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We investigate the square variation operator $V^2$ (which majorizes thepartial 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 $O(\ln(N))$ onany ONS. This result is sharp and refines the classical Rademacher-Menshovtheorem. We show that this can be improved to $O(\sqrt{\ln(N)})$ for thetrigonometric system, which is also sharp. We show that for any choice ofcoefficients, this truncation of the trigonometric system can be rearranged sothat 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 sothat 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 onGarsia's conjecture, which is equivalent to the $V^{\infty}$ case. Severalother results on operators of this form are also obtained. The proofs rely oncombinatorial and probabilistic methods.

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