Abstract
Given a monogenic function on the quaternionic algebra \({{\mathbb {H}}}\), the Clifford algebra \({{\mathbb {R}}}_n\) or the octonionic algebra \({{\mathbb {O}}}\) we prove that \(|\nabla ^m f|^\alpha \) is subharmonic for some \(\alpha >0\) where \(\nabla ^m f\) is the mth order gradient of f. We find also the optimal value of \(\alpha \). This is a generalization of a result of Calderon and Zygmund.
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References
Baracco, L.: Extension of holomorphic functions from one side of a hypersurface. Canad. Math. Bull. 48(4), 500–504 (2005)
Baracco, L., Fassina, M., Pinton, S.: An extension theorem for regular functions of two quaternionic variables. J. Math. Anal. Appl. (2020). https://doi.org/10.1016/j.jmaa.2019.123573
Begehr, H.: Iterated integral operators in Clifford analysis. Z. Anal. Anwendungen 18(2), 361–377 (1999)
Calderón, A.-P., Zygmund, A.: On higher gradients of harmonic functions. Studia Math. 24, 211–226 (1964)
Conway, J.H., Smith, D.A.: On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry. A K Peters, Ltd., Natick, xii+159 pp (2003)
Delanghe, R., Sommen, F., Soucek, V.: Clifford Algebra and Spinor-Valued Functions. Kluwer, Dordrecht (1992)
Hörmander, L.: An Introduction to Complex Analysis in Several Variables. North-Holland Mathematical Library, vol. 7, 3rd edn. North-Holland Publishing Co., Amsterdam (1990)
Kauhanen, J., Orelma, H.: Cauchy–Riemann operators in octonionic analysis. Adv. Appl. Clifford Algebr. 28(1), 14 (2018)
Kheyfits, A., Tepper, D.: Subharmonicity of powers of octonion-valued monogenic functions and some applications. Bull. Belg. Math. Soc. Simon Stevin 13(4), 609–617 (2006)
Lávička, R.: Complete orthogonal appell systems for spherical monogenics. Complex Anal. Oper. Theory 6, 477–489 (2012)
Stein, E.M., Weiss, G.: On the theory of harmonic functions of several variables. I. The theory of \(H^p\)-spaces. Acta Math. 103, 25–62 (1960)
Stein, E.M., Weiss, G.: Generalization of the Cauchy–Riemann equations and representations of the rotation group. Am. J. Math. 90, 163–196 (1968)
Acknowledgements
The authors wish to thank Professor Irene Sabadini for useful discussions and the anonymous referees for many useful suggestions which have improved the quality of our paper.
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Baracco, L., Pinton, S. Higher Order Gradients of Monogenic Functions. J Geom Anal 32, 14 (2022). https://doi.org/10.1007/s12220-021-00734-w
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DOI: https://doi.org/10.1007/s12220-021-00734-w