Some properties of the higher spin Laplace operator

Ding, Chao; Ryan, John

doi:10.1090/tran/7404

Some properties of the higher spin Laplace operator
HTML articles powered by AMS MathViewer

by Chao Ding and John Ryan PDF

Trans. Amer. Math. Soc. 371 (2019), 3375-3395 Request permission

Abstract:

The higher spin Laplace operator has been constructed recently as the generalization of the Laplacian in higher spin theory. This acts on functions taking values in arbitrary irreducible representations of the Spin group. In this paper, we first provide a decomposition of the higher spin Laplace operator in terms of Rarita-Schwinger operators. With such a decomposition, a connection between the fundamental solutions for the higher spin Laplace operator and the fundamental solutions for the Rarita-Schwinger operators is provided. Further, we show that the two components in this decomposition are conformally invariant differential operators. An alternative proof for the conformal invariance property is also pointed out, which can be connected to Knapp-Stein intertwining operators. Last but not least, we establish a Borel-Pompeiu type formula for the higher spin Laplace operator. As an application, we give a Green type integral formula.

References

Hendrik De Bie, David Eelbode, and Matthias Roels, The higher spin Laplace operator, Potential Anal. 47 (2017), no. 2, 123–149. MR 3669261, DOI 10.1007/s11118-016-9609-3
H. Begehr, Iterated integral operators in Clifford analysis, Z. Anal. Anwendungen 18 (1999), no. 2, 361–377. MR 1701359, DOI 10.4171/ZAA/887
Heinrich Begehr, Zhang Zhongxiang, and Vu Thi Ngoc Ha, Generalized integral representations in Clifford analysis, Complex Var. Elliptic Equ. 51 (2006), no. 8-11, 745–762. MR 2234095, DOI 10.1080/17476930600672940
F. Brackx, Richard Delanghe, and F. Sommen, Clifford analysis, Research Notes in Mathematics, vol. 76, Pitman (Advanced Publishing Program), Boston, MA, 1982. MR 697564
J. Bureš, F. Sommen, V. Souček, and P. Van Lancker, Rarita-Schwinger type operators in Clifford analysis, J. Funct. Anal. 185 (2001), no. 2, 425–455. MR 1856273, DOI 10.1006/jfan.2001.3781
J. L. Clerc and B. Orsted, Conformal covariance for the powers of the Dirac operator, https://arxiv.org/abs/1409.4983
R. Delanghe, F. Sommen, and V. Souček, Clifford algebra and spinor-valued functions, Mathematics and its Applications, vol. 53, Kluwer Academic Publishers Group, Dordrecht, 1992. A function theory for the Dirac operator; Related REDUCE software by F. Brackx and D. Constales; With 1 IBM-PC floppy disk (3.5 inch). MR 1169463, DOI 10.1007/978-94-011-2922-0
C. Ding and J. Ryan, On some conformally invariant operators in Euclidean space, arXiv:1509.00098 [math.CV], Clifford Analysis and Related Topics, In Honor of Paul A.M. Dirac, CART 2014, Tallahassee, Florida, December 15-17. 53–72.
Chao Ding, Raymond Walter, and John Ryan, Construction of arbitrary order conformally invariant operators in higher spin spaces, J. Geom. Anal. 27 (2017), no. 3, 2418–2452. MR 3667436, DOI 10.1007/s12220-017-9766-7
Charles F. Dunkl, Junxia Li, John Ryan, and Peter Van Lancker, Some Rarita-Schwinger type operators, Comput. Methods Funct. Theory 13 (2013), no. 3, 397–424. MR 3102644, DOI 10.1007/s40315-013-0027-x
David Eelbode and Matthias Roels, Generalised Maxwell equations in higher dimensions, Complex Anal. Oper. Theory 10 (2016), no. 2, 267–293. MR 3451576, DOI 10.1007/s11785-014-0436-5
John E. Gilbert and Margaret A. M. Murray, Clifford algebras and Dirac operators in harmonic analysis, Cambridge Studies in Advanced Mathematics, vol. 26, Cambridge University Press, Cambridge, 1991. MR 1130821, DOI 10.1017/CBO9780511611582
A. W. Knapp and E. M. Stein, Intertwining operators for semisimple groups, Ann. of Math. (2) 93 (1971), 489–578. MR 460543, DOI 10.2307/1970887
Junxia Li and John Ryan, Some operators associated with Rarita-Schwinger type operators, Complex Var. Elliptic Equ. 57 (2012), no. 7-8, 885–902. MR 2955963, DOI 10.1080/17476933.2011.636430
Ian R. Porteous, Clifford algebras and the classical groups, Cambridge Studies in Advanced Mathematics, vol. 50, Cambridge University Press, Cambridge, 1995. MR 1369094, DOI 10.1017/CBO9780511470912
John Ryan, Iterated Dirac operators and conformal transformations in $\textbf {R}^n$, Proceedings of the XV International Conference on Differential Geometric Methods in Theoretical Physics (Clausthal, 1986) World Sci. Publ., Teaneck, NJ, 1987, pp. 390–399. MR 1023211
John Ryan, Cauchy-Green type formulae in Clifford analysis, Trans. Amer. Math. Soc. 347 (1995), no. 4, 1331–1341. MR 1249888, DOI 10.1090/S0002-9947-1995-1249888-8
J. Ryan, Dirac operators in analysis and geometry, lecture notes, 2008, http://comp.uark.edu/$\sim$jryan/notes.doc
M. V. Shapiro, On some boundary-value problems for functions with values in Clifford algebra, Mat. Vesnik 40 (1988), no. 3-4, 321–326 (English, with Serbo-Croatian summary). Third International Symposium on Complex Analysis and Applications (Hercegnovi, 1988). MR 1044377
Stuart Shirrell and Raymond Walter, Hermitian Clifford analysis and its connections with representation theory, Complex Var. Elliptic Equ. 62 (2017), no. 9, 1329–1342. MR 3662508, DOI 10.1080/17476933.2016.1250953
E. M. Stein and G. Weiss, Generalization of the Cauchy-Riemann equations and representations of the rotation group, Amer. J. Math. 90 (1968), 163–196. MR 223492, DOI 10.2307/2373431
Xu Zhenyuan, A function theory for the operator $D-\lambda$, Complex Variables Theory Appl. 16 (1991), no. 1, 27–42. MR 1092277, DOI 10.1080/17476938208814464
Zhongxiang Zhang, A revised higher order Cauchy-Pompeiu formulas in Clifford analysis and its application, J. Appl. Funct. Anal. 2 (2007), no. 3, 269–278. MR 2296007