The Hardy-Littlewood theorem for double Fourier-Haar series from Lebesgue spaces $L_{\bar{p}}[0,1]$ with mixed metric and from net spaces $N_{\bar{p}, \bar{q}}(M)$

Abstract

In terms of the Fourier-Haar coefficients, a criterion is obtained for the function $f (x_1,x_2)$ to belong to the net space $N_{\bar{p},\bar{q}}(M)$ and to the Lebesgue space $L_{\bar{p}}[0,1]^2$ with mixed metric, where $1<\bar{p}<\infty$, $0<\bar{q}\leq\infty$, $\bar{p}=(p_1,p_2)$, $\bar{q}=(q_1,q_2)$, $M$ is the set of all rectangles in $\mathbb{R}^2$. We proved the Hardy-Littlewood theorem for multiple Fourier-Haar series.