Topologically invariant linear forms on spaces of abstract distributions on a locally compact Abelian group (Q1207668)

Let \(G\) be a locally compact abelian group. If \(1\leq p<\infty\), a space \({\mathcal F}_ p(G)\) of abstract distributions on \(G\) is introduced which has the property that the Fourier transform is an isometric isomorphism from \({\mathcal F}_ p(G)\) to \(L^ p(\widehat G)\), where \(\widehat G\) is the dual of \(G\). In particular, \({\mathcal F}_ 2(G)=L^ 2(G)\). The main result proved is that if \(G\) is locally compact, non-compact \(\sigma\)- compact, connected and abelian, then \({\mathcal F}_ p(G)=\{f-\varphi*f\): \(f\) is in \({\mathcal F}_ p(G)\), \(\varphi\geq 0\) and \(\int_ G \varphi=1\}\). Thus in this case, if \(L\) is a linear form on \({\mathcal F}_ p(G)\) such that \(L(f)=L(\varphi*f)\) for all \(f\) in \({\mathcal F}_ p(G)\) and for all \(\varphi\) in \(L^ 1(G)\) such that \(\varphi\geq 0\) and \(\int_ G \varphi=1\) (that is, if \(L\) is topologically invariant), it follows that \(L=0\). The idea of the proof is that there is \(\varphi\) in \(L^ 1(G)\), \(\varphi\geq 0\) and \(\int_ G \varphi=1\), such that if \(\widehat\varphi\) is the Fourier transform of \(\varphi\), then \(1-\widehat\varphi\) converges to 0 at the identity of \(\widehat G\) as slowly as any continuous function on \(\widehat G\) which is given in advance and which vanishes at the identity of \(\widehat G\).