The Hardy–Littlewood Theorem for Fourier–Haar Series

An interpolation theorem for a class of net spaces is proved. In terms of Fourier–Haar coefficients, we obtain a test for a function to belong to the net space $N_p^q (M)$, where 1 and M is the set of all closed intervals in $[0,1]$. As a corollary, we derive an analog of the Hardy–Littlewood theorem for Fourier–Haar series.