Densely defined non‐closable curl on carpet‐like metric measure spaces

Hinz, Michael; Teplyaev, Alexander

doi:10.1002/mana.201600467

Cited by 5 publications

(5 citation statements)

References 52 publications

(152 reference statements)

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“…Thus, in some sense, they are analogues of lines which have zero curvature. From a different perspective this corresponds to the Hodge-type theorems in [42,43] and Liouvilletype theorems [36,45] and [67,Introduction and Section 4]. The curvature interpretation of wBE(κ) will only manifest itself in higher dimension when d H > d W , and this is why Subsection 3.3 is important, as it allows us to construct higher dimensional examples satisfying wBE(κ).…”

Section: Weak Bakry-émery Nonnegative Curvature Conditionmentioning

confidence: 99%

Besov class via heat semigroup on Dirichlet spaces III: BV functions and sub-Gaussian heat kernel estimates

Ruiz¹,

Baudoin²,

Chen³

et al. 2019

Preprint

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With a view toward fractal spaces, by using a Korevaar-Schoen space approach, we introduce the class of bounded variation (BV) functions in a general framework of strongly local Dirichlet spaces with a heat kernel satisfying sub-Gaussian estimates. Under a weak Bakry-Émery curvature type condition, which is new in this setting, this BV class is identified with a heat semigroup based Besov class. As a consequence of this identification, properties of BV functions and associated BV measures are studied in detail. In particular, we prove co-area formulas, global L 1 Sobolev embeddings and isoperimetric inequalities. It is shown that for nested fractals or their direct products the BV class we define is dense in L 1 . The examples of the unbounded Vicsek set, unbounded Sierpinski gasket and unbounded Sierpinski carpet are discussed.

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Section: Weak Bakry-émery Nonnegative Curvature Conditionmentioning

confidence: 99%

Besov class via heat semigroup on Dirichlet spaces III: BV functions and sub-Gaussian heat kernel estimates

Ruiz¹,

Baudoin²,

Chen³

et al. 2019

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Thus, in some sense, they are analogues of lines which have zero curvature. From a different perspective this corresponds to the Hodge-type theorems in [49,50] and Liouville-type theorems [39,52] and [76,Introduction and Section 4]. The curvature interpretation of w B E(κ) will only manifest itself in higher dimension when d H > d W , and this is why Sect.…”

Section: Weak Bakry-émery Nonnegative Curvature Conditionmentioning

confidence: 98%

Besov class via heat semigroup on Dirichlet spaces III: BV functions and sub-Gaussian heat kernel estimates

et al. 2021

Self Cite

View full text Add to dashboard Cite

With a view toward fractal spaces, by using a Korevaar-Schoen space approach, we introduce the class of bounded variation (BV) functions in the general framework of strongly local Dirichlet spaces with a heat kernel satisfying sub-Gaussian estimates. Under a weak Bakry-Émery curvature type condition, which is new in this setting, this BV class is identified with a heat semigroup based Besov class. As a consequence of this identification, properties of BV functions and associated BV measures are studied in detail. In particular, we prove co-area formulas, global L 1 Sobolev embeddings and isoperimetric inequalities. It is shown that for nested fractals or their direct products the BV class we define is dense in L 1 . The examples of the unbounded Vicsek set, unbounded Sierpinski gasket and unbounded Sierpinski carpet are discussed.

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“…Apart from these motivations, objective (a) should also be seen as a key step towards a more general theory of 'differential forms' based on Dirichlet forms (local, non-local or mixed). In view of known results for first order forms, [27,53,56,57,58,84,85,95] and related results for higher order forms under somewhat different hypotheses, [43], such a theory seems desirable.…”

Section: Introductionmentioning

confidence: 99%

A tensor product approach to non-local differential complexes

Hinz¹,

Kommer²

2022

Preprint

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We study differential complexes of Kolmogorov-Alexander-Spanier type on metric measure spaces associated with unbounded non-local operators, such as operators of fractional Laplacian type. We define Hilbert complexes, observe invariance properties and obtain self-adjoint non-local analogues of Hodge Laplacians. For d-regular measures and operators of fractional Laplacian type we provide results on removable sets in terms of Hausdorff measures. We prove a Mayer-Vietoris principle and a Poincaré lemma and verify that in the compact Riemannian manifold case the deRham cohomology can be recovered. Contents1. Introduction 1 2. Preliminaries 4 3. Complexes of elementary functions 5 4. Kernels and measures 9 5. Non-local Hilbert complexes 13 5.1. Regularity and density 13 5.2. Closed extensions 14 5.3. Remarks on invariance 17 5.4. Non-local Hodge Laplacians 19 5.5. Approximation results 21 5.6. Removable sets 23 6. Covers and cohomology 27 6.1. Partitions of unity 27 6.2. Mayer-Vietoris sequences 28 6.3. Poincaré lemma 31 6.4. Recovering deRham cohomology 36 7. Some basic examples 37 References 39

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Densely defined non‐closable curl on carpet‐like metric measure spaces

Cited by 5 publications

References 52 publications

Besov class via heat semigroup on Dirichlet spaces III: BV functions and sub-Gaussian heat kernel estimates

Besov class via heat semigroup on Dirichlet spaces III: BV functions and sub-Gaussian heat kernel estimates

Besov class via heat semigroup on Dirichlet spaces III: BV functions and sub-Gaussian heat kernel estimates

A tensor product approach to non-local differential complexes

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