Invariance of basic Hodge numbers under deformations of Sasakian manifolds

We show that the Hodge numbers of Sasakian manifolds are invariant under arbitrary deformations of the Sasakian structure. We also present an upper semi continuity Theorem for the dimensions of kernels of a smooth family of transversely elliptic operators on manifolds with transversely Riemannian foliations. We use this to prove that the $\mathrm{\partial }\bar{\mathrm{\partial }}$-lemma and being transversely K"{a}hler are rigid properties under small deformations of the transversely holomorphic structure which preserve the foliation. Finally, we study an example which shows that this is not the case for arbitrary deformations of the transversely holomorphic foliation.