Hermitian Clifford Analysis on Euclidean Domains with Smooth and Fractal Boundaries: A Survey of Matrix Cauchy and Hilbert Transforms, Boundary Value Problems, Helmholtz-Type Dirac Operators and Duality
Abstract
Hermitian Clifford analysis is a refinement of Euclidean Clifford analysis obtained by imposing a complex structure on R2n and considering the simultaneous null solutions (the h-monogenic functions) of two mutually adjoint Hermitian Dirac operators. This survey collects, in a unified notation and without proofs, the main results obtained along this line of research between 2008 and 2026. We review the matrix Borel--Pompeiu, Koppelman and Cauchy formulae, first for domains with Liapunov or Ahlfors--David regular boundary and then for Jordan domains whose boundary is fractal in the sense of Mandelbrot; the associated Cauchy transform, Hilbert transform, Hardy projections and their mapping properties, including Hölder norm estimates in terms of the d-sum of a Whitney decomposition; the jump decomposition, Dirichlet and Riemann boundary value problems in each setting, including the linear Riemann problems with variable circulant coefficients and their nonlinear counterparts. We also review a duality theory for the module of germs of H-monogenic matrix functions, together with its Euclidean model (the annihilators of harmonic differential forms) and its algebraic prerequisites (a Poincaré lemma and a Mittag--Leffler theorem for circulant matrices); the function theory associated with a Hermitian version of the Helmholtz equation, the corresponding ∂̅-problem and its application to a formal two-dimensional Maxwell-type system; and finally, the parallel development in the quaternionic Hermitian setting, where four Dirac operators and circulant (4× 4) matrices take over.
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BibTeX
@misc{abreublaya2026hermitian,
title = {{Hermitian Clifford Analysis on Euclidean Domains with Smooth and Fractal Boundaries: A Survey of Matrix Cauchy and Hilbert Transforms, Boundary Value Problems, Helmholtz-Type Dirac Operators and Duality}},
author = {Ricardo Abreu-Blaya and Juan Bory-Reyes and Carlos Daniel Tamayo-Castro},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39087},
eprint = {2610.39087},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39087},
}
RIS
TY - UNPB
TI - Hermitian Clifford Analysis on Euclidean Domains with Smooth and Fractal Boundaries: A Survey of Matrix Cauchy and Hilbert Transforms, Boundary Value Problems, Helmholtz-Type Dirac Operators and Duality
AU - Ricardo Abreu-Blaya
AU - Juan Bory-Reyes
AU - Carlos Daniel Tamayo-Castro
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39087
UR - https://arxiv.org/abs/2610.39087
AB - Hermitian Clifford analysis is a refinement of Euclidean Clifford analysis obtained by imposing a complex structure on R^2n and considering the simultaneous null solutions (the h-monogenic functions) of two mutually adjoint Hermitian Dirac operators. This survey collects, in a unified notation and without proofs, the main results obtained along this line of research between 2008 and 2026. We review the matrix Borel--Pompeiu, Koppelman and Cauchy formulae, first for domains with Liapunov or Ahlfors--David regular boundary and then for Jordan domains whose boundary is fractal in the sense of Mandelbrot; the associated Cauchy transform, Hilbert transform, Hardy projections and their mapping properties, including Hölder norm estimates in terms of the d-sum of a Whitney decomposition; the jump decomposition, Dirichlet and Riemann boundary value problems in each setting, including the linear Riemann problems with variable circulant coefficients and their nonlinear counterparts. We also review a duality theory for the module of germs of H-monogenic matrix functions, together with its Euclidean model (the annihilators of harmonic differential forms) and its algebraic prerequisites (a Poincaré lemma and a Mittag--Leffler theorem for circulant matrices); the function theory associated with a Hermitian version of the Helmholtz equation, the corresponding ∂̅-problem and its application to a formal two-dimensional Maxwell-type system; and finally, the parallel development in the quaternionic Hermitian setting, where four Dirac operators and circulant (4× 4) matrices take over.
ER -