Fourier Transform for \(L^p\)-Functions with a Vector Measure on a Homogeneous Space of Compact Groups

Let G be a compact group and G/H a homogeneous space where H is a closed subgroup of G. Define an operator \(T_H:C(G) \rightarrow C(G/H)\) $T_H:C(G)\to C(G/H)$ by \(T_Hf(tH)=\int _H f(th) \, dh\) $T_{H}f(tH)={\int }_{H}f(th)dh$ for each \(tH \in G/H\) $tH\in G/H$ . In this paper, we extend \(T_H\) $T_H$ to a norm-decreasing operator between \(L^p\) $L^p$ -spaces with a vector measure for each \(1 \le p <\infty \) $1\le p<\infty$ . This extension will be used to derive properties of invariant vector measures on G/H. Moreover, a definition of the Fourier transform for \(L^p\) $L^p$ -functions with a vector measure is introduced on G/H. We also prove the uniqueness theorem and the Riemann–Lebesgue lemma.