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## Studia Mathematica

2005 | 167 | 1 | 63-98
Tytuł artykułu

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

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Let $(X,ϱ,μ)_{d,θ}$ be a space of homogeneous type, i.e. X is a set, ϱ 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,
$|ϱ(x,y) - ϱ(x',y)| ≤ C₀ϱ(x,x')^{θ} [ϱ(x,y) + ϱ(x',y)]^{1-θ}$,
and μ is a nonnegative Borel regular measure on X such that for some d > 0 and all x ∈ X,
$μ({y ∈ X: ϱ(x,y) < r}) ∼ r^{d}$.
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 $F⁰_{∞2}(X)$.
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Tom
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63-98
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wydano
2005
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• Department of Mathematics, Beijing Normal University, Beijing 100875, People's Republic of China
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