The space of multipliers and convolutors of Orlicz spaces on a locally compact group (Q2866755)

Let \(G\) be a locally compact group with left Haar measure \(\lambda\). Denote by \(L^0(G)\) the set of all equivalence classes of \(\lambda\)-measurable complex-valued functions. Let \(\varphi\) be a Young function. For \(f \in L^0(G)\) define NEWLINE\[NEWLINE \rho_{\varphi} (f) = \int_G \varphi (| f(x) | ) dx. NEWLINE\]NEWLINE Define the \textit{Orlicz space} \(L^{\varphi}(G)\) to be the set that consists of all \(f\) in \(L^0(G)\) that satisfy \(\rho_{\varphi} (af) < \infty\) for some \(a > 0\). With respect to the Luxemburg-Nakano norm \(N_{\varphi} (\cdot ), L^{\varphi}(G)\) is a Banach space. We note that if the Young function \(\varphi\) is finite and has strictly positive right derivative \(\varphi'\) at the origin, then \(L^{\varphi}(G)\) is a Banach algebra with multiplication given by convolution. Let \(X\) be a left \(L^{\varphi}(G)\)-submodule of \(L^{\psi}(G)\), where \(\psi\) is the complementary Young function to \(\varphi\). Indicate the norm on the Banach space \(X\) by \(\| \cdot \|_X\). For every \(u \in L^{\varphi}(G)\) and \(x \in X, ux\) is a bounded linear functional on \(\text{Hom}_{L^{\varphi}(G)} ( L^{\varphi}(G), X^{\ast})\), the space of all bounded right \(A\)-module homomorphisms from \(L^{\varphi}(G)\) into \(X^{\ast}\) with operator norm. This linear functional is defined by \(\langle ux, T \rangle = \langle x, T(u) \rangle\), where \(T \in \text{Hom}_{L^{\varphi}(G)} (L^{\varphi}(G), X^{\ast}).\) Denote by \(L^{\varphi}(G) \bullet X\) the norm closed linear span of \(L^{\varphi}(G)X\) in \(\text{Hom}_{L^{\varphi}(G)} (L^{\varphi}(G), X^{\ast})^{\ast}\). One of the main results proved by the authors in this paper is:NEWLINENEWLINELet \(G\) be a locally compact group and \(\varphi\) a finite Young function with \(\varphi'(0) >0\). If \((X, \| \cdot \|_X)\) is a left Banach \(L^{\varphi}(G)\)-submodule of \(L^{\psi}(G)\), then \(\text{Hom}_{L^{\varphi}(G)}(L^{\varphi}(G), X^{\ast}) = (L^{\varphi}(G)\bullet X)^{\ast}.\)NEWLINENEWLINENow let \(M^{\varphi}(G)\) be the set of all functions in \(L^0(G)\) that satisfy \(\rho_{\varphi} (af) < \infty\) for all \(a >0\). With the norm \(N_{\varphi}(\cdot), M^{\varphi}(G)\) is also a Banach space. A bounded linear operator on \(M^{\varphi}(G)\) is said to be a \textit{convolutor} if \(T(f \ast g) = T(f) \ast g\) whenever \(f,g \in C_c(G)\), the space of continuous functions with compact support. Write \(C_{V_{\varphi}}(G)\) for the space of convolutors. The space \(C_{V_{\varphi}}(G)\) is a closed subspace of the set of bounded linear operators on \(M^{\varphi}(G)\). The second main result of the paper identifies \(C_{V_{\varphi}}(G)\) with the dual of a Banach algebra of functions. Before we give this result we need some definitions. A Young function \(\varphi\) is said to be \(\Delta_2\)-regular if \(G\) is not compact and if there exists a \(k > 0\) such that for \(x \geq 0, \varphi(2x) \leq k \varphi(x)\). In addition, \(\varphi\) is also called a \(N\)-function if it satisfies certain limit conditions. Now let \(\mathcal{K}(G)\) be the set of all compact subsets of \(G\) with nonvoid interiors and such that each \(K \in \mathcal{K}(G)\) contains the identity element of \(G\). A normed space \(\check{A}_{\varphi, K}(G)\) is defined for each \(K \in \mathcal{K}(G)\). Then the union of all the \(\check{A}_{\varphi. K}(G)\), denoted by \(\check{A}_{\varphi}(G)\), is a normed algebra under pointwise multiplication. The other main result proved in this paper is:NEWLINENEWLINEIf \(G\) is a locally compact group and \(\varphi\) is a \(\Delta_2\)-regular \(N\)-function, then the dual of \(\check{A}_{\varphi}(G)\) can be identified with \(C_{V_{\varphi}}(G)\).