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

Caetano, António M.; Lopes, Sofia

doi:10.4064/dm460-0-1

Cited by 3 publications

(3 citation statements)

References 24 publications

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“…There are only few forerunners dealing with non-smooth atomic decompositions in function spaces so far. We refer to the papers [44], [26], and [4], all mainly considering the different Fourier-analytic approach for Besov spaces and having in common that they restrict themselves to the technically simpler case when p = q. Our approach generalizes and extends these results and seems to be the first one covering the full range of indices 0 < p, q ≤ ∞.…”

Section: Introductionmentioning

confidence: 80%

Non-smooth atomic decompositions, traces on Lipschitz domains, and pointwise multipliers in function spaces

Schneider

Vybíral

2013

Journal of Functional Analysis

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We provide non-smooth atomic decompositions for Besov spaces B s p,q (R n ), s > 0, 0 < p, q ≤ ∞, defined via differences. The results are used to compute the trace of Besov spaces on the boundary Γ of bounded Lipschitz domains Ω with smoothness s restricted to 0 < s < 1 and no further restrictions on the parameters p, q. We conclude with some more applications in terms of pointwise multipliers.Math Subject Classifications (MSC2010): 46E35, 42B35, 47B38.

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Section: Introductionmentioning

confidence: 80%

Non-smooth atomic decompositions, traces on Lipschitz domains, and pointwise multipliers in function spaces

Schneider

Vybíral

2013

Journal of Functional Analysis

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“…For the definition and several characterisations and properties of Besov spaces of generalised smoothness on R n we refer to [3,8,10,11,15,18] by Bricchi [2,19,21] and [7].…”

Section: Remark 316mentioning

confidence: 99%

“…The spaces presented in Definition 3.15 are included in the Besov spaces on h-sets considered in [13], which are particular class of Besov spaces on fractal h-sets considered in [3,16] and [7]. Generally, given an admissible sequence σ with s(σ > 0) (we refer to Definition 3.7 and Remark 3.8), 0 < p, q ≤ ∞ and an h-set ⊂ R n such that s(h) > −n, then…”

Section: Remark 316mentioning

confidence: 99%

The fractal Dirichlet Laplacian

Caetano

Lopes

2010

Rev Mat Complut

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An h-set is a non-empty compact subset of the Euclidean n-space which supports a finite Radon measure for which the measure of balls centered on the subset is essentially given by the image of their radius by a suitable function h. In most cases of interest such a subset has Lebesgue measure zero and has a fractal structure.We prove the existence of solutions for the so-called fractal Dirichlet problem for such h-sets.The "construction" of the solutions is based on the consideration of complete o.n. systems for convenient function spaces on the fractals. These systems are obtained combining properties of traces with integral representations (obtained with the help of Green's functions) for the so-called fractal Laplacian.

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Introduction

Schneider

2021

Beyond Sobolev and Besov

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

Cited by 3 publications

References 24 publications

Non-smooth atomic decompositions, traces on Lipschitz domains, and pointwise multipliers in function spaces

Non-smooth atomic decompositions, traces on Lipschitz domains, and pointwise multipliers in function spaces

The fractal Dirichlet Laplacian

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