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

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^