Real-variable characterizations and their applications to endpoint formulae of fractional Sobolev spaces associated with ball averages and ball Banach function spaces

Let 𝑋 be a ball Banach function space on R 𝑛 , 𝑠 ∈ (0, ∞), and 𝑞 ∈ (0, ∞].In this article, by replacing difference operators with ball average operators, we introduce the fractional Sobolev space 𝑊 𝑠 ,𝑞 associated with 𝑋.Under some mild assumptions on 𝑋, we show that 𝑊 𝑠 ,𝑞 coincides with the Besov space 𝐵 𝑠 𝑋,𝑞 and, as a consequence, we characterize 𝑊 𝑠 ,𝑞 , respectively, in terms of fractional heat semigroups, 𝐾-functionals, and differences.Furthermore, as applications, we obtain the Bourgain-Brezis-Mironescu formula that, for any given 𝑞 ∈ (0, ∞) and 𝛼 ∈ (0, ∞) and for any 𝑓 ∈ 𝑋 with ∥ (-Δ) 2 𝑓 ∥ 𝑋 < ∞,with the positive equivalence constants independent of 𝑓 .In particular, when 𝛼 := 2ℓ with ℓ ∈ N, it holds that, for any 𝑓 ∈ 𝑊 2ℓ,𝑋 ,𝑋 , where 𝑎 ℓ,𝑞,𝑛 is a positive constant depending only on ℓ, 𝑞, and 𝑛.This extends the well-known results obtained by Karadzhov, Milman, and Xiao in [ J. Funct.Anal.221 (2005), 323-339] from Laplacian to the fractional Laplacian.These results are of broad generality and can be applied to various specific function spaces and, even if applied to the Lebesgue space, most results are new.Their proofs fully utilize the close connections among ball average operators, difference operators, and fractional heat semigroups in frequency space, combined with refined kernel estimates in physical space.