Third-Order Fermionic and Fourth-Order Bosonic Operators

Abstract

This paper continues the work of our previous paper (Ding et al., Higher Order Fermionic and Bosonic Operators, Springer Series), where we generalize kth-powers of the Euclidean Dirac operator Dx to higher spin spaces in the case the target space is a degree one homogeneous polynomial space. To generalize the results in (Ding et al., Higher Order Fermionic and Bosonic Operators, Springer Series) to more general cases, i.e., the target space is a degree k homogeneous polynomial space, we reconsider the generalizations of \(D_x^3\) and \(D_x^4\) to higher spin spaces in the case the target space is a degree k homogeneous polynomial space in this paper. Constructions of 3rd- and 4th-order conformally invariant operators in higher spin spaces are given; these are the 3rd-order fermionic and 4th-order bosonic operators. They are consistent with the 3rd- and 4th-order conformally invariant differential operators obtained in our paper (Ding et al., J. Geometric Anal. 27(3), 2418–2452 (2017)) with a different technique. Further, we point out that the generalized symmetry technique used in (De Bie et al., Potential Analysis 47(2), 123–149 (2017); Eelbode and Roels, Compl. Anal. Oper. Theory, 1–27 (2014)) is not applicable for higher order cases because the computations are infeasible. Fundamental solutions and intertwining operators of both operators are also presented here. These results can be easily generalized to cylinders and Hopf manifolds as in Ding et al. (J. Indian Math. Soc. 83(3-4), 231–240 (2016)). To conclude this paper, we investigate ellipticity property for our 3rd- and 4th-order conformally invariant operators.