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 $\R^{2n}$ 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 $\bar\partial$-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 $(4\times 4)$ matrices take over.