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

Dachun Yang. "Some new inhomogeneous Triebel-Lizorkin spaces on metric measure spaces and their various characterizations." Studia Mathematica 167.1 (2005): 63-98. <http://eudml.org/doc/285308>.

@article{DachunYang2005,
abstract = {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 $F⁰_\{∞2\}(X)$.},
author = {Dachun Yang},
journal = {Studia Mathematica},
keywords = {space of homogeneous type; Plancherel-Pólya inequality; Triebel-Lizorkin space; Calderón reproducing formula; bmo},
language = {eng},
number = {1},
pages = {63-98},
title = {Some new inhomogeneous Triebel-Lizorkin spaces on metric measure spaces and their various characterizations},
url = {http://eudml.org/doc/285308},
volume = {167},
year = {2005},
}

TY - JOUR
AU - Dachun Yang
TI - Some new inhomogeneous Triebel-Lizorkin spaces on metric measure spaces and their various characterizations
JO - Studia Mathematica
PY - 2005
VL - 167
IS - 1
SP - 63
EP - 98
AB - 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 $F⁰_{∞2}(X)$.
LA - eng
KW - space of homogeneous type; Plancherel-Pólya inequality; Triebel-Lizorkin space; Calderón reproducing formula; bmo
UR - http://eudml.org/doc/285308
ER -