Lorentz spaces as \(L_ 1\)-modules and multipliers (Q1326594)

Let \(G\) be a locally compact group with Haar measure \(h\), and let \(\Omega\) be a Hausdorff space with space of Radon measures \(C^*_ c (\Omega)\). A set \(A\) is said to be a convolution algebra on \(G\) if \(u \in A\), \(v \in A\), implies that \((u*v) \in A\) and \(\| u*v \|_ A \leq \| u \|_ A \| v \|_ A\), where \(u*v\) represents the convolution of \(u\) and \(v\) on \(G\) and \(\| \cdot \|_ A\) is a norm on \(A\). A Banach space \(B\) is said to be an \(A\)-module if \(u \in A\), \(v \in B\), implies that \((u*v) \in B\) and \(\| u*v \|_ B \leq \| u \|_ A \| v \|_ B\). In the main results of this paper the authors consider the set \(M(A,B)\) of linear operators of the form \(T:A \to B\), such that \(T(u*v) = u*T(v)\), \(u \in A\), \(v \in A\). Conditions are stated in the main theorem which confirm the representation \(T(v) = \rho *v\) for operators in \(M(A,B)\). One of the applications of the main theorem relates to results involving Lorentz spaces. In particular, if \(\| f \|_{p,q} = ({q \over p} \int^ \infty_ 0 (f^{**} (t)^ qt^{(q/p) - 1})^{1/q}\), where \(sf^{**} (s) = \int^ s_ 0f^* (t)dt\), and \(f^*\) is the non- increasing rearrangement of \(f\), and if \(\|\;\|_{p,q}\) is the norm on \(L_{p,q}\), then it is shown that \(M(L_ 1(w), L_{p,q})\) is isomorphic to \(L_{p,q}\), where \(1 < p < \infty\), \(1 \leq q \leq \infty\), and \(L_ 1(w)\) is a weighted space of integrable functions.