Since 1970’s B. Fuglede and others have been studying finely holomorhic functions, i.e., ‘holomorphic’ functions defined on the so-called fine domains which are not necessarily open in the usual sense. This note is a survey of finely monogenic functions which were introduced in (Lávička, R., A generalisation of monogenic functions to fine domains, preprint.) like a higher dimensional analogue of finely holomorphic functions.

Euclidean Clifford analysis is a higher dimensional function theory studying so–called monogenic functions, i.e. null solutions of the rotation invariant, vector valued, first order Dirac operator $\underline{\partial }$. In the more recent branch Hermitean Clifford analysis, this rotational invariance has been broken by introducing a complex structure $J$ on Euclidean space and a corresponding second Dirac operator ${\underline{\partial }}_J$, leading to the system of equations $\underline{\partial }f=0={\underline{\partial }}_{J}f$ expressing so-called Hermitean monogenicity. The invariance of this...

It is shown that there is a one-to-one correspondence between uniformly bounded holomorphic functions of n complex variables in sectors of ℂⁿ, and uniformly bounded functions of n+1 real variables in sectors of ${ℝ}^{n+1}$ that are monogenic functions in the sense of Clifford analysis. The result is applied to the construction of functional calculi for n commuting operators, including the example of differentiation operators on a Lipschitz surface in ${ℝ}^{n+1}$.

We consider the Dirac-type operators D + a, a is a paravector in the Clifford algebra. For this operator we state a Cauchy-Green formula in the spaces $C^1\left(G\right)$ and $W_p^1\left(G\right)$. Further, we consider the Cauchy problem for this operator.