Some properties of Fourier transform for operators on homogeneous Banach spaces (Q1196280)

The authors define the space \({\mathcal L}^ 1_ s(B)\) of strongly right translation invariant operators on a homogeneous Banach space contained in the algebra \(L^ 1(G)\) of a locally compact abelian group \(G\). Such an operator \(T\) is bounded on \(B\) and satisfies \(\int_ G\| T\cdot R_ x(f)\|_ B dx<\infty\) for all \(f\in B\) where \(R_ x\) denotes translation by \(x\). The Fourier transform of \(T\) is defined by \[ T^ \wedge(\gamma)f= \int_ G \gamma(x)T\cdot R_ x(f)dx \quad \text{ for } \gamma\in G^ \wedge \text{ and } f\in B. \] The authors show that \(T^ \wedge\) is an element of \(C_ 0(G^ \wedge,{\mathcal L}^ 1_ s(B))\). They also define a convolution of a measure \(\mu\) with \(T\in{\mathcal L}^ 1_ s(B)\) by \(\mu*T(f)=\int_ G T\cdot R_ x f d\mu(x)\) and they derive some easy properties of the Fourier transform with respect to convolution.