Estimates for the square variation of partial sums of Fourier series and their rearrangements

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( 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( 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 ∞ case. Several other results on operators of this form are also obtained. The proofs rely on combinatorial and probabilistic methods.

Dimension-Free Estimates for Discrete Hardy-Littlewood Averaging Operators Over the Cubes in ℤd

“…Inequality (1.21) replaces the fractional integration argument from [8] (as it is not clear if this argument is available in the discrete setting) and allows us to obtain (1.12) for p ∈ (3/2, 2]. A variant of this inequality was proven by Lewko-Lewko [12,Lemma 13] in the context of variational Rademacher-Menshov type results for orthonormal systems and it was also obtained independently by the second author and Trojan [15,Lemma 1] in the context of variational estimates for discrete Radon transforms, see also [14]. Inequality (1.21) reduces estimates for a supremum or an r-variation restricted to a dyadic block to the situation of certain square functions, where the division intervals over which differences are taken (in these square functions) are all of the same size.…”

A bootstrapping approach to jump inequalities and their applications

“…The variation norm on the left-hand side of (2.6) can be extended to all t ∈ [0, 2 k ] if g : [0, 2 k ] → C is continuous. Lemma 2.5 originates in the paper of Lewko and Lewko [LL12], where it was observed that the 2-variation norm of a sequence of length N can be controlled by the sum of log N square functions and this observation was used to obtain a variational version of the Rademacher-Menshov theorem. Inequality (2.6), essentially in this form, was independently proved by the first author and Trojan in [MT16] and used to estimate r-variations for discrete Radon transforms.…”

ℓp(Zd)-estimates for discrete operators of Radon type: Maximal functions and vector-valued estimates

“…A variational variant of this inequality was proven by Lewko-Lewko [11,Lemma 13] in the context of variational Rademacher-Menshov theorems. It was also obtained independently by the first two authors in [15] in the context of variational estimates for discrete Radon transforms, see also [14].…”

“…(ii) A maximal estimate in terms of dyadic sub-blocks (see (2.7)). It is a consequence of a numerical maximal estimate (see Lemma 2.3), which in turn is an outgrowth of the idea implicit in the proof of the classical Rademacher-Menshov theorem (see [21] and also [11] and [15]). (iii) A refinement of the estimates for multi-dimensional Weyl's sums in [19], where the previous limitations N ǫ ≤ q ≤ N k−ǫ are replaced by the weaker restrictions (log N ) β ≤ q ≤ N k (log N ) −β for suitable β.…”