Geometric Characterizations of Embedding Theorems: For Sobolev, Besov, and Triebel–Lizorkin Spaces on Spaces of Homogeneous Type—via Orthonormal Wavelets

Han, Yanchang; Han, Yongsheng; He, Ziyi; Li, Ji; Pereyra, Cristina

doi:10.1007/s12220-020-00536-6

Cited by 16 publications

(5 citation statements)

References 38 publications

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“…It was shown in [32] that M s p,q (R n ) coincides with the classical Triebel-Lizorkin space F s p,q (R n ) for any s ∈ (0, 1), p ∈ ( n n+s , ∞), and q ∈ ( n n+s , ∞], and N s p,q (R n ) coincides with the classical Besov space B s p,q (R n ) for any s ∈ (0, 1), p ∈ ( n n+s , ∞), and q ∈ (0, ∞]. We also refer the reader to [14,23,24,27,28,1,21,4] for more information on Triebel-Lizorkin and Besov spaces on quasi-metric measure spaces.…”

Section: Triebel-lizorkin Besov and Sobolev Spacesmentioning

confidence: 99%

A Measure Characterization of Embedding and Extension Domains for Sobolev, Triebel-Lizorkin, and Besov Spaces on Spaces of Homogeneous Type

Alvarado¹,

Yang²,

Wen³

2022

Preprint

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In this article, for an optimal range of the smoothness parameter s that depends (quantitatively) on the geometric makeup of the underlying space, the authors identify purely measure theoretic conditions that fully characterize embedding and extension domains for the scale of Hajłasz-Triebel-Lizorkin spaces M s p,q and Hajłasz-Besov spaces N s p,q in general spaces of homogeneous type. Although stated in the context of quasi-metric spaces, these characterizations improve related work even in the metric setting. In particular, as a corollary of the main results in this article, the authors obtain a new characterization for Sobolev embedding and extension domains in the context of general doubling metric measure spaces.

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Section: Triebel-lizorkin Besov and Sobolev Spacesmentioning

confidence: 99%

A Measure Characterization of Embedding and Extension Domains for Sobolev, Triebel-Lizorkin, and Besov Spaces on Spaces of Homogeneous Type

Alvarado¹,

Yang²,

Wen³

2022

Preprint

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“…Moreover, Wang et al [82] and Alvarado et al [2] obtained, respectively, the difference and the pointwise characterizations of Besov and Triebel-Lizorkin spaces on X . All these results in this article, together with [44], [29], [82], and [2], give a complete real-variable theory of Besov and Triebel-Lizorkin spaces on an arbitrary space X of homogeneous type.…”

Section: Introductionmentioning

confidence: 99%

“…To limit the length of this article, the first, the third, and the fourth authors of this article and Yuan, in another article [44], established wavelet characterizations of these Besov and Triebel-Lizorkin spaces on X and then, via first obtaining the boundedness of almost diagonal operators on sequence Besov and Triebel-Lizorkin spaces on spaces of homogeneous type, they further established the molecular and various Littlewood-Paley function characterizations of Besov and Triebel-Lizorkin spaces on X . Han et al [29] showed that the embedding theorems for Besov and Triebel-Lizorkin spaces on X hold true if and only if the measure µ under study of X has a (local) lower bound. Moreover, Wang et al [82] and Alvarado et al [2] obtained, respectively, the difference and the pointwise characterizations of Besov and Triebel-Lizorkin spaces on X .…”

Section: Introductionmentioning

confidence: 99%

Besov and Triebel–Lizorkin spaces on spaces of homogeneous type with applications to boundedness of Calderón–Zygmund operators

Wang

Han

et al. 2021

Dissertationes Math.

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In this article, the authors introduce Besov and Triebel-Lizorkin spaces on spaces of homogeneous type in the sense of Coifman and Weiss, prove that these (inhomogeneous) Besov and Triebel-Lizorkin spaces are independent of the choices of both (inhomogeneous) approximations of the identity with exponential decay and underlying spaces of distributions, and give some basic properties of these spaces. As applications, the authors show that some known function spaces coincide with certain special cases of Besov and Triebel-Lizorkin spaces and, moreover, obtain the boundedness of Calderón-Zygmund operators on these Besov and Triebel-Lizorkin spaces. All these results strongly depend on the geometrical properties, reflected via dyadic cubes, of the relevant space of homogeneous type. Compared with the known theory of these spaces on metric measure spaces, a major novelty of this article is that all results presented in this article get rid of the dependence on the reverse doubling assumption of the measure under study of the underlying space.

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“…It was recently shown in [2] the lower measure bound (1.2) is actually equivalent to the existence of the Sobolev embeddings for M 1,p . We also refer the reader to [14,19,20,21,27,29,33,35,36,50] for partial or related results.…”

Section: Introductionmentioning

confidence: 99%

Optimal Embeddings for Triebel-Lizorkin and Besov Spaces on Quasi-Metric Measure Spaces

Alvarado¹,

Yang²,

Wen³

2022

Preprint

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In this article, via certain lower bound conditions on the measures under consideration, the authors fully characterize the Sobolev embeddings for the scales of Hajłasz-Triebel-Lizorkin and Hajłasz-Besov spaces in the general context of quasi-metric measure spaces for an optimal range of the smoothness parameter s. An interesting facet of this work is how the range of s for which the above characterizations of these embeddings hold true is intimately linked (in a quantitative manner) to the geometric makeup of the underlying space. Moreover, although stated for Hajłasz-Triebel-Lizorkin and Hajłasz-Besov spaces in the context of quasi-metric spaces, the main results in this article improve known work even for Sobolev spaces in the metric setting.

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Geometric Characterizations of Embedding Theorems: For Sobolev, Besov, and Triebel–Lizorkin Spaces on Spaces of Homogeneous Type—via Orthonormal Wavelets

Cited by 16 publications

References 38 publications

A Measure Characterization of Embedding and Extension Domains for Sobolev, Triebel-Lizorkin, and Besov Spaces on Spaces of Homogeneous Type

A Measure Characterization of Embedding and Extension Domains for Sobolev, Triebel-Lizorkin, and Besov Spaces on Spaces of Homogeneous Type

Besov and Triebel–Lizorkin spaces on spaces of homogeneous type with applications to boundedness of Calderón–Zygmund operators

Optimal Embeddings for Triebel-Lizorkin and Besov Spaces on Quasi-Metric Measure Spaces

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