Critical $\mathrm{L}^p$-differentiability of $\mathrm{BV}^{\mathbb{A}}$-maps and canceling operators

Abstract

We give a generalization of Dorronsoro's Theorem on critical $\mathrm{L}^p$-Taylor expansions for $\mathrm{BV}^k$-maps on $\mathbb{R}^n$, i.e., we characterize homogeneous linear differential operators $\mathbb{A}$ of $k$-th order such that $D^{k-j}u$ has $j$-th order $\mathrm{L}^{n/(n-j)}$-Taylor expansion a.e. for all $u\in\mathrm{BV}^\mathbb{A}_{\text{loc}}$ (here $j=1,\ldots, k$, with an appropriate convention if $j\geq n$). The space $\mathrm{BV}^\mathbb{A}_{\text{loc}}$ consists of those locally integrable maps $u$ such that $\mathbb{A} u$ is a Radon measure on $\mathbb{R}^n$. A new $\mathrm{L}^\infty$-Sobolev inequality is established to cover higher order expansions. Lorentz refinements are also considered. The main results can be seen as pointwise regularity statements for linear elliptic systems with measure-data.