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

TL;DR

This paper establishes a criterion for functions to belong to certain Lebesgue and net spaces based on Fourier-Haar coefficients, extending the Hardy-Littlewood theorem to double Fourier-Haar series with mixed metrics.

Contribution

It provides a new criterion for membership in Lebesgue and net spaces using Fourier-Haar coefficients, extending classical theorems to multiple series with mixed metrics.

Findings

Criterion for function space membership based on Fourier-Haar coefficients

Extension of Hardy-Littlewood theorem to double Fourier-Haar series

Applicable to functions in Lebesgue and net spaces with mixed metrics

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.