Fourier Transform for Lp-Functions with a Vector Measure on a Homogeneous Space of Compact Groups

Let G be a compact group and G/Ha homogeneous space where H is a closed sub-group of G. Define an operator TH:C(G)-> C(G/H)by THf(tH)=integral Hf(th)dh for each tH is an element of G/H. In this paper, we extend TH to a norm-decreasing operator between Lp-spaces with a vector measure for each 1 <= p<infinity. This extension will be used to derive properties of invariant vector measures on G/H. Moreover, a definition of the Fourier transform for Lp-functions with a vector measure is introduced on G/H. We also prove the uniqueness theorem and the Riemann-Lebesgue lemma