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Critical l^p differentiability of bv^a maps and canceling operators
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Raita, Bogdan (2019) Critical l^p differentiability of bv^a maps and canceling operators. Transactions of the American Mathematical Society, 372 . pp. 7297-7326. doi:10.1090/tran/7878 ISSN 0002-9947.
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Official URL: https://doi.org/10.1090/tran/7878
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
$\displaystyle \Vert D^{k-n}u\Vert _{\mathrm {L}^\infty }\leq C\Vert\mathbb{A} u\Vert _{\mathrm {L}^1}$ (1)
holds for all vector fields $ u\in \mathrm {C}^\infty _c$. Surprisingly, the (computable) condition on $ \mathbb{A}$ such that (1) 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
$\displaystyle \mathbb{A} u=\mu ,$
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 befor
| Item Type: | Journal Article | |||||||||
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| Subjects: | Q Science > QA Mathematics | |||||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | |||||||||
| Library of Congress Subject Headings (LCSH): | Approximation theory, Differential equations, Convolutions (Mathematics), Operator theory | |||||||||
| Journal or Publication Title: | Transactions of the American Mathematical Society | |||||||||
| Publisher: | American Mathematical Society | |||||||||
| ISSN: | 0002-9947 | |||||||||
| Official Date: | 13 August 2019 | |||||||||
| Dates: |
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| Volume: | 372 | |||||||||
| Page Range: | pp. 7297-7326 | |||||||||
| DOI: | 10.1090/tran/7878 | |||||||||
| Status: | Peer Reviewed | |||||||||
| Publication Status: | Published | |||||||||
| Reuse Statement (publisher, data, author rights): | First published in Transactions of the American Mathematical Society in 372, 2019, published by the American Mathematical Society. | |||||||||
| Access rights to Published version: | Restricted or Subscription Access | |||||||||
| Copyright Holders: | © Copyright 2019 American Mathematical Society | |||||||||
| Date of first compliant deposit: | 29 April 2019 | |||||||||
| Date of first compliant Open Access: | 27 January 2020 | |||||||||
| RIOXX Funder/Project Grant: |
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