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Let (X, %, μ)d,θ be a space of homogeneous type, i.e. X is a set, % is a quasi-metric on X with the property that there are constants θ ∈ (0, 1] and C0 > 0 such that for all x, x′, y ∈ X, |%(x, y)− %(x′, y)| ≤ C0%(x, x′)θ[%(x, y) + %(x′, y)]1−θ, and μ is a nonnegative Borel regular measure on X such that for some d > 0 and all x ∈ X, μ({y ∈ X : %(x, y) < r}) ∼ r. Let ε ∈ (0, θ], |s| < ε and max{d/(d+ ε), d/(d+ s+ ε)} < q ≤ ∞. The author introduces new inhomogeneous Triebel–Lizorkin spaces F s ∞q(X) and establishes their frame characterizations by first establishing a Plancherel–Pólya-type inequality related to the norm ‖ · ‖F s ∞q(X), which completes the theory of function spaces on spaces of homogeneous type. Moreover, the author establishes the connection between the space F s ∞q(X) and the homogeneous Triebel–Lizorkin space Ḟ s ∞q(X). In particular, he proves that bmo(X) coincides with F 0 ∞2(X).

Some new inhomogeneous Triebel–Lizorkin spaces on metric measure spaces and their various characterizations