Title
Decoupling estimates for fractal curves
Abstract
Decoupling estimates are an important tool in harmonic analysis with far-reaching applications to problems in geometric measure theory, PDEs and number theory. The main idea is to take a function with frequencies near a set with special arithmetic structure, and decompose the Fourier support into disjoint boxes to study the cancellation between the different frequency pieces caused by destructive interference. In their foundational work, Bourgain and Demeter developed decoupling theory for the parabola. Curvature plays an important role here, as it gives the parabola a non-linear structure which promotes a lot of cancellation. The theory had been extended to other curves, but under strong assumptions on the regularity as well as the curvature.
I will discuss a recent work, where I extend the Bourgain--Demeter decoupling theorem to arbitrary convex curves in the plane, in a meaningful way. The proof uses a robust technique known as the high/low method, which has seen success across many related problems in harmonic analysis and incidence geometry. I will also discuss examples of self-similar fractal curves, that satisfy better decoupling estimates than the parabola due to stronger cancellation phenomena.
