On some structural sets and a quaternionic ‐hyperholomorphic function theory

Abstract

Quaternionic analysis is regarded as a broadly accepted branch of classical analysis referring to many different types of extensions of the Cauchy‐Riemann equations to the quaternion skew field double-struckH. It relies heavily on results on functions defined on domains in R4(or R3) with values in double-struckH. This theory is centred around the concept of ψ‐hyperholomorphic functions related to a so‐called structural set ψ of H4(or H3) respectively. The main goal of this paper is to develop the nucleus of the (φ,ψ)‐hyperholomorphic function theory, i.e., simultaneous null solutions of two Cauchy‐Riemann operators associated to a pair φ,ψ of structural sets of double-struckH4. Following a matrix approach, a generalized Borel‐Pompeiu formula and the corresponding Plemelj‐Sokhotzki formulae are established.

On the Π-operator in Clifford analysis

“…The well-known theorem of Hadamard [2], which states, that a function w(z), where the Jacobian determinant is not equal to zero in any point and w(z) → ∞ if |z| → ∞, realizes a global homeomorphism, together with our norm estimate of the ϕ,ψ Π-operator (34) allows us to establish the following theorem.…”

“…Let ϕ, ψ, ϑ be three arbitrary structural sets and f ∈ W 1 p (Ω, R 0,n ), 1 < p < ∞. Then, the following equalities 2 …”

“…If we denote the space of k-vectors by R (k) 0,n , then R 0,n = n k=0 ⊕R (k) 0,n . In such a way, the spaces R and R n will be identified with R (0) 0,n and R (1) 0,n respectively. Moreover, each element x = (x 0 , x 1 , .…”

“…Recently, in [1] there was given such a Borel-Pompeiu representation formula, but in the context of quaternionic analysis. In this paper we offer direct extension of the formula to the Clifford analysis setting.…”