Multipliers on some weighted \(L^p\)-spaces (Q1578349)

Let \(G\) be a locally compact abelian group. Multipliers (that is, bounded linear transformations which commute with translations) on weighted function spaces on \(G\) have been studied by several authors. The paper under review appears to be motivated and strongly influenced by [\textit{G. N. K. Murthy} and \textit{K. R. Unni}, Multipliers on weighted spaces, in: Functional analysis and its applications, Lect. Notes Math. Vol. 399 (Berlin etc. 1974; Zbl 0298.46031); pp. 272-291], where, for a Beurling weight function \(w\), the space of multipliers from \(L^p_w(G)\) to \(L^q_w(G)\) has been identified as a certain dual Banach space. Naturally, replacing \(L^p_w(G)\) by an intersection of two \(L^p\)-spaces and \(L^q_w(G)\) by some other function space makes things technically more complicated. The author obtains three results, the proofs of which are very much the same. We therefore mention only one of the results. Let \(1<p_i\), \(q_i<\infty\). Endow \(L_w^{p_1} (G)\cap L^{p_2}_w(G)\) with the norm \(\|f\|_{D_p}= \max(\|f\|_{p_1,w}, \|f\|_{p_2,w})\). Let \(S(q_1,q_2,w)\) be the space of all functions \(g=g_1+g_2\), where \(g_i\in L_w^{q_i} (G)\) \((i=1,2)\), and define a norm on \(S(q_1,q_2,w)\) by \(\|g \|_{D_q}= \inf\{\|g_1\|_{q_1,w} +\|g_2\|_{q_2,w}\}\), where the infimum is taken over all such decompositions of \(g\). Moreover, let \(K(p_1,p_2,q_1,q_2,w)\) be the space of functions \(h\) on \(G\) which can be written as \(h= \sum^\infty_{j=1} f_j*g_j\), where \(f_j\in C_c(G) \subseteq L_w^{p_1}(G)\cap L_w^{p_2} (G)\) and \(g_j\in L_w^{q_1} (G)\cap L_w^{q_2} (G)\) with \(\sum^\infty_{j=1} \|f_j\|_{D_p} \|g_j\|_{D_q} <\infty\). The norm on \(K(p_1,p_2, q_1,q_2,w)\) is given as \(\|h\|= \inf\{\sum^\infty_{j=1} \|f_j\|_{D_p} \|g_j\|_{D_q}\}\), where the infimum is taken over all such representations of \(h\). Suppose that \({1\over p_i}+ {1\over q_i}\geq 1\), and let \(q_i'\) be defined by \({1\over q_i'} +{1\over q_i}=1\) \((i=1,2)\). Then the space of multipliers from \(L_w^{p_1}(G)\cap L_w^{p_2}(G)\) to \(S(q_1',q_2',w^{-1})\) turns out to be isometrically isomorphic to the dual space of \(K(p_1,p_2, q_1,q_2,w)\) (Theorem 4.1).