Weak inequalities for Cesàro and Riesz summability of Walsh–Fourier series

AbstractThe maximal operators for Cesàro or (C,α) 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 Hp to Lp for all 1/(α+1)<p<∞. In this work we prove that this boundedness result does not hold anymore if p⩽1/(α+1). However, for p=1/(α+1) the maximal operators are bounded from H1/(α+1) to the weak L1/(α+1) space. To the proof some known estimations for the Cesàro and Riesz kernels have to be sharpened