Vol. 14, No. 2, 2021

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Pointwise differentiability of higher-order for distributions

Ulrich Menne

Vol. 14 (2021), No. 2, 323–354

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.

distribution, higher-order pointwise differentiability, \Lojasiewicz point value, asymptotic expansion, higher-order rectifiability, Rademacher–Stepanov-type theorem, Lusin-type approximation, Poincaré inequality
Mathematical Subject Classification
Primary: 46F10
Secondary: 26B05, 41A58
Received: 28 March 2018
Revised: 17 August 2019
Accepted: 21 November 2019
Published: 20 March 2021
Ulrich Menne
Department of Mathematics
National Taiwan Normal University
Taipei City
Taiwan (R. O. C.)
Mathematical Division
National Center for Theoretical Sciences
Taipei City
Taiwan (R. O. C.)