Abstract
The aim of this paper is to obtain a generalized CK-extension theorem in superspace for the biaxial Dirac operator \(\partial _{\mathbf{x}} +\partial _{\mathbf{y}}\). In the classical commuting case, this result can be written as a power series of Bessel type of certain differential operators acting on a single initial function. In the superspace setting, novel structures appear in the cases of negative even superdimensions. In these cases, the CK-extension depends on two initial functions on which two power series of differential operators act. These series are not only of Bessel type but they give rise to an additional structure in terms of Appell polynomials. This pattern also is present in the structure of the Pizzetti formula, which describes integration over the supersphere in terms of differential operators. We make this relation explicit by studying the decomposition of the generalized CK-extension into plane waves integrated over the supersphere. Moreover, these results are applied to obtain a decomposition of the Cauchy kernel in superspace into monogenic plane waves, which shall be useful for inverting the super Radon transform.
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Acknowledgements
The author gratefully acknowledges the many helpful suggestions of Frank Sommen, Irene Sabadini, Juan Bory-Reyes, Michael Wutzig and Hendrik De Bie during the preparation of the paper. The author is supported by a B.O.F. postdoctoral grant from Ghent University with grant number BOF18/PDO/073.
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Guzmán Adán, A. Generalized Cauchy–Kovalevskaya extension and plane wave decompositions in superspace. Annali di Matematica 200, 1417–1450 (2021). https://doi.org/10.1007/s10231-020-01043-9
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DOI: https://doi.org/10.1007/s10231-020-01043-9