A Bourgain–Brezis–Mironescu characterization of higher order Besov–Nikol^{\prime }skii spaces

Abstract

We study a class of nonlocal functionals in the spirit of the recent characterization of the Sobolev spaces W 1,p derived by Bourgain, Brezis and Mironescu. We show that it provides a common roof to the description of the BV (R N ), W 1,p (R N ), B s p,∞ (R N ) and C 0,1 (R N ) scales and we obtain new equivalent characterizations for these spaces. We also establish a noncompactness result for sequences and new (non-)limiting embeddings between Lipschitz and Besov spaces which extend the previous known results.2010 Mathematics Subject Classification. 46E35.

Jump detection in Besov spaces via a new BBM formula. Applications to Aviles–Giga-type functionals

“…We should mention that (1.16) of Proposition 1.1 can be deduced from a more general result, obtained independent by Brasseur in [7], that characterizes the Besov spaces B s p,∞ (R N ) via a BBM-type formula, for all values of s ∈ (0, 1) and p ∈ [1, ∞). Remark 1.4.…”

On restrictions of Besov functions

“…By [8,Lemma 8.2] (in fact in [8] it is implicitly supposed that 1 p < ∞ but the proof still works when 0 < p < 1) and (6.4), we have sup h∈K j ∆ M h f (·, x N ) L p (R N −1 ) c 2 −js Ψ(2 −j ) −1 Λ j (x N ) c 2 −js Ψ(2 −j ) −1 2 j/p λ j, 2 j x N , (6.6) for any j 0 and some c > 0 independent of j. Recall that…”

A Remake on the Bourgain-Brezis-Mironescu Characterization of Sobolev Spaces

“…Our approach in this paper manly rely on elementary calculi and closely follows techniques from [BBM01,Bre02,Dáv02,Pon03]. In the same vein, different results on the characterization of Sobolev spaces can be found in [Pon04, PS17, LS11, Lud14, MS02] see also [Bra18] for characterization of Besov type spaces. We frequently use without any mentioning the convex inequality (a + b) p ≤ 2 p−1 (a p + b p ) for a, b ∈ R. We denote a ∧ b = min(a, b).…”