Quasiconvexity and self-improving size estimates

We show that Müller’s $L\log L$ bound $$F(Du)\ge 0,Du\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^p({\mathbb{R}}^n)⟹F(Du)\in L\log L_{\mathrm{l}\mathrm{o}\mathrm{c}}({\mathbb{R}}^n)$$ for $F=\det$ and $p=n$ holds for quasiconcave $F$ which are homogeneous of degree $p>1$. This contrasts similar Hardy space bounds which hold only for null Lagrangians.