Abstract
A bosonic Laplacian is a conformally invariant second order differential operator acting on smooth functions defined on domains in Euclidean space and taking values in higher order irreducible representations of the special orthogonal group, in this case, the irreducible representation spaces of homogeneous harmonic polynomials. In this paper, we study boundary value problems involving bosonic Laplacians in the upper-half space and the unit ball. Poisson kernels in the upper-half space and the unit ball are constructed, which give us solutions to the Dirichlet problems with \(L^p\) boundary data, \(1 \le p \le \infty \). We also prove the uniqueness for solutions to the Dirichlet problems with continuous data for bosonic Laplacians and provide analogs of some properties of harmonic functions for null solutions of bosonic Laplacians, for instance, Cauchy’s estimates, the mean-value property, Liouville’s Theorem, etc.
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References
Axler, S., Bourdon, P., Ramey, W.: Harmonic Function Theory, 2nd edn. Graduate Texts in Mathematics. Springer, New York (2001)
Stein, E.M., Weiss, G.: Generalization of the Cauchy-Riemann equations and representations of the rotation group. Amer. J. Math. 90, 163–196 (1968)
Bargmann, V., Wigner, E.P.: Group theoretical discussion of relativistic wave equations. Proc. Natl. Acad. Sci. USA 34(5), 211–223 (1948)
De Bie, H., Eelbode, D., Roels, M.: The higher spin Laplace operator. Potential Anal. 47(2), 123–149 (2017)
Ding, C., Walter, R., Ryan, J.: Construction of arbitrary order conformally invariant operators in higher spin spaces. J. Geomet. Anal. 27(3), 2418–2452 (2017)
Eelbode, D., Roels, M.: Generalised Maxwell equations in higher dimensions. Complex Anal. Oper. Theo. 10(2), 267–293 (2016)
Bureš, J., Sommen, F., Souček, V., Van Lancker, P.: Rarita-Schwinger Type Operators in Clifford Analysis. J. Funct. Anal. 185(2), 425–455 (2001)
Dunkl, C.F., Li, J., Ryan, J., Van Lancker, P.: Some Rarita-Schwinger type operators. Comput. Methods Funct. Theo. 13(3), 397–424 (2013)
Clerc, J.L., Ørsted, B.: Conformal covariance for the powers of the Dirac operator. Preprint at https://arxiv.org/abs/1409.4983v1 (2014)
Ding, C.: Integral formulas for higher order conformally invariant fermionic operators. Adv. Appl. Clifford Algebras 29(37) (2019). https://doi.org/10.1007/s00006-019-0953-4
Ding, C., Ryan, J.: Some properties of the higher spin Laplace operator. Trans. Am. Math. Soc. 371(5), 3375–3395 (2019)
Ding, C., Nguyen, P.T., Ryan, J.: Polynomial null solutions to bosonic laplacians, bosonic Bergman and Hardy spaces. Proc. Edinb. Math. Soc. (2022). https://doi.org/10.1017/S0013091522000426
Ding, C., Ryan, J.: Green’s formulas and Poisson’s equation for bosonic Laplacians. Math. Methods Appl. Sci. (2020). https://doi.org/10.1002/mma.6922
Bañuelos, R., Moore, C.N.: Probabilistic Behavior of Harmonic Functions. Birkhäuser Basel, Switzerland (1991)
Li, J., Ryan, J., Vanegas, C.J.: Rarita-Schwinger type operators on spheres and real projective space. Arch. Math. 48(4), 271–289 (2012)
Ryan, J.: Dirac operators, conformal transformations and aspects of classical harmonic analysis. J. Lie Theoy 8, 67–82 (1998)
Gilbert, J., Murray, M.: Clifford Algebras and Dirac Operators in Harmonic Analysis. Cambridge University Press, Cambridge (1991)
Narasimhan, R.: Analysis on Real and Complex Manifolds. Elsevier, Amsterdam (1985)
Acknowledgements
Chao Ding and Phuoc-Tai Nguyen are supported by Czech Science Foundation, project GJ19-14413Y.
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C.D. wrote the main manuscript. All authors revised and reviewed the manuscript.
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Ding, C., Nguyen, PT. & Ryan, J. Boundary value problems in Euclidean space for bosonic Laplacians. Complex Anal Synerg 10, 6 (2024). https://doi.org/10.1007/s40627-024-00132-2
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DOI: https://doi.org/10.1007/s40627-024-00132-2