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 $u\in \mathrm {BV}^\mathbb {A}_{\operatorname {loc}}$ (here $j=1,\ldots , k$, with an appropriate convention if $j\geq n$). The space $\mathrm {BV}^\mathbb {A}_{\operatorname {loc}}$, a single framework covering $\mathrm {BV}$, $\mathrm {BD}$, and $\mathrm {BV}^k$, consists of those locally integrable maps $u$ such that $\mathbb {A} u$ is a Radon measure on $\mathbb {R}^n$.

For $j=1,\ldots ,\min \{k, n-1\}$, we show that the $\mathrm {L}^p$-differentiability property above is equivalent to Van Schaftingen’s elliptic and canceling condition for $\mathbb {A}$. For $j=n,\ldots , k$, ellipticity is necessary, but cancellation is not. To complete the characterization, we determine the class of elliptic operators $\mathbb {A}$ such that the estimate \begin{align}\tag {1} \|D^{k-n}u\|_{\mathrm {L}^\infty }\leqslant C\|\mathbb {A} u\|_{\mathrm {L}^1} \end{align} holds for all vector fields $u\in \mathrm {C}^\infty _c$. Surprisingly, the (computable) condition on $\mathbb {A}$ such that \eqref{eq:abs} holds is strictly weaker than cancellation.

The results on $\mathrm {L}^p$-differentiability can be formulated as sharp pointwise regularity results for overdetermined elliptic systems \begin{align*} \mathbb {A} u=\mu , \end{align*} where $\mu$ is a Radon measure, thereby giving a variant for the limit case $p=1$ of a theorem of Calderón and Zygmund which was not covered before.