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Transactions of the American Mathematical Society

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

 
 

 

Critical $ \mathrm{L}^p$-differentiability of $ \mathrm{BV}^\mathbb{A}$-maps and canceling operators


Author: Bogdan Raiţă
Journal: Trans. Amer. Math. Soc. 372 (2019), 7297-7326
MSC (2010): Primary 26B05; Secondary 35J48
DOI: https://doi.org/10.1090/tran/7878
Published electronically: August 13, 2019
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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 before.

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Additional Information

Bogdan Raiţă
Affiliation: University of Warwick, Zeeman Building, Coventry CV4 7HP, United Kingdom
Email: bogdanraita@gmail.com

DOI: https://doi.org/10.1090/tran/7878
Keywords: Approximate differentiability, $\lebe^p$-Taylor expansions, convolution operators, functions with bounded variation, Sobolev spaces, critical embeddings, Sobolev inequalities, linear elliptic systems, canceling operators.
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)
Article copyright: © Copyright 2019 American Mathematical Society