Abstract
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.
Original language | English |
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Pages (from-to) | 323-354 |
Number of pages | 32 |
Journal | Analysis and PDE |
Volume | 14 |
Issue number | 2 |
DOIs | |
Publication status | Published - 2021 |
Keywords
- Lusin-type approximation
- Poincaré
- Rademacher–Stepanov-type theorem
- asymptotic expansion
- distribution
- higher-order pointwise differentiability
- higher-order rectifiability
- inequality
- Łojasiewicz point value
ASJC Scopus subject areas
- Analysis
- Numerical Analysis
- Applied Mathematics