Marcinkiewicz-Fejér means of \(d\)-dimensional Walsh--Fourier series
It is proved that the Marcinkiewicz maximal operator of a \(d\)-dimensional Walsh-Fourier series is bounded from the Hardy-Lorentz space \(H_{p,q}\) to \(L_{p,q}\) \((p>d/(d+1), 0<q\leq \infty)\) and it is of weak type \((1,1)\). From this it follows that the Marcinkiewicz means of a function \(f \in L_1\) converge to \(f\) a.e. All these results can be found for \(d=2\) in [\textit{F. Weisz}, ''Convergence of double Walsh-Fourier series and Hardy spaces''. Approximation Theory Appl. 17, No.~2, 32--44 (2001; Zbl 0992.42013)].
scientific article; zbMATH DE number 2174933