Characterizations of Besov-Type and Triebel–Lizorkin–Type Spaces via Averages on Balls

Abstract

AbstractLet ℓ ∊ ℕ and α > (§, 2ℓ). In this article, the authors establish equivalent characterizations of Besov-type spaces, Triebel–Lizorkin-type spaces, and Besov–Morrey spaces via the sequence {ƒ-Bl,2-kƒ}k consisting of the diòerence between f and the ball average Bl,2-kƒ. These results lead to the introduction of Besov-type spaces, Triebel–Lizorkin-type spaces, and Besov–Morrey spaceswith any positive smoothness order onmetricmeasure spaces. As special cases, the authors obtain a new characterization ofMorrey–Sobolev spaces and Qα spaces with ∈ > (0, 1), which are of independent interest.