On the inequality of different metrics for multiple Fourier–Haar series

Let $1<p<q<\infty$, $f\in L_p[0, 1]$. Then, according to the inequality of different metrics due to S.M. Nikol'skii, for the sequence of norms of partial sums of the Fourier–Haar series $\{||S_{2^k}(f)||_{L_q}\}_{k=0}^\infty$ the following relation is true $||S_{2^k}(f)||_{L_q}=O\left(2^{k\left(\frac1p-\frac1q\right)}\right)$. In this paper, we study the asymptotic behavior of partial sums in the Lorentz spaces. In particular, it is obtained that $||S_{2^{k_1}2^{k_2}}(f)||_{L_{\overline{q}}}=o\left(2^{k_1\left(\frac1{p_1}-\frac1{q_1}\right)+k_2\left(\frac1{p_2}-\frac1{q_2}\right)}\right)$ for $f\in L_{\overline{p},\overline{\tau}}[0, 1]^2$.