We prove strong jump inequalities for a large class of operators of Radon type in the discrete and ergodic theoretical settings. These inequalities are the r = 2 endpoints of the r-variational estimates studied in [MST17], and are formulated in terms of the jump quasi-seminorm. Here we rely on the facts about jumps derived in [MSZ18a], and also used in [MSZ18b].

The aim of this paper is to present an abstract and general approach to jump inequalities in harmonic analysis. Our principal conclusion is the refinement of r-variational estimates, previously known for r > 2, to end-point results for the jump quasi-seminorm corresponding to r = 2. This is applied to the dimension-free results recently obtained by the first two authors in collaboration with Bourgain, and Wróbel, and also to operators of Radon type treated by Jones, Seeger, and Wright.

Jump inequalities are the r = 2 endpoint of Lépingle's inequality for r-variation of martingales. Extending earlier work by Pisier and Xu [PX88] we interpret these inequalities in terms of Banach spaces which are real interpolation spaces. This interpretation is used to prove endpoint jump estimates for vectorvalued martingales and doubly stochastic operators as well as to pass via sampling from R d to Z d for jump estimates for Fourier multipliers.The quantity N λ (f (t) : t ∈ I) is monotonically increasing in I and does not changeFor a measure space (X, B, m) we consider the family of jump quasi-seminorms on functions f :for 0 < p < ∞, 0 < q ≤ ∞, and 1 < < ∞. The quantity J p,q is monotonically decreasing in q. We omit the index q if q = p.The jump quasi-seminorms J p r are dominated by L p norms of r-variations (which are recalled in (2.1)) in view of (2.3). On the other hand, L p bounds for r-variations can be deduced from estimates for J p only when r > , using Lemma 2.12 and interpolation. Since in many situations one does not have r-variational control for r = 2, having jump control for = 2 is then an end-point refinement.We now briefly highlight the main results of this paper.(1) We prove that the quantities J p,q are in fact equivalent to Banach space norms which arise via real interpolation (see Lemma 2.7 and Corollary 2.11).

ABSTRACT. We prove q-variation estimates, q > 2, on ℓ p spaces for averages along primes (with 1 < p < ∞) and polynomials (with |1/p − 1/2| < 1/2(d + 1), where d is the degree of the polynomial). This improves the pointwise ergodic theorems for these averages in the corresponding ranges of L p spaces.

We prove p L p decoupling inequalities for a class of moment manifolds. These inequalities imply optimal mean value estimates for multidimensional Weyl sums of the kind considered by Arkhipov, Chubarikov, and Karatsuba and by Parsell.In our proofs we take a new point of view on the Bourgain-Demeter-Guth induction on scales argument. This point of view substantially simplifies even the proof of 2 L p decoupling for the moment curve.