TY - JOUR
T1 - Pointwise differentiability of higher-order for distributions
AU - Menne, Ulrich
N1 - Funding Information:
The author is grateful to Professor Guido De Philippis for a discussion on Allard’s strong constancy lemma (seeRemark 3.7), to Professor Bernd Kirchheim for a discussion leading him to find Łojasiewicz’s fundamental papers [Łojasiewicz 1957,1958], to Professor Ricardo Estrada for pointing out the connection to asymptotic expansions (see in particular Remarks2.14and2.15) and for an alternative idea to prove Theorem E, see Remark 3.12, to Dr. Sławomir Kolasi n´ski for his comments on this manuscript, to Mr. Du˜ng Tran The for a very useful list of errata, and to Professor David Preiss for a correspondence on the Radon– Nikodým property. The initial version of this paper (seearXiv:1803.10855v1) was written while the author worked at the University of Leipzig and the Max Planck Institute for Mathematics in the Sciences.
Publisher Copyright:
© 2021
PY - 2021
Y1 - 2021
N2 - For distributions, we build a theory of higher-order pointwise differentiability comprising, for order zero, Łojasiewicz’s notion of point value. Results include Borel regularity of differentials, higher-order rectifiability of the associated jets, a Rademacher–Stepanov-type differentiability theorem, and a Lusin-type approximation. A substantial part of this development is new also for zeroth order. Moreover, we establish a Poincaré inequality involving the natural norms of negative order of differentiability. As a corollary, we characterise pointwise differentiability in terms of point values of distributional partial derivatives.
AB - For distributions, we build a theory of higher-order pointwise differentiability comprising, for order zero, Łojasiewicz’s notion of point value. Results include Borel regularity of differentials, higher-order rectifiability of the associated jets, a Rademacher–Stepanov-type differentiability theorem, and a Lusin-type approximation. A substantial part of this development is new also for zeroth order. Moreover, we establish a Poincaré inequality involving the natural norms of negative order of differentiability. As a corollary, we characterise pointwise differentiability in terms of point values of distributional partial derivatives.
KW - Lusin-type approximation
KW - Poincaré
KW - Rademacher–Stepanov-type theorem
KW - asymptotic expansion
KW - distribution
KW - higher-order pointwise differentiability
KW - higher-order rectifiability
KW - inequality
KW - Łojasiewicz point value
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U2 - 10.2140/apde.2021.14.323
DO - 10.2140/apde.2021.14.323
M3 - Article
AN - SCOPUS:85105179387
SN - 2157-5045
VL - 14
SP - 323
EP - 354
JO - Analysis and PDE
JF - Analysis and PDE
IS - 2
ER -