Quasiuniversal Fourier–Walsh Series for the Classes $L^p[0,1]$, $p>1$

It is proved that, for each number $p>1$, there exists a function $L^1[0,1]$ whose Fourier–Walsh series is quasiuniversal with respect to subseries-signs in the class $L^p[0,1]$ in the sense of $L^p$-convergence.