The classification problem for the capacities associated with the Besov and Triebel-Lizorkin spaces

David R. Adams

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Displaying similar documents to “The classification problem for the capacities associated with the Besov and Triebel-Lizorkin spaces”

Brézis-Gallouët-Wainger type inequality for Besov-Morrey spaces

Yoshihiro Sawano (2010)

Studia Mathematica

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The aim of the present paper is to obtain an inequality of Brézis-Gallouët-Wainger type for Besov-Morrey spaces. We investigate these spaces in a self-contained manner. Also, we verify that our result is sharp.

Tractable embeddings of Besov spaces into Zygmund spaces

Hans Triebel (2011)

Banach Center Publications

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The paper deals with dimension-controllable (tractable) embeddings of Besov spaces on n-dimensional cubes into Zygmund spaces. This can be expressed in terms of tractability envelopes.

Tractable embeddings of Besov spaces into Zygmund spaces, II

Hans Triebel (2014)

Banach Center Publications

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The paper deals with dimension-controllable (tractable) embeddings of Besov spaces on n-dimensional cubes into Zygmund spaces.

Gelfand and Kolmogorov numbers of embedding of radial Besov and Sobolev spaces

Alicja Gąsiorowska (2011)

Banach Center Publications

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We prove asymptotic formulas for the behavior of Gelfand and Kolmogorov numbers of Sobolev embeddings between Besov and Triebel-Lizorkin spaces of radial distributions. Our method works also for Weyl numbers.

The Besov capacity in metric spaces

Juho Nuutinen (2016)

Annales Polonici Mathematici

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We study a capacity theory based on a definition of Hajłasz-Besov functions. We prove several properties of this capacity in the general setting of a metric space equipped with a doubling measure. The main results of the paper are lower bound and upper bound estimates for the capacity in terms of a modified Netrusov-Hausdorff content. Important tools are γ-medians, for which we also prove a new version of a Poincaré type inequality.

A problem of invariance for Lebesgue measure

James Fickett, Jan Mycielski (1979)

Colloquium Mathematicae

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Homogeneity, non-smooth atoms and Besov spaces of generalised smoothness on quasi-metric spaces

António M. Caetano, Sofia Lopes

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An h-space is a compact set with respect to a quasi-metric and endowed with a Borel measure such that the measure of a ball of radius r is equivalent to h(r), for some function h. Applying an approach introduced by Triebel in [28] we define Besov spaces of generalised smoothness on h-spaces. We describe the techniques and tools used in this construction, namely snowflaked transforms and charts. This approach relies on using what is known for function spaces on some fractal sets, which...

Sharp estimates of the embedding constants for Besov spaces.

David E. Edmunds, W. Desmond Evans, Georgi E. Karadzhov (2006)

Revista Matemática Complutense

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Sharp estimates are obtained for the rates of blow up of the norms of embeddings of Besov spaces in Lorentz spaces as the parameters approach critical values.

Pointwise multiplication in Triebel-Lizorkin spaces.

Winfried Sickel (1993)

Forum mathematicum

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Multilinear analysis on metric spaces

Loukas Grafakos, Liguang Liu, Diego Maldonado, Dachun Yang

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The multilinear Calderón-Zygmund theory is developed in the setting of RD-spaces which are spaces of homogeneous type equipped with measures satisfying a reverse doubling condition. The multiple-weight multilinear Calderón-Zygmund theory in this context is also developed in this work. The bilinear T1-theorems for Besov and Triebel-Lizorkin spaces in the full range of exponents are among the main results obtained. Multilinear vector-valued T1 type theorems on Lebesgue spaces, Besov spaces,...

Carleson embeddings.

Heiming, Helmut J. (1996)

Abstract and Applied Analysis

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Hölder quasicontinuity of Sobolev functions on metric spaces.

Piotr Hajlasz, Juha Kinnunen (1998)

Revista Matemática Iberoamericana

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We prove that every Sobolev function defined on a metric space coincides with a Hölder continuous function outside a set of small Hausdorff content or capacity. Moreover, the Hölder continuous function can be chosen so that it approximates the given function in the Sobolev norm. This is a generalization of a result of Malý [Ma1] to the Sobolev spaces on metric spaces [H1].

Separation conditions on controlled Moran constructions

Antti Käenmäki, Markku Vilppolainen (2008)

Fundamenta Mathematicae

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It is well known that the open set condition and the positivity of the t-dimensional Hausdorff measure are equivalent on self-similar sets, where t is the zero of the topological pressure. We prove an analogous result for a class of Moran constructions and we study different kinds of Moran constructions in this respect.

Variable Sobolev capacity and the assumptions on the exponent

Petteri Harjulehto, Peter Hästö, Mika Koskenoja, Susanna Varonen (2005)

Banach Center Publications

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In a recent article the authors showed that it is possible to define a Sobolev capacity in variable exponent Sobolev space. However, this set function was shown to be a Choquet capacity only under certain assumptions on the variable exponent. In this article we relax these assumptions.

Traces of Besov spaces on fractal h-sets and dichotomy results

António M. Caetano, Dorothee D. Haroske (2015)

Studia Mathematica

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We study the existence of traces of Besov spaces on fractal h-sets Γ with a special focus on assumptions necessary for this existence; in other words, we present criteria for the non-existence of traces. In that sense our paper can be regarded as an extension of Bricchi (2004) and a continuation of Caetano (2013). Closely connected with the problem of existence of traces is the notion of dichotomy in function spaces: We can prove that-depending on the function space and the set Γ-there...

Traces of Besov, Triebel-Lizorkin and Sobolev Spaces on Metric Spaces

Eero Saksman, Tomás Soto (2017)

Analysis and Geometry in Metric Spaces

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We establish trace theorems for function spaces defined on general Ahlfors regular metric spaces Z. The results cover the Triebel-Lizorkin spaces and the Besov spaces for smoothness indices s < 1, as well as the first order Hajłasz-Sobolev space M1,p(Z). They generalize the classical results from the Euclidean setting, since the traces of these function spaces onto any closed Ahlfors regular subset F ⊂ Z are Besov spaces defined intrinsically on F. Our method employs the definitions...

Function algebras of Besov and Triebel-Lizorkin-type

Fares Bensaid, Madani Moussai (2023)

Czechoslovak Mathematical Journal

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We prove that in the homogeneous Besov-type space the set of bounded functions constitutes a unital quasi-Banach algebra for the pointwise product. The same result holds for the homogeneous Triebel-Lizorkin-type space.

Hausdorff operator on Morrey spaces and Campanato spaces

Jianmiao Ruan, Dashan Fan, Hongliang Li (2020)

Czechoslovak Mathematical Journal

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We study the high-dimensional Hausdorff operators on the Morrey space and on the Campanato space. We establish their sharp boundedness on these spaces. Particularly, our results solve an open question posted by E. Liflyand (2013).

Approximation by functions of compact support in Besov-Triebel-Lizorkin spaces on irregular domains

António Caetano (2000)

Studia Mathematica

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General Besov and Triebel-Lizorkin spaces on domains with irregular boundary are compared with the completion, in those spaces, of the subset of infinitely continuously differentiable functions with compact support in the same domains. It turns out that the set of parameters for which those spaces coincide is strongly related to the fractal dimension of the boundary of the domains.