The martingale Hardy type inequality for Marcinkiewicz-Fejér means of two-dimensional conjugate Walsh-Fourier series (Q353544)

The author proves that the operator \(\tilde{\sigma}_n^{(t)}\) of the Marcinkiewicz-Fejér means of the two-dimensional conjugate Walsh-Fourier series is not uniformly bounded from the Hardy space \(H_p(G\times G)\) to the space \(L_p(G\times G)\), when \(0<p<2/3\). This result approves the conjecture made by Weisz, which states that for the uniform boundedness of the operator \(\tilde{\sigma}_n^{(t)}\) from the Hardy space \(H_p(G\times G)\) to the space \(L_p(G\times G)\), the assumption \(p>2/3\) is essential.