Failure of uniform Sobolev estimates for concave nonlocal equations with rough kernels

We consider interior full-order Sobolev estimates for concave, translation-invariant nonlocal equations with even kernels comparable to that of the fractional Laplacian. Such estimates are required to be uniform over the kernel family and to depend on the forcing only through its $L^p$ norm. We show that they fail for every strict ellipticity ratio and every finite $p>\max\{2,n/\sigma\}$, where $0<\sigma<2$ is the order of the equation. The counterexamples have smooth bounded solutions, smooth bounded forcing, and kernels smooth away from the origin. Independent controls on disjoint jump annuli convert small oscillations into a prescribed nonlinear average, while the linear oscillations tend to zero in $L^p$. Concentrated negative Riesz potentials then contradict the proposed uniform Sobolev bound.