On the Fejér means of bounded Ciesielski systems (Q2717723)

Let \(\sigma^{(m,k)}_nf\) denote the Fejér means of the Ciesielski-Fourier series of \(f\), and let \(f^{(m,k)}_*(x)\) be the non-tangential maximal function of a tempered distribution \(f\). The main result states that: 1) \(\|\sigma^{(m,k)}_*f\|_{L^p}\leq C_p\|f\|_{H_p}\) for \(f\in H_p\) (Hardy space), \(\frac 12<p<\infty\), \(m\geq 0\), \(|k|\leq m+1\), 2) \(\lambda(\sigma_*^{(m,k)}f>\varrho)\leq \frac c\varrho\|f\|_{L_1}\) for \(f\in L_1\).