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

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.