Geodesic
Martingale transforms and uniformly convergent orthogonal series
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XIV, Tome 141 (1985), pp. 18-38

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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 $$ \operatorname{mes}\Bigl\{\sup_N\Bigl|\sum_{n\le2N}F(w_n)w_n\Bigr|>\lambda\Bigr\}\le\mathrm{const}\,\lambda^{-1}\|F\|_{U(W)^*}.$$ 
@article{ZNSL_1985_141_a1,
     author = {S. V. Kislyakov},
     title = {Martingale transforms and uniformly convergent orthogonal series},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {18--38},
     publisher = {mathdoc},
     volume = {141},
     year = {1985},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1985_141_a1/}
}
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S. V. Kislyakov. Martingale transforms and uniformly convergent orthogonal series. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XIV, Tome 141 (1985), pp. 18-38. http://geodesic.mathdoc.fr/item/ZNSL_1985_141_a1/