A nonlocal equation with $p$-growth with $1 < p < \infty$ is modeled by
$\mathrm{P.V.} \int_{\mathbb{R}^n} |u(x)-u(y)|^{p-2}(u(x)-u(y)) K(x, y)\, dy = 0, \quad x\in\Omega.$
If the kernel $K : \mathbb{R}^n\times\mathbb{R}^n \to [0,\infty)$ satisfies the $s$-order uniform ellipticity condition for some $s \in (0, 1)$:
$\frac{1}{\Lambda} \frac{1}{|x-y|^{n+sp}} \le K(x, y) \le \frac{\Lambda}{|x-y|^{n+sp}},$
where $\Lambda\ge 1$ is a constant, then weak solutions to these nonlocal equations are known to be H\"older continuous and satisfy a nonlocal version of the Harnack inequality.
Our main interest is to investigate conditions on kernels that do not satisfy the above uniform ellipticity condition but still lead to these regularity results.
In this talk, I will introduce two classes of such kernels. The first class consists of degenerate or singular kernels associated with the Muckenhoupt class.
The second class includes general kernels that cover both the above uniformly elliptic cases and borderline cases as $s \to 1$.
This is based on joint work with Linus Behn, Lars Diening, and Julian Rolfes (Bielefeld University), and Kyeong Song (KIAS, Seoul).