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

Dachun Yang

Studia Mathematica, Tome 166 (2005), p. 63-98 / Harvested from The Polish Digital Mathematics Library

Résumé

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, $|ϱ(x,y) - ϱ(x^{\prime },y)| ≤ C₀ϱ(x,x^{\prime })^{θ} [ϱ(x,y) + ϱ(x^{\prime },y)]^{1-θ}$, 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 maxd/(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 $F⁰_{∞2}(X)$.

Détail

Publié le : 2005-01-01

Zbl 1060.42013

EUDML-ID : urn:eudml:doc:285308