Sobolev martingales

Abstract

We introduce new spaces of martingales that behave like L_1 -based Sobolev spaces on \mathbb R^d . We prove a martingale analog of Van Schaftingen’s theorem and give sharp estimates on the lower Hausdorff dimension of the terminal distributions of these martingales. We also provide martingale analogs of trace theorems for Sobolev functions.

Hausdorff dimension of measures with arithmetically restricted spectrum

“…In the first case, the extremal points are ±(1, −1, 1, −1), while for the second choice they are ±(1, 1, −1, −1), ±(1, −1, −1, 1). This gives δ 4 ≥ 1 2 .…”

On finite configurations in the spectra of singular measures

“…Then, if µ P M `pTq is a measure such that p µ is supported on integers of the form q N m, where the residue of m modulo q belongs to B, then dim H pµq ě cpBq ą 0, where cpBq is a constant depending on B only. This result was derived from a theorem proved in [ASW21b], concerning dimensional estimates for measures satisfying martingale analogs of constraints given by bundles.…”

Hardy--Littlewood--Sobolev inequality for $p=1$

“…The paper [3] suggested a discrete model for the problems mentioned above: the spaces BV Ω have relatives in the world of discrete time martingales over regular filtrations. In the discrete model, the problems are simpler, and the paper [3] contains solutions to them. The approach is based upon four ingredients: the monotonicity formula (in the world of discrete martingales it reduces to a simple form of convexity), the splitting into convex and flat atoms, an improvement of the monotonicity formula in the case the corresponding cancellation condition holds true, and a combinatorial argument.…”