Multipliers and tensor products of weighted \(L^p\)-spaces (Q5946690)

Let \(G\) be a locally compact unimodular group and \(\omega\) a real-valued, continuous function on \(G\) such that \(1\leq \omega(x)\) and \(\omega(x +y) \leq \omega(x) \omega(y)\) for \(x,y\in G\). The set \(L^p_\omega(G)\), \(1\leq p< \infty\), of functions \(f\) such that \(f\omega \in L^p(G)\) with norm \(|f|_{p,\omega} = |f\omega |_p\), is a left Banach \(L^1_\omega(G)\)-module. In the same way, the right Banach \(L^1_\omega(G)\)-module \(\overline{L}^p_\omega(G)\) can be defined. For \(\frac{1}{p} + \frac{1}{q} > 1\), a family of Banach spaces denoted by \(A^{p,q}_\omega(G)\) is introduced. The paper is devoted to study some of their properties. The authors determine the locally compact groups for which \(A^{p,q}_\omega(G)\) is isometrically isomorphic to the \(L^1_\omega(G)\)-module tensor product \(L^p_\omega(G)\bigotimes_{L^1_\omega(G)}\overline{L}^q_\omega(G)\), defined as the quotient Banach space \(L^p_\omega(G)\bigotimes_\gamma\overline{L}^q_\omega(G)/K\). Here \(L^p_\omega(G)\bigotimes_\gamma\overline{L}^q_\omega(G)\) denotes the projective tensor product of Banach spaces and \(K\) is the closed linear subspace spanned by all elements of the form \(f\star g\bigotimes h-g\bigotimes \widetilde{f}\star h\), for every \(f \in L^1_\omega(G)\), \(g \in L^p_\omega(G)\) and \(h \in \overline{L}^q_\omega(G)\). Another description of \(A^{p,q}_\omega(G)\) is obtained in the abelian case.