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.