By Cinzia Bisi, Caterina Stoppato (auth.), Graziano Gentili, Irene Sabadini, Michael Shapiro, Franciscus Sommen, Daniele C. Struppa (eds.)

This quantity is meant to assemble vital study effects to the lectures and discussions which came about in Rome, on the INdAM Workshop on diverse Notions of Regularity for capabilities of Quaternionic Variables in September 2010. This quantity will gather contemporary and new effects, that are hooked up to the subject coated in the course of the workshop. The paintings goals at bringing jointly foreign top experts within the box of Quaternionic and Clifford research, in addition to younger researchers drawn to the topic, with the assumption of offering and discussing fresh effects, reading new developments and strategies within the sector and, as a rule, of selling clinical collaboration. specific realization is paid to the presentation of alternative notions of regularity for features of hypercomplex variables, and to the learn of the most positive factors of the theories that they originate.

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42 F. Colombo et al. The domains on which we will consider slice regular functions are described in the definitions below. Definition 2 Let Ω ⊆ H be a domain. We say that Ω is a slice domain (s-domain for short) if Ω ∩ R is non empty and if Ω ∩ C(i) is a domain in C(i) for all i ∈ S2 . Definition 3 Let Ω ⊆ H. We say that Ω is axially symmetric if for every x + iy ∈ Ω we have that x + jy ∈ Ω for all j ∈ S2 . Any axially symmetric open set Ω can be uniquely associated with an open set OΩ ⊆ R2 defined by OΩ := (x, y) ∈ R2 x + yi ∈ Ω, i ∈ S2 .

We indeed have ∂ −m−n E−m−n = ∂ −m−n δ ∗ ∂ m+n δ = Em+n ∗ ∂ m+n δ = δ. So the operator ∂ μ [ . ] eventually is defined for all μ ∈ C, and there holds in distributional sense ∂ μ [Eμ ] = ∂ μ ∂ −μ δ = δ, or, at the level of the operators: ∂ μ ∂ −μ μ∈C = 1. 5 A New Operator Recalling the following distributional boundary values of the conjugate harmonic potentials studied in [7] ⎧ 1 Γ ( m−2k ⎪ 2 ) ∗ ⎪ T−m+2k , k ∈ Z, 2k < m ⎪ m+2k ⎨ a2k−1 = 22k π 2 ⎪ 1 Γ ( m−2k ⎪ 2 ) ∗ ⎪ ⎩ b2k = U−m+2k+1 , k ∈ Z, 2k < m 22k+1 π m+2k+2 2 it becomes clear, in view of the results in Sect.

Sci. 34, 1896–1909 (2011) 13. : The Bergman and Szegö kernels for separately monogenic functions. Z. Anal. Anwend. 9, 97–103 (1990) 14. : Bergman kernels for rectangular domains and multiperiodic functions in Clifford analysis. Math. Methods Appl. Sci. 25, 1509–1526 (2002) 15. : Bergman spaces of higher-dimensional hyperbolic polyhedrontype domains I. Math. Methods Appl. Sci. 29, 85–98 (2006) 54 F. Colombo et al. 16. : On Hilbert modules with reproducing kernel. In: Functional and Theoretical Methods: Partial Differential Equations, Proceedings of the International Symposium, Darmstadt, 1976.

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