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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. 72977326. doi:10.1090/tran/7878

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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^{kj}u$ has $ j$th order $ \mathrm {L}^{n/(nj)}$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, n1\}$, 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
$\displaystyle \Vert D^{kn}u\Vert _{\mathrm {L}^\infty }\leq C\Vert\mathbb{A} u\Vert _{\mathrm {L}^1}$ (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  

Subjects:  Q Science > QA Mathematics  
Divisions:  Faculty of 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:  00029947  
Official Date:  13 August 2019  
Dates: 


Date of first compliant deposit:  29 April 2019  
Volume:  372  
Page Range:  pp. 72977326  
DOI:  10.1090/tran/7878  
Status:  Peer Reviewed  
Publication Status:  Published  
Publisher Statement:  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  
RIOXX Funder/Project Grant: 


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