Regularity estimates for continuous solutions of $\alpha$-convex balance laws

The paper proves new regularity estimates for continuous solutions to balance equation\udu_t+f(u)_x=g\udwhere the source g is bounded and the flux f is 2n-times continuously differentiable and it satisfies a convexity assumption that we denote as 2n-convexity.\udThe results are known in the case of the quadratic flux by very different arguments in the literature. We prove that the continuity of u must be in fact 1/2n-Holder continuity and that the distributional source term g is determined by the classical derivative of u along any characteristics; part of the proof consists in showing that this classical derivative is well defined at any `Lebesgue point' of g for suitable coverings. These two regularity statements fail in general for strictly convex fuxes, see [3]