Extension and tangential CRF conditions in quaternionic analysis
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Extension and tangential CRF conditions in quaternionic analysis Marco Maggesi1 · Donato Pertici1 · Giuseppe Tomassini2 Received: 10 November 2019 / Accepted: 22 February 2020 © Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We prove some extension theorems for quaternionic holomorphic functions in the sense of Fueter. Starting from the existence theorem for the nonhomogeneous Cauchy–Riemann–Fueter problem, we prove that an ℍ-valued function f on a smooth hypersurface in ℍ2 , satisfying suitable tangential conditions, is locally a jump of two ℍ-holomorphic functions. From this, we obtain, in particular, the existence of the solution for the Dirichlet problem with smooth data. We extend these results to the continuous case. In the final part, we discuss the octonion case. Keywords Quaternionic analysis · Cauchy–Riemann–Fueter operator · ℍ-holomorphic functions · Nonhomogeneous Cauchy–Riemann–Fueter system Mathematics Subject Classification 30G35
1 Introduction This paper aims to set forth the methods of complex analysis in the quaternionic analysis in several variables. The main objects of such a theory are the ℍ-holomorphic functions, i.e., those functions f = f (q1 , … , qn ) , q1 , … , qn ∈ ℍ , which are (left) regular in the sense of Fueter with respect to each variable. For the basic results in the quaternionic analysis in one and several variables, we refer to the articles by Sudbery [22] and Pertici [19], respectively. As for a more geometric aspect of the theory, we refer to the book [11] and the rich bibliography quoted there. Coming to the content of the paper, we are dealing with the boundary values and extension problems for ℍ-holomorphic functions. As it is well known, this is one of the * Marco Maggesi [email protected] Donato Pertici [email protected] Giuseppe Tomassini [email protected] 1
Dipartimento di Matematica e Informatica ‘Ulisse Dini’, Viale Morgagni, 67/a, 50134 Florence, Italy
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Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56126 Pisa, Italy
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central themes in complex analysis, which motivated the study of overdetermined systems of linear partial differential equations, the CR geometry, and the theory of extension of “holomorphic objects”. For the sake of simplicity, we restrict ourselves to the case n = 2 , even if most of the main results proved in the paper hold in any dimension. The paper is organized into three sections. In Sect. 2, after fixing the main notations, we define the differential forms dq𝛼 , Dq𝛼 that play a fundamental role, and the Cauchy–Riemann–Fueter operator 𝔇 . As an application of the Cauchy–Fueter formula in one variable [10, 22], we prove a result of “Carleman type” (Proposition 2.1). We also recall the Bochner–Martinelli formula proved in [19], and we show that the Bochner–Martinelli kernel 𝐊BM (q, q0 ) writes as a sum 𝐊BM (q, q0 ) + 𝐊BM (q, q0 )𝗃 , where 𝐊BM (q, q0 ) and 𝐊BM (q, q0 ) are complex differential 1
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