2015/07/29 by Dai, Feng, Gogatishvili, Amiran, Yang, Dachun +1
#42B35 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Primary 46E35 #Secondary 42B25
paper · doi:10.48550/arxiv.1507.08004
Let ℓ∈ℕ and p∈(1,∞]. In this article, the authors prove that the sequence \f-Bℓ,2-kf\k∈ℤ consisting of the differences between f and the ball average Bℓ,2-kf characterizes the Besov space Bαp,q(\rn) with q∈ (0, ∞] and the Triebel-Lizorkin space Fαp,q(\rn) with q∈ (1,∞] when the smoothness order α∈(0,2ℓ). More precisely, it is proved that f-Bℓ,2-kf plays the same role as the approximation to the identity φ2-k∗ f appearing in the definitions of Bαp,q(\rn) and Fαp,q(\rn). The corresponding results for inhomogeneous Besov and Triebel-Lizorkin spaces are also obtained. These results, for the first time, give a way to introduce Besov and Triebel-Lizorkin spaces with any smoothness order in (0, 2ℓ) on spaces of homogeneous type, where ℓ∈\mathbb N.