We study the higher Hölder regularity of local weak solutions to a class of nonlinear nonlocal elliptic equations with kernels that satisfy a mild continuity assumption. An interesting feature of our main result is that the obtained regularity is better than one might expect when considering corresponding results for local elliptic equations in divergence form with continuous coefficients. Therefore, in some sense our result can be considered to be of purely nonlocal type, following the trend of various such purely nonlocal phenomena observed in recent years. Our approach can be summarized as follows. First, we use certain test functions that involve discrete fractional derivatives in order to obtain higher Hölder regularity for homogeneous equations driven by a locally translation invariant kernel, while the global behaviour of the kernel is allowed to be more general. This enables us to deduce the desired regularity in the general case by an approximation argument.

我们研究一类非线性，满足局部连续性假设的椭圆形非局部椭圆型方程的局部弱解的较高的Hölder正则性。我们的主要结果的一个有趣特征是，当考虑具有连续系数的发散形式的局部椭圆方程的相应结果时，获得的正则性好于人们预期的正则性。因此，在某种意义上，我们的结果可以被认为是纯粹的非局部类型，这是随着近年来观察到的各种这种纯粹的非局部现象的趋势而得出的。我们的方法可以总结如下。首先，我们使用某些涉及离散分数导数的测试函数，以便针对由局部平移不变核驱动的齐次方程获得更高的Hölder正则性，同时允许内核的全局行为更通用。这使我们能够通过逼近参数推断出一般情况下所需的规律性。