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

This quantity is meant to gather vital examine effects to the lectures and discussions which came about in Rome, on the INdAM Workshop on varied Notions of Regularity for capabilities of Quaternionic Variables in September 2010. This quantity will acquire contemporary and new effects, that are hooked up to the subject lined through the workshop. The paintings goals at bringing jointly overseas best experts within the box of Quaternionic and Clifford research, in addition to younger researchers attracted to the topic, with the belief of offering and discussing fresh effects, reading new developments and methods within the quarter and, often, of selling medical collaboration. specific awareness is paid to the presentation of other notions of regularity for capabilities of hypercomplex variables, and to the examine of the most positive factors of the theories that they originate.

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**Additional info for Advances in Hypercomplex Analysis**

**Example text**

Indeed, the purpose of this paper is to prove an analog of the Bloch-Landau Theorem for slice regular functions. In the complex case, this result is an important fact in the study of the range of holomorphic functions defined on the open unit disc D.

Proof It follows by standard arguments as in the complex case. Remark 4 Let Ω be a given bounded axially symmetric s-domain in H. • Theorem 3 implies that the Bergman spaces A (Ωi ), for i ∈ S2 , are quaternionic right linear Hilbert spaces. • Since f|Ωi = F + Gj where F, G : Ωi → C(i) are holomorphic functions and j ⊥ i, then for f|Ωi ∈ A (Ωi ) there holds: F, G ∈ A (Ωi ). This follows from the inequalities ⎫ ⎪ |F |2 dσi ⎪ ⎬ Ωi ≤ f|Ωi 2A (Ωi ) . ⎪ 2 ⎪ |G| dσi ⎭ Ωi Proposition 1 and Remark 4 imply that given any q ∈ Ωi the evaluation functional φq : A (Ωi ) → H, given by φq [f ] := f (q), ∀f ∈ A (Ωi ), is a bounded quaternionic right-linear functional on A (Ωi ) for every i ∈ S2 .

By a straightforward calculation, similar to the one in the proof of Proposition 2, the following fundamental property is proven. Proposition 7 For α, β ∈ C such that α, β and α + β are different from − m2 − n, n = 0, 1, 2, . . one has (−Δm )α δ ∗ (−Δm )β δ = (−Δm )α+β δ m+4 Corollary 3 For β ∈ C\{± m2 , ± m+2 2 , ± 2 , . } one has (−Δm )β δ ∗ (−Δm )−β δ = δ m+4 Now putting for β ∈ C\{ m2 , m+2 2 , 2 , . } Kβ = (−Δm )−β δ = 2−β Γ ( m−2β 2 ) π m+2β 2 ∗ T−m+2β Boundary Values of Harmonic Potentials in Half-Space 33 and in particular for integer k ⎧ 1 Γ ( m−2k ⎪ −k 2 ) ∗ ⎪ = (−Δ ) δ = T−m+2k , 2k < m when m is even K ⎪ k m ⎪ m+2k 2k ⎪ 2 ⎨ π 2 ) ∗ 1 Γ ( m−2k−1 −k− 12 2 ⎪ ⎪ K δ = T−m+2k+1 , 1 = (−Δm ) ⎪ m+2k+1 k+ 2 2k+1 ⎪ 2 ⎪ π 2 ⎩ 2k < m − 1 when m is odd the above Corollary 3 implies that (−Δm )β [Kβ ] = δ, β ∈ C\ ± (9) m m+2 m+4 ,± ,± ,...