We develop a global Calderón–Zygmund theory for a quasilinear divergence form parabolic operator with discontinuous entries which exhibit nonlinearities both with respect to the weak solution u and its spatial gradient Du in a nonsmooth domain. The nonlinearity behaves as the parabolic p -Laplacian in Du , its discontinuity with respect to the independent variables is measured in terms of small-BMO, while only Hölder continuity is required with respect to u and the underlying domain is assumed to be $$\delta $$ δ -Reifenberg flat. We introduce and employ essentially a new concept of the intrinsic parabolic maximal function in order to overcome the main difficulties stemming from both the parabolic scaling deficiency and the nonlinearity of u -variable of such a very general parabolic operator, obtaining optimal $$L^q$$ L q -estimates for the spatial gradient under a minimal geometric condition on the domain.

中文翻译：

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一般非线性抛物线方程的最优正则估计

我们为具有不连续项的拟线性发散形式抛物线算子开发了全局 Calderón-Zygmund 理论，该算子在非光滑域中表现出关于弱解 u 及其空间梯度 Du 的非线性。非线性表现为 Du 中的抛物线 p-拉普拉斯算子，其相对于自变量的不连续性是根据小 BMO 来衡量的，而对于 u 只需要 Hölder 连续性，并且假设基础域为 $$\ delta $$ δ -Reifenberg 持平。为了克服抛物线标度缺陷和这种非常通用的抛物线算子的 u 变量的非线性所产生的主要困难，我们引入并采用了本质上抛物线极大函数的新概念，