On the almost everywhere convergence of Fejér means of functions on the group of 2-adic integers
\textit{M. H. Taibleson} [``Fourier analysis on local fields'' (1975; Zbl 0319.42011)] conjectured that if \(f\) is integrable on the ring of integers in a 2-adic field, then the Cesàro means \(\sigma_nf\) of the Fourier series of \(f\) converge to \(f\) almost everywhere. This conjecture, which stood for more than two decades, has finally been solved. Using a decomposition of the ``2-adic Dirichlet kernel'' due to \textit{F. Schipp} and \textit{W. R. Wade} [Lect. Notes Pure Appl. Math. 138, 437-452 (1992; Zbl 0770.42007)], the author shows that the Cesàro maximal operator \(\sigma^*\) is of type \((H^1,L^1)\) and of weak type \((1,1)\). Since \textit{F. Schipp} and \textit{W. R. Wade} (loc. cit.) had shown that \(\sigma^*\) is of type \((\infty,\infty)\), it follows from interpolation that \(\sigma^*\) is of type \((p,p)\) for \(1<p\leq\infty\), and from the usual density argument that \(\sigma_nf\to f\) a.e. as \(n\to\infty\) for all \(f\in L^1\).
Cesàro summability of Fourier series on the group of integers of a 2-adic field
scientific article; zbMATH DE number 1042515
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