Characterizations of Besov and Triebel–Lizorkin spaces via averages on balls

Let ℓ∈Nℓ∈N and p∈(1,∞]p∈(1,∞]. In this article, the authors prove that the sequence {f−Bℓ,2−kf}k∈Z{f−Bℓ,2−kf}k∈Z consisting of the differences between f   and the ball average Bℓ,2−kfBℓ,2−kf characterizes the Besov space B˙p,qα(Rn) with q∈(0,∞]q∈(0,∞] and the Triebel–Lizorkin space F˙p,qα(Rn) with q∈(1,∞]q∈(1,∞] when the smoothness order α∈(0,2ℓ)α∈(0,2ℓ). More precisely, it is proved that f−Bℓ,2−kff−Bℓ,2−kf plays the same role as the approximation to the identity φ2−k⁎fφ2−k⁎f appearing in the definitions of B˙p,qα(Rn) and F˙p,qα(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)(0,2ℓ) on spaces of homogeneous type, where ℓ∈Nℓ∈N.