Characterizations of Besov-type and Triebel–Lizorkin-type spaces via maximal functions and local means

Let s∈Rs∈R. In this paper, the authors first establish the maximal function characterizations of the Besov-type space Ḃp,qs,τ(Rn) with p,q∈(0,∞] and τ∈[0,∞)τ∈[0,∞), the Triebel–Lizorkin-type space Ḟp,qs,τ(Rn) with p∈(0,∞)p∈(0,∞), q∈(0,∞]q∈(0,∞] and τ∈[0,∞)τ∈[0,∞), the Besov–Hausdorff space BḢp,qs,τ(Rn) with p∈(1,∞)p∈(1,∞), q∈[1,∞)q∈[1,∞) and τ∈[0,1(max{p,q})′] and the Triebel–Lizorkin–Hausdorff space FḢp,qs,τ(Rn) with p,q∈(1,∞) and τ∈[0,1(max{p,q})′], where t′t′ denotes the conjugate index of t∈[1,∞]t∈[1,∞]. Using this characterization, the authors further obtain the local mean characterizations of these function spaces via functions satisfying the Tauberian condition and establish a Fourier multiplier theorem on these spaces. All these results generalize the existing classical results on Besov and Triebel–Lizorkin spaces by taking τ=0τ=0 and are also new even for QQ spaces and Hardy–Hausdorff spaces.