Gaussian upper and lower bounds and H\"older continuity are established for the heat kernel associated to the prolate spheroidal wave functions (PSWFs) of order zero. These results are obtained by application of a general perturbation principle using the fact that the PSWF operator is a perturbation of the Legendre operator. Consequently, the Gaussian bounds and H\"older inequality for the PSWF heat kernel follow from the ones in the Legendre case. % As an application of the general perturbation principle, we also establish Gaussian bounds for the heat kernels associated to generalized univariate PSWFs and PSWFs on the unit ball in Rd. Further, we develop the related to the PSWFs of order zero smooth functional calculus, which in turn is the necessary groundwork in developing the theory of Besov and Triebel-Lizorkin spaces associated to the PSWFs. One of our main results on Besov and Triebel-Lizorkin spaces associated to the PSWFs asserts that they are the same as the Besov and Triebel-Lizorkin spaces generated by the Legendre operator.