Zero divisors and Orlicz spaces
Let \(G\) be a countable discrete group and let \(\Phi\) be a Young function on \(G\). Let \(\alpha\) be a nonzero element in \(\ell^1(G)\) and denote convolution of functions on \(G\) by \(\ast\). We shall say that \(\alpha\) is a \(\Phi\)-zero divisor if there exists a nonzero function \(\beta\) in the Orlicz space \(\ell^{\Phi}(G)\) that satisfies \(\alpha \ast \beta = 0\). Let \(F_k\) be the free group on \(k\) generators, where \(k \geq 2\). Denote by \(\vert g \vert\) the length of \(g \in F_k\). Let \(\chi_1\) be the characteristic function on elements of length one in \(F_k\). The main result of the paper under review is: Assume that \(\Phi\) is a Young function for which there exist \(s > 2\), \(C>0\), \(\delta > 0\) for which \(\Phi (x) \leq C \vert x \vert^s\) when \(\vert x \vert \leq \delta\). Then the following hold. \begin{itemize} \item[1.] \(\chi_1\) is a \(\Phi\)-zero divisor. \item[2.] Let \(k\) be an even integer greater than three and let \(\{ x_1, \dots, x_k\}\) be a set of generators for \(F_k\). Then the characteristic function of \(\{ x_1, \dots ,x_k\}\) is a \(\Phi\)-zero divisor for \(F_k\). \end{itemize} Now let \(p\) be a real number strictly greater than two. Young functions that satisfy the hypothesis of the above result are: \(\vert x \vert^p\), \(\vert x \vert^p (\log \vert x \vert + c)\) where \( c > \frac{2p-1}{p(p-1)}\), \(\frac{\vert x \vert^p}{\ln (e + \vert x\vert)}\) and \({e^{\vert x \vert^p}-1}\).
scientific article; zbMATH DE number 7608901