Frame Characterizations of Besov and Triebel–Lizorkin Spaces on Spaces of Homogeneous Type and their Applications

2002 ◽  
Vol 9 (3) ◽  
pp. 567-590
Author(s):  
Dachun Yang
Keyword(s):  
Besov Spaces ◽  
Interpolation Method ◽  
Real Interpolation ◽  
Homogeneous Type ◽  
Spaces Of Homogeneous Type ◽  
Entropy Numbers ◽  
Compact Embeddings

Abstract The author first establishes the frame characterizations of Besov and Triebel–Lizorkin spaces on spaces of homogeneous type. As applications, the author then obtains some estimates of entropy numbers for the compact embeddings between Besov spaces or between Triebel–Lizorkin spaces. Moreover, some real interpolation theorems on these spaces are also established by using these frame characterizations and the abstract interpolation method.

Real Interpolation of Sobolev Spaces on Subdomains of Rn

1978 ◽  
Vol 30 (01) ◽  
pp. 190-214 ◽  
Author(s):  
R. A. Adams ◽  
J. J. F. Fournier
Keyword(s):  
Sobolev Space ◽  
Fractional Order ◽  
Sobolev Spaces ◽  
Besov Spaces ◽  
Interpolation Method ◽  
A Priori ◽  
Real Interpolation ◽  
Convenient Tool ◽  
Real Interpolation Method ◽  
Nikolskii Spaces

The real interpolation method is a very convenient tool in the study of imbedding relationships among Sobolev spaces and some of their fractional order generalizations, (Besov spaces, Nikolskii spaces etc.) Central to the application of these methods is the a priori determination that a given Sobolev space Wk'p(Ω) belongs to an appropriate class of spaces intermediate between two other “extreme” spaces.

scholarly journals Besov spaces on spaces of homogeneous type and fractals

2003 ◽  
Vol 156 (1) ◽  
pp. 15-30 ◽  
Author(s):  
Dachun Yang
Keyword(s):  
Besov Spaces ◽  
Homogeneous Type ◽  
Spaces Of Homogeneous Type

A Plancherel–Polya Inequality in Besov Spaces on Spaces of Homogeneous Type

2018 ◽  
Vol 29 (2) ◽  
pp. 1571-1582
Author(s):  
Philippe Jaming ◽  
Felipe Negreira
Keyword(s):  
Besov Spaces ◽  
Homogeneous Type ◽  
Spaces Of Homogeneous Type

scholarly journals Smooth Decompositions of Triebel-Lizorkin and Besov Spaces on Product Spaces of Homogeneous Type

2014 ◽  
Vol 2014 ◽  
pp. 1-15
Author(s):  
Fanghui Liao ◽  
Zongguang Liu ◽  
Xiaojin Zhang
Keyword(s):  
Besov Spaces ◽  
Homogeneous Type ◽  
Product Spaces ◽  
Spaces Of Homogeneous Type ◽  
Calderón’S Reproducing Formula

We introduce Triebel-Lizorkin and Besov spaces by Calderón’s reproducing formula on product spaces of homogeneous type. We also obtain smooth atomic and molecular decompositions for these spaces.

scholarly journals Trace operators on fractals, entropy and approximation numbers

2011 ◽  
Vol 18 (3) ◽  
pp. 549-575
Author(s):  
Cornelia Schneider
Keyword(s):  
Besov Spaces ◽  
Trace Operator ◽  
Approximation Numbers ◽  
Atomic Decompositions ◽  
Sharp Estimates ◽  
Compact Embeddings ◽  
Trace Space

Abstract First we compute the trace space of Besov spaces – characterized via atomic decompositions – on fractals Γ, for parameters 0 < p < ∞, 0 < q ≤ min(1, p) and s = (n – d)/p. New Besov spaces on fractals are defined via traces for 0 < p, q ≤ ∞, s ≥ (n – d)/p and some embedding assertions are established. We conclude by studying the compactness of the trace operator TrΓ by giving sharp estimates for entropy and approximation numbers of compact embeddings between Besov spaces. Our results on Besov spaces remain valid considering the classical spaces defined via differences. The trace results are used to study traces in Triebel–Lizorkin spaces as well.

Entropy numbers of embedding maps between Besov spaces with an application to eigenvalue problems

1981 ◽  
Vol 90 (1-2) ◽  
pp. 63-70 ◽  
Author(s):  
Bernd Carl
Keyword(s):  
Banach Space ◽  
Asymptotic Behaviour ◽  
Besov Spaces ◽  
Eigenvalue Problems ◽  
Function Spaces ◽  
Sequence Spaces ◽  
Entropy Numbers ◽  
Diagonal Operators

SynopsisIn this paper we determine the asymptotic behaviour of entropy numbers of embedding maps between Besov sequence spaces and Besov function spaces. The results extend those of M. Š. Birman, M. Z. Solomjak and H. Triebel originally formulated in the language of ε-entropy. It turns out that the characterization of embedding maps between Besov spaces by entropy numbers can be reduced to the characterization of certain diagonal operators by their entropy numbers.Finally, the entropy numbers are applied to the study of eigenvalues of operators acting on a Banach space which admit a factorization through embedding maps between Besov spaces.The statements of this paper are obtained by results recently proved elsewhere by the author.

Hardy spaces on ZN

2002 ◽  
Vol 132 (1) ◽  
pp. 25-43 ◽  
Author(s):  
Santiago Boza ◽  
María J. Carro
Keyword(s):  
Maximal Function ◽  
Hardy Spaces ◽  
Atomic Decomposition ◽  
Classical Case ◽  
Positive Answer ◽  
Riesz Transforms ◽  
Homogeneous Type ◽  
Spaces Of Homogeneous Type ◽  
Definition Of ◽  
Open Question

The work of Coifman and Weiss concerning Hardy spaces on spaces of homogeneous type gives, as a particular case, a definition of Hp(ZN) in terms of an atomic decomposition.Other characterizations of these spaces have been studied by other authors, but it was an open question to see if they can be defined, as it happens in the classical case, in terms of a maximal function or via the discrete Riesz transforms.In this paper, we give a positive answer to this question.

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