Nonlocal equations with kernels of general order

We consider a broad class of nonlinear integro-differential equations with a kernel whose differentiability order is described by a general function $\phi$. This class includes not only the fractional $p$-Laplace equations, but also borderline cases when the fractional order approaches $1$. Under mild assumptions on $\phi$, we establish sharp Sobolev-Poincar\'e type inequalities for the associated Sobolev spaces, which are connected to a question raised by Brezis (Russian Math. Surveys 57:693--708, 2002). Using these inequalities, we prove H\"older regularity and Harnack inequalities for weak solutions to such nonlocal equations. All the estimates in our results remain stable as the associated nonlocal energy functional approaches its local counterpart.