Cesàro summability of one- and two-dimensional Walsh-Fourier series. (English) Zbl 0866.42020

The author introduces \(p\)-quasi-local operators (the case \(p=1\) reduces to Schipp’s concept of a quasi-local operator) and using Hardy-Lorentz spaces and interpolation proves the following extrapolation result: If \(T\) is sublinear, \(p\)-quasilocal for \(0<p\leq 1\), and of type \((\infty,\infty)\), then \(T\) is bounded from \(H_{p,q}\) to \(L_{p,q}\) for every \(0<p<\infty\) and \(0<q\leq\infty\). He applies this result to Cesàro means of Walsh and double Walsh-Fourier series. He proves the maximal operator \(\sigma^*f\equiv \sup|\sigma_n f|\) is bounded from \(H_{p,q}\) to \(H_{p,q}\) for every \(1/2<p<\infty\) and \(0<q\leq\infty\) (so \(\sigma^*\) is of weak type \((1,1)\)). He obtains a similar result for the maximal operator associated with dyadic Cesàro means without the restriction \(p>1/2\), and for double Cesàro means of Walsh-Fourier series when \(f\) belongs to the hybrid dyadic Hardy space \(H^\#_1\). It follows that if \(f\in H^\#_1\) then \(\sigma_{n,m}f\to f\) a.e. (as \(\min(n,m)\to\infty\)). This improves an earlier result of F. Móricz, F. Schipp and the reviewer [Trans. Am. Math. Soc. 329, No. 1, 131-140 (1992; Zbl 0795.42016)] which was valid for the smaller space \(L\log^+ L\).


42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
42B25 Maximal functions, Littlewood-Paley theory
42B30 \(H^p\)-spaces


Zbl 0795.42016
Full Text: DOI


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