Estimates of the spherical and ultraspherical heat kernel

Abstract

In this thesis we establish an upper bound for the spherical heat kernel on the N-dimensional unit sphere SN for N = 1; 2; 3. The strategy is to use the fact that the spherical heat kernel is completely determined by the ultraspherical heat kernel. By techniques from Fourier analysis, explicit formulas for the ultraspherical heat kernel with parameter = 1=2; 1=2 are deduced. Also, an integral formula for the kernel with parameter = 0 is introduced. By estimating these formulas for the ultraspherical heat kernels, the estimates of the spherical heat kernel are obtained. Furthermore, we prove that the periodized Gauss-Weierstrass kernel is strictly decreasing on [0; ]. Both an analytic and a probabilistic proof are given. A generalization of this result is also established for small t, saying that the spherical heat kernel on S2 and S3 is strictly decreasing as a function of the spherical distance between its two arguments.