Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Critical $\mathrm {L}^p$-differentiability of $\mathrm {BV}^\mathbb {A}$-maps and canceling operators
HTML articles powered by AMS MathViewer

by Bogdan Raiţă PDF
Trans. Amer. Math. Soc. 372 (2019), 7297-7326 Request permission


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.

Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 26B05, 35J48
  • Retrieve articles in all journals with MSC (2010): 26B05, 35J48
Additional Information
  • Bogdan Raiţă
  • Affiliation: University of Warwick, Zeeman Building, Coventry CV4 7HP, United Kingdom
  • Email:
  • Received by editor(s): October 10, 2018
  • Received by editor(s) in revised form: March 27, 2019
  • Published electronically: August 13, 2019
  • Additional Notes: The author was supported by the Engineering and Physical Sciences Research Council Award No. EP/L015811/1. This project received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program under Grant Agreement No. 757254 (SINGULARITY)
  • © Copyright 2019 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 372 (2019), 7297-7326
  • MSC (2010): Primary 26B05; Secondary 35J48
  • DOI:
  • MathSciNet review: 4024554