Some new Triebel-Lizorkin spaces on spaces of homogeneous type and their frame characterizations

Abstract

Let (X,ρ,µ)d,θ be a space of homogeneous type, e ∈ (0, θ], |s| < e and max {d/(d + e),d/(d + s + e)} < q ≤ ∞. The author introduces the new Triebel-Lizorkin spaces F∞q s(X) and establishes the frame characterizations of these spaces by first establishing a Plancherel-Polya-type inequality related to the norm of the spaces F∞q s(X). The frame characterizations of the Besov space Bs pq(X) with |s| < e, max{d/(d + e),d/(d+ s + e)} < p ≤ ∞ and 0 < q ≤ ∞ and the Triebel-Lizorkin space Fs pq(X) with |s| < e, max {d/(d + e),d/(d + s + e)}<p<∞ and max {d/(d + e),d/(d + s + e)} < q ≤ ∞; are also presented. Moreover, the author introduces the new Triebel-Lizorkin spaces bF∞q s(X) and HF∞q s(X) associated to a given para-accretive function b. The relation between the space bF∞q s(X) and the space HF∞q s(X) is also presented. The author further proves that if s = 0 and q = 2, then HF∞q s(X) = F∞q s(X), which also gives a new characterization of the space BMO(X), since F∞q s(X) = BMO(X).