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

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Abstract

Let (X,ρ,µ)d,θ be a space of homogeneous type, ε ∈ (0, θ], |s| < ε and max {d/(d + ε),d/(d + s + ε)} < 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-Pôlya-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| < ε, max{d/(d + ε),d/(d+ s + ε)} < p ≤ ∞ and 0 < q ≤ ∞ and the Triebel-Lizorkin space Fs pq(X) with |s| < ε, max {d/(d + ε),d/(d + s + ε)}<p<∞ and max {d/(d + ε),d/(d + s + ε)} < 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).

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Authors and Affiliations

  1. Department of Mathematics, Beijing Normal University, 100875, Beijing, China

    Dachun Yang

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  1. Dachun Yang
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Correspondence to Dachun Yang.

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Yang, D. Some new Triebel-Lizorkin spaces on spaces of homogeneous type and their frame characterizations. Sci. China Ser. A-Math. 48, 12–39 (2005). https://doi.org/10.1007/BF02942219

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  • DOI: https://doi.org/10.1007/BF02942219

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