Eigenvalue estimate of the basic Dirac operator on a Kähler foliation

Let FF be a Kähler spin foliation of codimension q=2nq=2n on a compact Riemannian manifold M with the transversally holomorphic mean curvature form κ. It is well known [S.D. Jung, T.H. Kang, Lower bounds for the eigenvalue of the transversal Dirac operator on a Kähler foliation, J. Geom. Phys. 45 (2003) 75–90] that the eigenvalue λ   of the basic Dirac operator DbDb satisfies the inequality λ2⩾n+14ninfM{σ∇+|κ|2}, where σ∇σ∇ is the transversal scalar curvature of FF. In this paper, we introduce the transversal Kählerian twistor operator and prove that the same inequality for the eigenvalue of the basic Dirac operator by using the transversal Kählerian twistor operator. We also study the limiting case. In fact, FF is minimal and transversally Einsteinian of odd complex codimension n with nonnegative constant transversal scalar curvature.