Fourier transform for \(L^p\)-functions with a vector measure on a homogeneous space of compact groups (Q6550713)

Throughout this review \(G\) will always denote a compact, Hausdorff topological group. Let \(H\) be a closed subgroup of \(G\) and let \(T_H\) be an operator from \(C(G) \rightarrow C(G/H)\) given by\N\[\NT_H f(tH) = \int_H f(th)\, dh.\N\]\NA vector measure is a measure that takes values in a Banach space. Let \(\mu\) be a vector measure on \(G/H\). The authors show that a vector measure \(\breve{\mu}\) can be constructed on \(G\) such that the extension \(T_H\colon L^p(G, \breve{\mu}) \rightarrow L^p(G, \mu)\) is a norm-decreasing operator from \(L^p(G, \breve{\mu})\) to \(L^p(G/H, \mu )\) for any \(1 \leq p < \infty\), and\N\[\N\int_G f\, d\breve{\mu} = \int_{G/H} T_H f \, d\mu\N\]\Nfor \(f \in L^1(G, \breve{\mu})\). The following theorem is proved:\N\NLet \(1 \leq p < \infty\). The extension \(T_H \colon L^p(G, \breve{\mu}) \rightarrow L^p(G/H, \mu)\) satisfies the formula \(T_H f(tH) = \int_H f(th)\, dh\) \(\mu\)-a.e. for all \(f \in L^p(G, \breve{\mu})\). Moreover, the extension \(T_H \colon L^p(G, \breve{\mu}) \rightarrow L^p(G/H, \mu)\) is surjective.\N\NIt then follows that Weil's formula\N\[\N\int_{G/H} \int_H f(th)\, dh\, d\mu (tH) = \int_G f\, d\breve{\mu}\N\]\Nholds for all \(f \in L^1(G, \breve{\mu})\).\N\NRelations between \(\mu\) and \(\breve{\mu}\) in terms of invariance properties are given. For example, let \(\mu\) be a vector measure on \(G/H\). Definitions for \(\mu\) to be \(L_a\)-invariant, where \(a \in G\); and semivariation \(\tau\)-invariant, where \(\tau\) is a homeomorphism from \(G/H\) to \(G/H\) are given. The following proposition is proved:\N\begin{itemize}\N\item[1.] \(\mu\) is \(L_a\)-invariant if and only if \(\breve{\mu}\) is \(L_a\)-invariant.\N\item[2.] \(\mu\) is norm integral \(L_a\)-invariant if and only if \(\breve{\mu}\) is norm integral \(L_a\)-invariant.\N\item[3.] \(\mu\) is semivariation \(L_a\)-invariant if and only if \(\breve{\mu}\) is semivariation \(L_a\)-invariant.\N\end{itemize}