Multipliers for the weighted Orlicz spaces of a locally compact abelian group
Throughout this review \(G\) will always denote a locally compact abelian group, and \(\omega\) will always be a weight function on \(G\). Denote by \(L^{\Phi}(G, \omega)\) the weighted Orlicz space with Young function \(\Phi\). All Young functions in this review will satisfy the \(\Delta_2\)-condition. We will use \(\ast\) to indicate convolution of measurable functions on \(G\). With respect to convolution, \(L^{\Phi} (G, \omega)\) is an essential Banach \(L^1(G, \omega)\)-module. The left translation of a function \(f\) is defined to be \(L_af(x) = f(x-a)\), where \(x\) and \(a\) are in \(G\). Suppose that \(\Phi_1\) and \(\Phi_2\) are Young functions on \(G\). A bounded linear operator \(T\) from \(L^{\Phi_2}(G, \omega)\) to \(L^{\Phi_1}(G, \omega)\) is called a left multiplier for the pair \((L^{\Phi_2}(G, \omega), L^{\Phi_1}(G, \omega))\) if \(T(L_x f) = L_x(Tf)\) for all \(f \in L^{\Phi_2} (G, \omega)\) and \(x \in G\). The set of left multipliers for the pair \((L^{\Phi_2}(G, \omega), L^{\Phi_1}(G, \omega))\) will be denoted by \(M(L^{\Phi_2}(G, \omega), L^{\Phi_1}(G, \omega))\). In the paper under review, the author investigates \(M(L^{\Phi_2}(G, \omega), L^{\Phi_1}(G, \omega))\) and proves several interesting results. For example, it is shown that \(T\) is in \(M(L^{\Phi_2}(g, \omega), L^{\Phi_1}(G, \omega))\) if and only if \(T( h \ast f) = h \ast T(f)\) for all \(h \in L^1(G, \omega)\), and \(f \in L^{\Phi_2}(g, \omega)\). Using this result the author then proves the following: For a bounded linear operator \(T\) from \(L^1(G, \omega)\) to \(L^{\Phi}(G, \omega)\), the following are equivalent: \begin{itemize} \item[1.] \(T \in M(L^1(G, \omega), L^{\Phi}(G, \omega))\). \item[2.] There exists an \(f \in L^{\Phi}(G, \omega)\) such that \(T(h) = h \ast f\) for \(h \in L^1(G, \omega)\). \end{itemize}
scientific article; zbMATH DE number 7383833
scientific article