Continuity for Sobolev mappings with null Lagrangian bounds

We prove the continuity of Sobolev functions $\varphi \in W^{1,n}_{\mathrm{loc}}(\Omega)$, $\Omega \subset \mathbb{R}^n$, that satisfy $$\lvert\nabla \varphi(x)\rvert^n \le K(x)\bigl(\langle \nabla \varphi(x), \xi(x)\rangle + A(x)\bigr),$$ where $\xi \in L_{\mathrm{loc}}^{n/(n-1)}(\Omega, \mathbb{R}^n)$ is weakly divergence-free, and $K \in L^p_{\mathrm{loc}} (\Omega)$, $A \in L^q_{\mathrm{loc}} (\Omega)$ are non-negative with $p^{-1}+q^{-1}<1$. The result is applicable to a broad class of differential inequalities of null Lagrangian type. As our principal application, we obtain a sharp continuity theorem for $f \in W^{1,n}_{\mathrm{loc}} (\Omega, \mathbb{R}^n)$ satisfying the distortion inequality with defect $\lvert Df(x)\rvert^n \le K(x)\det Df(x) + \Sigma(x)$; this result is new even in the planar case, and closes a significant gap between existing methods and known counterexamples. The proof relies on an overlooked Sobolev-type inequality formulated in terms of measures of superlevel sets.