Localization of nonlocal gradients in various topologies

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Fractional Piola identity and polyconvexity in fractional spaces (2018, methodology): “In [24], a definition of a nonlocal gradient in bounded or unbounded domains, for which (1) is a particular case, is introduced.”
On a new class of fractional partial differential equations (2014, background): “: Brasco and Lindgren [5], Di Castro, Kuusi, and Palatucci [14,15], Korvenpaa, Kuusi, and Palatucci [28], Kuusi, Mingione and Sire [30,31] and Schikorra [39]. 1 Technically the non-local gradients in [33] were dened with integration over a bounded domain, though with suitable modication much of the ”