Journal ArticleDOI
A Martinelli-Bochner formula on fractal domains
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A new perspective on a Cauchy integral formula for Clifford algebras valued functions on domains with quite smooth boundaries was discussed in this article, where a new perspective was proposed for Clifford integral formulas for valued functions.Abstract:
A new perspective on a Cauchy integral formula for Clifford algebras valued functions on domains with quite smooth boundaries was discussed in [5]. On the other hand, the Cauchy transform associated to Clifford analysis has been involved recently with fractional metric dimensions and fractals, see [1, 2, 3].read more
Citations
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Boundary value problems for Dirac operators and Maxwell's equations in fractal domains
TL;DR: In this paper, boundary value problems for time-harmonic electromagnetic fields on fractal domains were solved in the framework of hypercomplex function theory, in the context of exploiting hypercomplex functions.
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Hermitean Téodorescu transform decomposition of continuous matrix functions on fractal hypersurfaces
TL;DR: In this article, a matricial Hermitean Teodorescu transform is used to solve the problem of -monogenicity of a matrix function defined on the fractal boundary of a domain.
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Extension theorem for complex Clifford algebras-valued functions on fractal domains.
TL;DR: In this article, a monogenic extension theorem of complex Clifford algebras-valued functions over a bounded domain with fractal boundary is obtained, dealing with the class of Holder continuous functions.
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Hölder norm estimate for the Hilbert transform in Clifford analysis
TL;DR: In this paper, the Holder norm of a fractal version of the Hilbert transform in the Clifford analysis context acting from Holder spaces of Clifford algebra valued functions defined on a Jordan domain is estimated.
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Criteria for monogenicity of Clifford algebra-valued functions on fractal domains
TL;DR: In this paper, the results of Abreu Blaya et al. are extended by using a new Clifford Cauchy transform for Jordan domains in the quadratic space with fractal boundaries.
References
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Book
Singular Integrals and Differentiability Properties of Functions.
TL;DR: Stein's seminal work Real Analysis as mentioned in this paper is considered the most influential mathematics text in the last thirty-five years and has been widely used as a reference for many applications in the field of analysis.
Book
Geometric Measure Theory
TL;DR: In this article, Grassmann algebras of a vectorspace have been studied in the context of the calculus of variations, and a glossary of some standard notations has been provided.
Book
Analysis of and on uniformly rectifiable sets
Guy David,Stephen Semmes +1 more
TL;DR: The notion of uniform rectifiability of sets (in a Euclidean space), which emerged only recently, can be viewed in several different ways as mentioned in this paper, as a quantitative and scale-invariant substitute for the classical notion of Rectifiability; as the answer (sometimes only conjecturally) to certain geometric questions in complex and harmonic analysis; as a condition which ensures the parametrizability of a given set, with estimates, but with some holes and self-intersections allowed, as an achievable baseline for information about the structure of a set.
Journal ArticleDOI
The Gauss-Green theorem for fractal boundaries
Jenny Harrison,Alec Norton +1 more
TL;DR: In this paper, the authors extend the validity of (1) to much more general boundaries while still using the ordinary Lebesgue integral; this is the topic of the present paper.
Journal ArticleDOI
Jump problem and removable singularities for monogenic functions
TL;DR: In this article, the jump problem for monogenic functions on the boundary of a Jordan domain in Euclidean spaces is investigated, and sufficient conditions to extend monogenically continuous Clifford algebra valued functions across a hypersurface are proved.