We prove that the local version of the chain rule cannot hold for the fractional variation defined in in our previous article (2019). In the case � = 1 n=1, we prove a stronger result, exhibiting a function � ∈ � � � ( � ) f∈BV α (R) such that ∣ � ∣ ∉ � � � ( � ) ∣f∣∈ / BV α (R). The failure of the local chain rule is a consequence of some surprising rigidity properties for non-negative functions with bounded fractional variation which, in turn, are derived from a fractional Hardy inequality localized to half-spaces. Our approach exploits the distributional techniques developed in our previous works (2019–2022). As a byproduct, we refine the fractional Hardy inequality obtained in works of Shieh and Spector (2018) and Spector (2020) and we prove a fractional version of the closely related Meyers–Ziemer trace inequality.

Failure of the local chain rule for the fractional variation

Comi, Giovanni E.
;
2023

Abstract

We prove that the local version of the chain rule cannot hold for the fractional variation defined in in our previous article (2019). In the case � = 1 n=1, we prove a stronger result, exhibiting a function � ∈ � � � ( � ) f∈BV α (R) such that ∣ � ∣ ∉ � � � ( � ) ∣f∣∈ / BV α (R). The failure of the local chain rule is a consequence of some surprising rigidity properties for non-negative functions with bounded fractional variation which, in turn, are derived from a fractional Hardy inequality localized to half-spaces. Our approach exploits the distributional techniques developed in our previous works (2019–2022). As a byproduct, we refine the fractional Hardy inequality obtained in works of Shieh and Spector (2018) and Spector (2020) and we prove a fractional version of the closely related Meyers–Ziemer trace inequality.
2023
Comi, Giovanni E.; Stefani, Giorgio
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11585/915524
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