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
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.

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Keywords
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
Milestones
Received: 28 March 2018
Revised: 17 August 2019
Accepted: 21 November 2019
Published: 20 March 2021
Authors
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.)