Slice Fueter-Regular Functions

Slice Fueter-regular functions, originally called slice Dirac-regular functions, are generalized holomorphic functions defined over the octonion algebra O, recently introduced by M. Jin, G. Ren and I. Sabadini. A function f : Omega(D) subset of O -> O is called (quaternionic) slice Fueter-regular if, given any quaternionic subalgebra H-I of O generated by a pair I = (I, J) of orthogonal imaginary units I and J (H-I is a 'quaternionic slice' of O), the restriction of f to H-I belongs to the kernel of the corresponding Cauchy-Riemann-Fueter operator partial derivative/partial derivative x(0) + I partial derivative/partial derivative x(1) + J partial derivative/partial derivative x(2) +(I J)partial derivative/partial derivative x(3). The goal of this paper is to show that slice Fueter-regular functions are standard (complex) slice functions, whose stem functions satisfy a Vekua system having exactly the same form of the one characterizing axially monogenic functions of degree zero. The mentioned standard sliceness of slice Fueter-regular functions is able to reveal their 'holomorphic nature': slice Fueter-regular functions have Cauchy integral formulas, Taylor and Laurent series expansions, and a version of Maximum Modulus Principle, and each of these properties is global in the sense that it is true on genuine 8-dimesional domains of O. Slice Fueter-regular functions are real analytic. Furthermore, we introduce the global concepts of spherical Dirac operator Gamma and of slice Fueter operator (theta) over bar (F) over octonions, which allow to characterize the slice Fueter-regular functions as the l(2)-functions in the kernel of (theta) over bar (F) satisfying a second order differential system associated with Gamma.