Fourier transforms of Dini-Lipschitz functions on Vilenkin groups (Q1196617)

The author extends some theorems on Fourier transforms of functions on \(R^ n\) for functions on Vilenkin groups. Let \(G\) be a Vilenkin group. Then there is a countable basic set of neighborhoods \(\{G_ n\}\) of the identity element \(e\) of \(G\) (\(G_ n\) is a closed subgroup of \(G\)) such that \(G=G_ 0\supset G_ 1\supset\dots\), \(\bigcap^{\infty}_{n=0}G_ n=\{e\}\) and \(V_ k/V_{k-1}\) is of prime order \(p_ k\) for every \(k\geq 1\), where \(V_ k\) is the annihilator of \(G_ k\). Put \(m_ 0=1\), \(m_{k+1}=p_ km_ k\) (\(k\geq 1\)). For \(f\in L^ p(G)\), the \(p\)-th modulus of continuity \(\omega_ p(f,k)\) is defined by \(\sup_{h\in G_ k}\| f(x+h)-f(x)\|_ p\). The author obtains the following Theorem: Let \(f\in L^ p(G)\), \(1<p\leq 2\), such that \(\omega_ p(f,k)=O(m_ k^{-\alpha}/(\text{Log }m_ k)^ r)\), \(0<\alpha\leq 1\). Then \(\widehat{f}\in L^ \beta(\widehat{G})\) for \(q=p/(p-1)\geq \beta > \max(p/(p+\alpha p-1),1/r)\).