On the restricted summability of Walsh-Kaczmarz-Fejér means (Q2260276)

The paper investigates the behaviour of Fejér means with respect to two-dimensional Walsh-Kaczmarz systems, where the set of indices is restricted to a cone-like set. The main results are that the maximal operator is bounded from the Hardy space \(H_p\) to the space \(L_p\) for \(1/2<p<1\) and is of weak type \((1,1)\) and of type \((p,p)\) if \(1<p\leq\infty\). Moreover, it is not bounded from \(H_{1/2}\) to \(L_{1/2}\). These results extend some previous results obtained by P. Simon in the case of indices inside a positive cone around the identical function.