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On $L_p$-integrability of functions with general monotone Fourier coefficients

Askhat Mukanov, Erlan Nursultanov

TL;DR

This work extends Hardy-Littlewood-type $L_p$-norm characterizations for Fourier series to two new coefficient classes, GM$^*$ and $\overline{GM}$, which allow partial failures of monotonicity but maintain enough structure for sharp norm control. By leveraging discrete net spaces $n_{p,q}$ and Paley-type functionals $J_p$ and $J_p^*$, the authors prove two main equivalences: $\|f\|_{L_p} \asymp J_p^*(f)$ for $a_k \in \text{GM}^*$ and $\|f\|_{L_p} \asymp J_p(f)$ for $a_k \in \overline{GM}$, for all $1<p<\infty$. They analyze the relationships among general monotone classes, demonstrate compensatory effects, and extend results to alternating-series settings using idempotent Fourier multipliers. The findings broaden the applicability of HL-type theorems, provide net-space tools for Fourier analysis, and yield new insights into Fourier multiplier theory for complex coefficient sequences.

Abstract

We introduce new classes of general monotone sequences and study their properties. For functions whose Fourier coefficients belong to these classes, we establish Hardy-Littlewood-type theorems.

On $L_p$-integrability of functions with general monotone Fourier coefficients

TL;DR

This work extends Hardy-Littlewood-type -norm characterizations for Fourier series to two new coefficient classes, GM and , which allow partial failures of monotonicity but maintain enough structure for sharp norm control. By leveraging discrete net spaces and Paley-type functionals and , the authors prove two main equivalences: for and for , for all . They analyze the relationships among general monotone classes, demonstrate compensatory effects, and extend results to alternating-series settings using idempotent Fourier multipliers. The findings broaden the applicability of HL-type theorems, provide net-space tools for Fourier analysis, and yield new insights into Fourier multiplier theory for complex coefficient sequences.

Abstract

We introduce new classes of general monotone sequences and study their properties. For functions whose Fourier coefficients belong to these classes, we establish Hardy-Littlewood-type theorems.
Paper Structure (11 sections, 19 theorems, 160 equations)

This paper contains 11 sections, 19 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. Hardy-Littlewood theorem and its analogues
  3. Main aim
  4. Main results
  5. Some remarks
  6. Auxiliary results
  7. The estimates of the net spaces norms of the Fourier coeffients and their applications
  8. The proof of the main results
  9. Comparison of classes of general monotone sequences
  10. Compensatory effect of sequences in $\overline{\textnormal{GM}}$
  11. Alternating series and idempotent multipliers

Key Result

Theorem 1

Let $1<p<\infty$ and $f \in L_1([-\pi, \pi])$ be a function with Fourier series $\sum\limits_{k=-\infty}^{\infty} a_{k} e^{i{kx}}$. Let also $a=\{a_{k}\}_{k=-\infty}^{\infty} \in \textnormal{GM}^*$, then

Theorems & Definitions (49)

  • Definition 1: Ti
  • Definition 2
  • Remark 1
  • Definition 3
  • Remark 2
  • Definition 4
  • Remark 3
  • Theorem 1
  • Theorem 2
  • Remark 4
  • ...and 39 more