Geodesic
A criterion for the almost-everywhere convergence of Fourier–Walsh square partial sums of integrable functions
Sbornik. Mathematics, Tome 186 (1995) no. 7, pp. 1057-1070 Cet article a éte moissonné depuis la source Math-Net.Ru

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S. V. Konyagin showed that if the one-dimensional Lebesgue constants $L_{n_k}$ for the Walsh–Paley system are unbounded, then the square partial sums $S_{n_k,n_k}(f)$ of some integrable function $f({x})=f(x_1,x_2)$ diverge almost everywhere. On the other hand the author constructed an example of sequence $\{n_k\}$ for which, sup $\sup L_{n_k}$ is finite, but for some integrable function $f({x})=f(x_1,x_2)$ the partial sums $S_{n_k,n_k}(f)$ diverge almost everywhere. Thus boundedness of the Lebesgue constants $L_{n_k}$ is not a necessary and sufficient condition for the convergence almost everywhere of the partial sums $S_{n_k,n_k}(f)$ of any integrable function. In this article we find such a necessary and sufficient condition. 
@article{SM_1995_186_7_a7,
     author = {S. F. Lukomskii},
     title = {A criterion for the~almost-everywhere convergence of {Fourier{\textendash}Walsh} square partial sums of integrable functions},
     journal = {Sbornik. Mathematics},
     pages = {1057--1070},
     year = {1995},
     volume = {186},
     number = {7},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1995_186_7_a7/}
}
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S. F. Lukomskii. A criterion for the almost-everywhere convergence of Fourier–Walsh square partial sums of integrable functions. Sbornik. Mathematics, Tome 186 (1995) no. 7, pp. 1057-1070. http://geodesic.mathdoc.fr/item/SM_1995_186_7_a7/

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