Modulation invariant operators

TL;DR

This thesis explores advanced harmonic analysis topics, including bounds for singular Brascamp-Lieb forms, polynomial Carleson operators, and directional square functions, advancing understanding of their boundedness and interactions.

Contribution

It provides new bounds and partial results for Brascamp-Lieb forms, polynomial Carleson operators, and directional square functions, extending previous work and addressing conjectures.

Findings

Improved bounds for truncations of Brascamp-Lieb forms

Partial results for dyadic trilinear models in 2D

Estimates for multidimensional polynomial Carleson operators

Abstract

The first three results in this thesis are motivated by a far-reaching conjecture on boundedness of singular Brascamp-Lieb forms. Firstly, we improve over the trivial estimate for their truncations, thus excluding potential trivial counterexamples. Secondly, in a dyadic model we give a partial result in the trilinear, 2-dimensional case when one of the functions depends only on one variable. Thirdly, we estimate the multidimensional version of the polynomial Carleson operator, whose boundedness would also be a consequence of the conejcture. The final pair of results concerns directional square functions. One of them concerns interaction of Lipschitz change of variable on the line with Littlewood--Paley decomposition and is used to extend the conditional result of Lacey and Li for Hilbert transform along $C^{1+\epsilon}$ vector fields to Lipschitz vector fields. The other concerns directional averaging operators and gives an alternative approach to a single scale maximal function with averages in N directions due to Katz away from the endpoint.