The Cauchy Transform for the Hodge/De Rham System and Some of its Properties
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Ricardo Abreu-Blaya
Abstract
We study the analogue of the Cauchy transform for the theory of solutions of the Hodge/de Rham system in the case of a rectifiable surface of integration which additionally satisfies an Ahlfors/David regularity condition and we prove the Cauchy integral formula, the Plemelj/Privalov theorem and the Sokhotski/Plemelj theorem for it, as well as the necessary and sufficient condition for the possibility to extend a given 𝑘-form from such a surface to a harmonic 𝑘-form in the domain. A formula for the square of the singular Cauchy transform is given. The proofs of all these facts are based on a close relation between algebra-valued null-solutions of the Dirac operator in the Euclidean space and hyperholomorphic functions of Clifford analysis.
© Heldermann Verlag
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- The Cauchy Transform for the Hodge/De Rham System and Some of its Properties
- Explicit Bounds for Composition Operators Preserving BMO(ℝ)
- A Dirichlet Problem for the Inhomogeneous Polyharmonic Equation in the Upper Half Plane
- The Monomiality Approach to Multi-Index Polynomials in Several Variables
- On the Global and Local Solution of the Multidimensional Darboux Problem for Some Nonlinear Wave Equations
- Completeness Theorems for Elliptic Equations of Higher Order with Constant Coefficients
- Generating Relations of Hermite–Tricomi Functions Using a Representation of Lie Algebra 𝑇3
- Near Field Representations of the Acoustic Green's Function in a Shallow Ocean with Fluid-Like Seabed
- An Initial-Boundary Value Problem for a Viscous Compressible Flow
- On the Measurability of Additive Functionals
- On BVPS for Strongly Elliptic Systems with Higher Order Boundary Conditions
- An Example of Fractal Singular Homogenization
- Solution of a Goursat Problem in Optimal Domains
Articles in the same Issue
- The Cauchy Transform for the Hodge/De Rham System and Some of its Properties
- Explicit Bounds for Composition Operators Preserving BMO(ℝ)
- A Dirichlet Problem for the Inhomogeneous Polyharmonic Equation in the Upper Half Plane
- The Monomiality Approach to Multi-Index Polynomials in Several Variables
- On the Global and Local Solution of the Multidimensional Darboux Problem for Some Nonlinear Wave Equations
- Completeness Theorems for Elliptic Equations of Higher Order with Constant Coefficients
- Generating Relations of Hermite–Tricomi Functions Using a Representation of Lie Algebra 𝑇3
- Near Field Representations of the Acoustic Green's Function in a Shallow Ocean with Fluid-Like Seabed
- An Initial-Boundary Value Problem for a Viscous Compressible Flow
- On the Measurability of Additive Functionals
- On BVPS for Strongly Elliptic Systems with Higher Order Boundary Conditions
- An Example of Fractal Singular Homogenization
- Solution of a Goursat Problem in Optimal Domains
