Pointwise differentiability of higher order for distributions

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.

33 pages, no figures. Additions and changes in version 2: (1) description of the relation to asymptotic expansions; (2) alternative proof of Theorem E; (3) minor corrections in 2.2, 2.16, 2.23, and 3.7; (4) updates of acknowledgements, references, and affiliations; (5) minor expository improvements. Comments of R. Estrada and a referee induced (1)+(2) and (5), respectively