New characterizations and applications of inhomogeneous Besov and Triebel-Lizorkin spaces on homogeneous type spaces and fractals

Yongsheng Han, and Dachun Yang. New characterizations and applications of inhomogeneous Besov and Triebel-Lizorkin spaces on homogeneous type spaces and fractals. 2002. <http://eudml.org/doc/285948>.

@book{YongshengHan2002,
abstract = {Let d > 0 and θ ∈ (0,1]. We consider homogeneous type spaces, $(X,ϱ,μ)_\{d,θ\}$, which are variants of the well known homogeneous type spaces in the sense of Coifman and Weiss. We introduce fractional integrals and derivatives, and prove that the Besov spaces $B^\{s\}_\{pq\}(X)$ and Triebel-Lizorkin spaces $F^\{s\}_\{pq\}(X)$ have the lifting properties for |s| < θ. Moreover, we give explicit representations for the inverses of these fractional integrals and derivatives. By using these representations, we prove that the fractional integrals and derivatives are independent of the choices of approximations to the identity, and obtain some Poincaré-type inequalities. We also establish frame decompositions of $B^\{s\}_\{pq\}(X)$ and $F^\{s\}_\{pq\}(X)$. Applying these, we obtain estimates of the entropy numbers of compact embeddings between $B^\{s\}_\{pq\}(X)$ or $F^\{s\}_\{pq\}(X)$ when μ(X) < ∞. Parts of these results are new even when $(X,ϱ,μ)_\{d,θ\}$ is the n-dimensional Euclidean space, or a compact d-set, Γ, in ℝⁿ, which includes various kinds of fractals. We also establish some limiting embeddings between these spaces, and by considering spaces $L^\{p\}(log L)_\{a\}(X)$, we then establish some limiting compact embeddings and obtain estimates of their entropy numbers when μ(X) < ∞. We also discuss the relationship between Hajłasz-Sobolev spaces of order 1 and the spaces defined by our methods. Finally, we give some applications of the estimates of the entropy numbers to estimates of eigenvalues of some positive-definite self-adjoint operators related to quadratic forms.},
author = {Yongsheng Han, Dachun Yang},
keywords = {spaces of homogeneous type; metric spaces; fractals; Besov spaces; Triebel-Lizorkin spaces; Sobolev spaces; Calderon reproducing formulae; Hardy-Littlewood maximal function; Calderon-Zygmund operators; fractional integration; fractional derivatives; entropy numbers; atoms; molecules},
language = {eng},
title = {New characterizations and applications of inhomogeneous Besov and Triebel-Lizorkin spaces on homogeneous type spaces and fractals},
url = {http://eudml.org/doc/285948},
year = {2002},
}

TY - BOOK
AU - Yongsheng Han
AU - Dachun Yang
TI - New characterizations and applications of inhomogeneous Besov and Triebel-Lizorkin spaces on homogeneous type spaces and fractals
PY - 2002
AB - Let d > 0 and θ ∈ (0,1]. We consider homogeneous type spaces, $(X,ϱ,μ)_{d,θ}$, which are variants of the well known homogeneous type spaces in the sense of Coifman and Weiss. We introduce fractional integrals and derivatives, and prove that the Besov spaces $B^{s}_{pq}(X)$ and Triebel-Lizorkin spaces $F^{s}_{pq}(X)$ have the lifting properties for |s| < θ. Moreover, we give explicit representations for the inverses of these fractional integrals and derivatives. By using these representations, we prove that the fractional integrals and derivatives are independent of the choices of approximations to the identity, and obtain some Poincaré-type inequalities. We also establish frame decompositions of $B^{s}_{pq}(X)$ and $F^{s}_{pq}(X)$. Applying these, we obtain estimates of the entropy numbers of compact embeddings between $B^{s}_{pq}(X)$ or $F^{s}_{pq}(X)$ when μ(X) < ∞. Parts of these results are new even when $(X,ϱ,μ)_{d,θ}$ is the n-dimensional Euclidean space, or a compact d-set, Γ, in ℝⁿ, which includes various kinds of fractals. We also establish some limiting embeddings between these spaces, and by considering spaces $L^{p}(log L)_{a}(X)$, we then establish some limiting compact embeddings and obtain estimates of their entropy numbers when μ(X) < ∞. We also discuss the relationship between Hajłasz-Sobolev spaces of order 1 and the spaces defined by our methods. Finally, we give some applications of the estimates of the entropy numbers to estimates of eigenvalues of some positive-definite self-adjoint operators related to quadratic forms.
LA - eng
KW - spaces of homogeneous type; metric spaces; fractals; Besov spaces; Triebel-Lizorkin spaces; Sobolev spaces; Calderon reproducing formulae; Hardy-Littlewood maximal function; Calderon-Zygmund operators; fractional integration; fractional derivatives; entropy numbers; atoms; molecules
UR - http://eudml.org/doc/285948
ER -