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 happened in Rome, on the INdAM Workshop on varied Notions of Regularity for services of Quaternionic Variables in September 2010. This quantity will acquire fresh and new effects, that are hooked up to the subject lined through the workshop. The paintings goals at bringing jointly foreign top experts within the box of Quaternionic and Clifford research, in addition to younger researchers attracted to the topic, with the belief of featuring and discussing contemporary effects, examining new traits and strategies within the region and, as a rule, of marketing clinical collaboration. specific realization is paid to the presentation of alternative notions of regularity for capabilities of hypercomplex variables, and to the learn of the most gains of the theories that they originate.
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This result assures that the following definition is well posed. Definition 8 (Slice Bergman kernel of the second kind associated with Ω) Let KΩj (·,·) : Ωj × Ωj → H be the Bergman kernel associated with Ωj . We will call slice Bergman kernel of the second kind associated with Ω the function KΩ : Ω × Ω → H, defined by 1 1 KΩ (x + yi, r) := (1 − ij)KΩj (x + yj, r) + (1 + ij)KΩj (x − yj, r). 2 2 Proposition 7 (Properties of the slice Bergman kernel of the second kind associated with Ω) Let Ω ⊂ H be an axially symmetric s-domain.
This is indeed the case since a2k = −F2k+1 = −∂ −2k−1 H, k ∈ Z, 2k + 1 < m b2k−1 = F2k = ∂ −2k H, k ∈ Z, 2k − 1 < m 32 F. Brackx et al. We conclude that all distributional boundary values of the sequence of conjugate harmonic potentials of Sect. 3 are fundamental solutions of ∂ μ and μ H for specific integer values of μ. 6 Powers of the Laplace Operator For complex powers of the Laplace operator the standard definition (see ) reads (−Δm )β [ . ] = (−Δm )β δ ∗ [ . ], where the convolution kernel (−Δm )β δ is given by (−Δm )β δ = 22β Γ ( m+2β 2 ) π m−2β 2 ∗ T−m−2β .
We conclude by noting that the slice Bergman kernel of the second kind KΩ can be computed in closed form whenever it is known the corresponding slice Bergman kernel KΩi . For example, it is possible to construct KΩ when Ω is the unit ball in H thus overcoming the difficulties shown in Corollary 1. The knowledge of a closed form of the kernel is useful for several reasons and has been used in  to construct the Bergman-Sce transform in the setting of slice monogenic functions. We will postpone to a forthcoming paper a more detailed study of the slice Bergman kernel of the second kind as well as the comparison of the two theories of first and second kind.
Advances in Hypercomplex Analysis by Cinzia Bisi, Caterina Stoppato (auth.), Graziano Gentili, Irene Sabadini, Michael Shapiro, Franciscus Sommen, Daniele C. Struppa (eds.)
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