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