Sparse domination was first introduced in the context of Calderón--Zygmund theory. Shortly after, the concept was successfully extended to many other operators in Harmonic Analysis, although many endpoint situations have remained unknown. In this talk, we will present new endpoint sparse bounds for oscillatory and Miyachi-type Fourier multipliers using Littlewood—Paley theory. Furthermore, the results can be extended to more general dilation-invariant classes of multiplier transformations via Hardy space techniques, yielding results, for instance, for multi-scale sums of radial bump multipliers.

This is joint work with Joris Roos and Andreas Seeger

https://sites.google.com/view/paw-seminar