Cesàro summability with respect to two-parameter Walsh systems (Q5931263)

The author examines summability of double Walsh-Fourier series in the Kaczmarz ordering. He shows that the restricted operator of the Cesàro means of Walsh-Fourier series in the Kaczmarz ordering is bounded from the diagonal Hardy space \(H^p\) into \(L^p\) for all \(1/2<p\leq 1\). The paper also contains a weak-type inequality for hybrid Hardy spaces, an example to show why the results do not extend to indices less than 1/2, and a discussion of consequences of the main results to certain Hardy-Lorentz spaces, and to almost everywhere summability of Walsh-Kaczmarz-Fourier series of functions in the hybrid Hardy space \(H^1_\sharp (G^2)\). As usual, the author is thorough and complete.