Martingale transforms and uniformly convergent orthogonal series

S. A. Vinogradov's method is adapted to prove for certain orthogonal product systems the analogue of his inequality concerning the trigonometric system. For example, for the Walsh system $W=\{w_n\}$ the following holds. Let $U(W)$ be the space of functions with a uniformly convergent Walsh–Fourier series. Then, for every functional $F$ on $U(W)$ we have the inequality
$$ \operatorname{mes}\Bigl\{\sup_N\Bigl|\sum_{n\le2N}F(w_n)w_n\Bigr|>\lambda\Bigr\}\le\mathrm{const}\,\lambda^{-1}\|F\|_{U(W)^*}.$$