On the Fourier–Haar Coefficients of Functions of Several Variables with Bounded Vitali Variation

In this paper, we study the behavior of the Fourier–Haar coefficients $a_{m_1,\dots,m_n}(f)$ of functions $f$ Lebesgue integrable on the $n$-dimensional cube $D_n=[0,1]^n$ and having a bounded Vitali variation $V_{D_n}f$ on it. It is proved that
$$ \sum _{m_1=2}^\infty\dotsi\sum _{m_n=2}^\infty |a_{m_1,\dots,m_n}(f)| \le\biggl(\frac{2+\sqrt 2}3\biggr)^n\cdot V_{D_n}f $$
and shown that this estimate holds for some function of bounded finite nonzero Vitali variation.