Roger L. Jones

Strong variational and jump inequalities in harmonic analysis

We prove variational and jump inequalities for a large class of linear operators arising in harmonic analysis.

Oscillation and variation for the Hilbert transform

It is well known that this limit exists a.e. for all f ∈ L, 1 ≤ p < ∞. In this paper, we will consider the oscillation and variation of this family of operators as goes to zero, which gives extra information on their convergence as well as an estimate on the number of λ-jumps they can have. For earlier results on oscillation and variation operators in analysis and ergodic theory, including some historical remarks and applications, the reader may look in [2], [3], [5], [4], and [6]. For each fixed sequence (ti) ↘ 0, we define the oscillation operator ( H∗f ) (x) = ( ∞ ∑

Oscillation and variation for singular integrals in higher dimensions

In this paper we continue our investigations of square function inequalities in harmonic analysis. Here we investigate oscillation and variation inequalities for singular integral operators in dimensions d > 1. Our estimates give quantitative information on the speed of convergence of truncations of a singular integral operator, including upcrossing and A jump inequalities.

Square functions in ergodic theory

Given the usual averages in ergodic theory, let n 1 ≤ n 2 ≤ … and . There is a strong inequality ‖ Sf ‖ 2 ≤ 25‖ f ‖ 2 and there is a weak inequality m { Sf > λ} ≤ (7000/λ)‖f‖ 1 . Related results and questions for other variants of this square function are also discussed.

Oscillation in ergodic theory: Higher dimensional results

In this paper we continue our investigations of square function inequalities. The results in [9] are primarily one dimensional, and here we extend all the results to multi-dimensional averages. Our basic tool is still a comparison of the ergodic averages with various dyadic (reversed) martingales, but the Fourier transform arguments are replaced by more geometric almost orthogonality arguments.The results imply the pointwise ergodic theorem for the action of commuting measure preserving transformations, and give additional information such as control of the number of upcrossings of the ergodic averages. Related differentiation results are also discussed.

The strong sweeping out property for lacunary sequences, Riemann sums, convolution powers, and related matters

In this paper we establish conditions on a sequence of operators which imply divergence. In fact, we give conditions which imply that we can find a set B of measure as close to zero as we like, but such that the operators applied to the characteristic function of this set have a lim sup equal to 1 and a lim inf equal to 0 a.e. (strong sweeping out). The results include the fact that ergodic averages along lacunary sequences, certain convolution powers, and the Riemann sums considered by Rudin are all strong sweeping out. One of the criteria for strong sweeping out involves a condition on the Fourier transform of the sequence of measures, which is often easily checked. The second criterion for strong sweeping out involves showing that a sequence of numbers satisfies a property similar to the conclusion of Kroneckers lemma on sequences linearly independent over the rationals.

Ergodic averages on spheres

AbstractLetU1,U2, …,Un denoten commuting ergodic invertible measure preserving flows on a probability space (X,Σ,m). LetSr denote the sphere of radiusr inRn, and αr the rotationally invariant unit measure onSr. WriteUtx to denote

Oscillation inequalities for rectangles

U_1^{t_1 } ...U_n^{t_n } x