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 $u\in {\mathrm{BV}}_{\text{loc}}^{\mathbb{A}}$ (here $j=1,\dots ,k$, with an appropriate convention if $j\ge n$). The space ${\mathrm{BV}}_{\text{loc}}^{\mathbb{A}}$ 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.

29 pages; to appear in Transactions of the American Mathematical Society

References in corpus (2)

• On Rank-One Convex Functions that are homogeneous of Degree One
• Continuity and canceling operators of order $n$ on ${\mathbb{R}}^n$

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