Conditions under Which the Sums of Absolute Values of Blocks in the Fourier–Walsh Series for Functions of Bounded Variation Belong to Spaces $L^p$

In this paper, the following question is considered: what conditions on a strictly increasing sequence of positive integers $\{n_j\}_{j=1}^{\infty}$ guarantee that the sum of the series
$$ \sum_{j=1}^{\infty}\bigg|\sum_{k=n_j}^{n_{j+1}-1}c_k(f) w_k(x)\bigg|,$$
where $c_k(f)$ are the Walsh–Fourier coefficients of a function $f$, belongs to the space $L^p[0,1)$, $p>1$, for any function $f$ of bounded variation? For $p=\infty$, it is proved that such a sequence does not exist. For finite $p>1$, sufficient conditions are obtained for the sequence $\{n_{j}\}$; these conditions are similar to the ones obtained by the first author in the trigonometric case.