Failure of interior $L^p$ estimates for nonlocal elliptic equations with continuous kernels

Abstract

Let $0<σ<2$ and $1\leq p<\infty$. We construct a fixed linear nonlocal elliptic operator $L$ and functions $u_m\in C^\infty(\mathbb R^n)\cap L^\infty(\mathbb R^n)$ such that \[ \|u_m\|_{L^\infty(\mathbb R^n)}+\|Lu_m\|_{L^p(B_2)}\longrightarrow0, \qquad \liminf_{m\to\infty}\|(-Δ)^{σ/2}u_m\|_{L^p(B_1)}>0. \] The kernel is symmetric in the increment variable, uniformly elliptic, and satisfies the standard bounds on its first two derivatives in that variable. Its dependence on $x$ has a uniform modulus of continuity of order $(\log(1/r))^{-1/2}$ as $r\downarrow0$. Moreover, $\|Du_m\|_{L^p(B_4)}\to0$ for $σ>1$, and these norms remain bounded for $σ=1$. Thus the interior $L^p$ estimates for translation-invariant operators do not extend to this class under mere continuity in $x$.