Quaternionic ψ-hyperholomorphic functions, singular integral operators and boundary value problems I. ψ-hyperholomorphic function theory

Bloch’s Theorem in the Context of Quaternion Analysis

“…Yet quaternion analysis has become a major research area in mathematics having connections with boundary value problems and partial differential equations theory or other fields of physics and engineering. For a thorough treatment of this function theory, the reader is referred to [14,15,22,23,40,41].…”

Hardy spaces of solutions of generalized Riesz and Moisil–Teodorescu systems

“…For an account of this theory of Hardy spaces in Euclidean space we refer to the monographs [28,35,18], to the lecture notes [34] and the papers [14,16,2,29]. In the same context, the Hilbert transform, as well as more general singular integral operators, have been studied in higher dimensional Euclidean space (see [38,30,20,21,5]), in particular on Lipschitz hypersurfaces (see [36,33,32]) and also on smooth closed hypersurfaces, in particular the unit sphere (see [19,6,13,12]). …”

Hölder norm estimate for a Hilbert transform in Hermitean Clifford analysis

“…Clifford analysis may also be considered as a refinement of harmonic analysis, since, as does the Cauchy-Riemann operator in the complex plane, the rotation-invariant Dirac operator factorizes the Laplacian. The theory of Hardy spaces in Clifford analysis is by now well established, see [16,35,19,7], and the multidimensional Hilbert transform, as well as more general singular integral operators have been studied intensively, see [28,26,35,39,29,18,20], in particular on Lipschitz hypersurfaces, see [32,31,34], and on smooth closed hypersurfaces, such as the unit sphere, see [19,13].…”