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Weak inequalities for Cesàro and Riesz summability of Walsh-Fourier series. (English) Zbl 1143.42032

Summary: The maximal operators for Cesàro or \((C,\alpha)\) and Riesz summability with respect to Walsh-Fourier series are investigated as mappings between dyadic Hardy and Lebesgue spaces. It is well known that they are bounded from \(H_p\) to \(L_p\) for all \(1/(\alpha+1)<p<\infty\). In this work we prove that this boundedness result does not hold anymore if \(p\leq 1/(\alpha+1)\). However, for \(p=1/(\alpha+1)\) the maximal operators are bounded from \(H_{1/(\alpha+1)}\) to the weak \(L_{1/(\alpha+1)}\) space. To the proof some known estimations for the Cesàro and Riesz kernels have to be sharpened.

MSC:

42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
60G42 Martingales with discrete parameter
60G46 Martingales and classical analysis
46B42 Banach lattices
46E40 Spaces of vector- and operator-valued functions
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