Variational and Jump Inequalities

Explore variational and jump inequalities in this 40-minute lecture by Pavel Zorin-Kranich from the Hausdorff Center for Mathematics. Delve into the theory of rough paths and discover how variational norms, as parametrization-invariant versions of Hölder norms, quantify convergence results for truncated singular integrals and ergodic averages. Examine endpoint versions of quantified results using jump norms, and learn about their applications in diffusion semigroups, ergodic averages, and stochastic integrals. Follow the lecture's progression from Lépingle's inequality to periodic multipliers, discrete Radon transforms, and variational estimates for martingale paraproducts, gaining insights from joint works with M. Mirek, E. Stein, and V. Kovač.

Intro
Lépingle's inequality
Proof of endpoint Lépingle inequality
Jumps as a real interpolation space
Periodic multipliers
Application: discrete Radon transforms
Variational estimate for martingale paraproduct
Application: stochastic integrals