The convergence of double Fourier-Haar series over homothetic copies of sets

The paper is concerned with the convergence of double Fourier-Haar series with partial sums taken over homothetic copies of a given bounded set $W\subset \mathbb{R}_+^2$ containing the intersection of some neighbourhood of the origin with $\mathbb{R}_+^2$. It is proved that for a set $W$ from a fairly broad class (in particular, for convex $W$) there are two alternatives: either the Fourier-Haar series of an arbitrary function $f\in L([0,1]^2)$ converges almost everywhere or $L\ln^+L([0,1]^2)$ is the best integral class in which the double Fourier-Haar series converges almost everywhere. Furthermore, a characteristic property is obtained, which distinguishes which of the two alternatives is realized for a given $W$.
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