, L. Diening, H.-S. Lee, J. Math. Pures Appl. (9) 177 (2023), 484-530] 已经观察到，在所谓的$\log$-$\mathrm{BMO}$条件以及边界上的最小正则性假设下，对于具有退化权的非线性方程获得 Calderón-Zygmund 估计。 本文也沿着这一方向前进，并将梯度解水平集的一般梯度估计扩展到更精细的函数空间。特别是，我们通过分数极大算子构造了一个关于大尺度参数的空间梯度$|\nabla u|$的超水平集的覆盖。 显示英文摘要

摘要： In this study, we deal with generalized regularity properties for solutions to $p$-Laplace equations with degenerate matrix weights. It has already been observed in previous interesting works [A. Kh. Balci, L. Diening, R. Giova, A. Passarelli di Napoli, SIAM J. Math. Anal. 54(2022), 2373-2412] and [A. Kh. Balci, S.-S. Byun, L. Diening, H.-S. Lee, J. Math. Pures Appl. (9) 177(2023), 484-530] that gaining Calder\'on-Zygmund estimates for nonlinear equations with degenerate weights under the so-called $\log$-$\mathrm{BMO}$ condition and minimal regularity assumption on the boundary. In this paper, we also follow this direction and extend general gradient estimates for level sets of the gradient of solutions up to more subtle function spaces. In particular, we construct a covering of the super-level sets of the spatial gradient $|\nabla u|$ with respect to a large scaling parameter via fractional maximal operators.