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 L2 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(√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 L2 norm of the associated V 2 operator is O(√ln ln(N) ). We also show that for p >2, a bounded ONS of size N can be rearranged so that the L2 norm of the V p operator is at most Op(ln ln(N)) uniformly for all choices of coefficients. This refines Bourgain’s work on Garsia’s conjecture, which is equivalent to the V∞ case. Several other results on operators of this form are also obtained. The proofs rely on combinatorial and probabilistic methods. © 2011 Elsevier Inc. All rights reserved.