Metric characterization of the sum of fractional Sobolev spaces

Rodiac, Rémy;Van Schaftingen, Jean
(2021) Studia Mathematica — Vol. 258, n° 1, p. 27-51 (2021)

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We introduce a non-linear criterion which allows us to determine when a function can be written as a sum of functions belonging to homogeneous fractional spaces: for ℓ∈N∗, s∈(0,1) and p∈[1,+∞), u:Ω→R can be decomposed as u=u1+⋯+uℓ with ui∈W^s,p(Ω) if and only if ∬Ω×Ω min1≤i≤ℓ|u(x)−u(y)|^p|x−y|^(n+sp) dx dy<+∞.
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Rodiac, R., & Van Schaftingen, J. (2021). Metric characterization of the sum of fractional Sobolev spaces. Studia Mathematica, 258(1), 27-51. https://doi.org/10.4064/sm190408-21-4 (Original work published 2021)