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Transactions of the American Mathematical Society

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A nonconstant coefficients differential operator associated to slice monogenic functions


Authors: Fabrizio Colombo, J. Oscar González-Cervantes and Irene Sabadini
Journal: Trans. Amer. Math. Soc. 365 (2013), 303-318
MSC (2010): Primary 30G35
DOI: https://doi.org/10.1090/S0002-9947-2012-05689-3
Published electronically: July 24, 2012
MathSciNet review: 2984060
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Abstract: Slice monogenic functions have had a rapid development in the past few years. One of the main properties of such functions is that they allow the definition of a functional calculus, called $ S$-functional calculus, for (bounded or unbounded) noncommuting operators. In the literature there exist two different definitions of slice monogenic functions that turn out to be equivalent under suitable conditions on the domains on which they are defined. Both the existing definitions are based on the validity of the Cauchy-Riemann equations in a suitable sense. The aim of this paper is to prove that slice monogenic functions belong to the kernel of the global operator defined by $ G(x):=\vert\underline {x}\vert^2\frac {\partial }{\partial x_0} \ + \ \underline {x} \ \sum _{j=1}^n x_j\frac {\partial }{\partial x_j}, $ where $ \underline {x}$ is the 1-vector part of the paravector $ x=x_0+\underline {x}$ and $ n\in \mathbb{N}$. Despite the fact that $ G$ has nonconstant coefficients, we are able to prove that a subclass of functions in the kernel of $ G$ have a Cauchy formula. Moreover, we will study some relations among the three classes of functions and we show that the kernel of the operator $ G$ strictly contains the functions given by the other two definitions.


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Additional Information

Fabrizio Colombo
Affiliation: Dipartimento di Matematica, Politecnico di Milano, Via E. Bonardi, 9, 20133 Milano, Italy
Email: fabrizio.colombo@polimi.it

J. Oscar González-Cervantes
Affiliation: Departamento de Matemáticas, E.S.F.M. del I.P.N., 07338 México D.F., México
Email: jogc200678@gmail.com

Irene Sabadini
Affiliation: Dipartimento di Matematica, Politecnico di Milano, Via E. Bonardi, 9, 20133 Milano, Italy
Email: irene.sabadini@polimi.it

DOI: https://doi.org/10.1090/S0002-9947-2012-05689-3
Keywords: Slice monogenic functions, global operator associated to slice monogenic functions, partial differential equation with nonconstant coefficients, Cauchy formula.
Received by editor(s): February 16, 2011
Published electronically: July 24, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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