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- Title
The Bourgain–Brezis–Mironescu formula on ball Banach function spaces.
- Authors
Dai, Feng; Grafakos, Loukas; Pan, Zhulei; Yang, Dachun; Yuan, Wen; Zhang, Yangyang
- Abstract
Let p ∈ [ 1 , ∞) and X be a ball Banach function space on R n with an absolutely continuous norm for which the Hardy–Littlewood maximal operator is bounded on (X 1 / p) ′ , the associate (dual) space of its 1/p-convexification. The purpose of this work is to establish the fundamental formula lim s → 1 - (1 - s) ∫ R n | f (·) - f (y) | p | · - y | n + s p d y 1 p X p = 2 π n - 1 2 Γ p + 1 2 p Γ p + n 2 | ∇ f | X p for any f ∈ X , where Γ is the Gamma function. This identity coincides with the celebrated classical formula of Bourgain, Brezis, and Mironescu [12, 22] when X = L p (R n) , but it is new for general X, in particular for X = L q (R n) ( 1 ≤ p < q < ∞ ). Translation invariance plays a vital role in the proof of this formula in its aforementioned standard proofs in [12, 22]. But translation invariance may not be valid for ball Banach function spaces, nor is there an explicit expression for the associate norm. The authors overcome these obstacles via a key weighted estimate, obtained using fine geometric properties of adjacent systems of dyadic cubes, and Poincaré's inequality. This estimate is then combined with harmonic analysis tools, such as extrapolation and the boundedness of the Hardy–Littlewood maximal operator, to derive the desired formula. Applications of this limiting identity yield new characterizations of ball Banach Sobolev spaces. Explicit spaces X for which these results apply are Morrey spaces, mixed-norm (resp., weighted or variable) Lebesgue spaces, Orlicz(-slice) (or generalized amalgam) spaces, and Lorentz spaces.
- Subjects
FUNCTION spaces; BANACH spaces; LORENTZ spaces; MAXIMAL functions; HARMONIC analysis (Mathematics); GAMMA functions; SOBOLEV spaces
- Publication
Mathematische Annalen, 2024, Vol 388, Issue 2, p1691
- ISSN
0025-5831
- Publication type
Article
- DOI
10.1007/s00208-023-02562-5