In this paper, we study quasilinear parabolic equations with the nonlinearity structure modeled after the p(x, t)-Laplacian on nonsmooth domains. The main goal is to obtain end point Calderón-Zygmund type estimates in the variable exponent setting. In a recent work [1], the estimates obtained were strictly above the natural exponent p(x, t) and hence there was a gap between the natural energy estimates and the estimates above p(x, t) (see (1.3) and (1.2)). Here, we bridge this gap to obtain the end point case of the estimates obtained in [1]. To this end, we make use of the parabolic Lipschitz truncation developed in [2] and obtain significantly improved a priori estimates below the natural exponent with stability of the constants. An important feature of the techniques used here is that we make use of the unified intrinsic scaling introduced in [3], which enables us to handle both the singular and degenerate cases simultaneously.

在本文中，我们研究了具有非线性结构的拟线性抛物线方程，该方程是根据非光滑域上的p ( x ,  t )-拉普拉斯算子建模的。主要目标是在可变指数设置中获得端点 Calderón-Zygmund 类型估计。在最近的一项工作 [1] 中，获得的估计值严格高于自然指数p ( x ,  t )，因此自然能量估计值与高于p ( x ,  t )的估计值之间存在差距)（见（1.3）和（1.2））。在这里，我们弥合了这一差距，以获得在 [1] 中获得的估计的终点情况。为此，我们利用 [2] 中开发的抛物线 Lipschitz 截断方法，在常数稳定的情况下，在自然指数以下获得显着改进的先验估计。这里使用的技术的一个重要特征是我们利用了 [3] 中引入的统一内在缩放，这使我们能够同时处理奇异和退化情况。