Estimates for the Square Variation of Partial Sums of Fourier Series and their Rearrangements

TL;DR

This paper provides sharp bounds for the square variation operator on orthonormal systems, improving classical results and showing how rearrangements can significantly reduce the operator norms, with implications for Fourier series analysis.

Contribution

It establishes sharp bounds for the $V^2$ operator on orthonormal systems and demonstrates how rearrangements can optimize these bounds, refining classical theorems and conjectures.

Findings

Bound of $O(\ln(N))$ for $V^2$ on any ONS, sharpness confirmed.

Improved bound of $O(\sqrt{\ln(N)})$ for trigonometric system, sharp.

Rearrangements can reduce bounds to $O(\sqrt{\ln\ln(N)})$ for $V^2$, and to $O(\ln\ln(N))$ for $V^p$, $p>2$.

Abstract

We investigate the square variation operator $V^2$ (which majorizes the partial sum maximal operator) on general orthonormal systems (ONS) of size $N$. We prove that the $L^2$ norm of the $V^2$ operator is bounded by $O(\ln(N))$ on any ONS. This result is sharp and refines the classical Rademacher-Menshov theorem. We show that this can be improved to $O(\sqrt{\ln(N)})$ for the trigonometric system, which is also sharp. We show that for any choice of coefficients, this truncation of the trigonometric system can be rearranged so that the $L^2$ norm of the associated $V^2$ operator is $O(\sqrt{\ln\ln(N)})$. We also show that for $p>2$, a bounded ONS of size $N$ can be rearranged so that the $L^2$ norm of the $V^p$ operator is at most $O_p(\ln \ln (N))$ uniformly for all choices of coefficients. This refines Bourgain's work on Garsia's conjecture, which is equivalent to the $V^{\infty}$ case. Several other results on operators of this form are also obtained. The proofs rely on combinatorial and probabilistic methods.