New perspectives on vectorial Hankel operators

For nearly two decades, the Ferguson-Lacey theorem (Ferguson and Lacey, Acta Mathematica 2002) has been a cornerstone in multi-parameter harmonic analysis.

However, a recent observation by Volberg illustrates a gap in the argument of Ferguson and Lacey, effectively  degrading their main result from theorem to conjecture. Attempts to rectify the error have so far been unsuccessful.The proposal has five subprojects. The first three are immediately connected to the Ferguson--Lacey theorem.

A key idea is to decompose the problem into questions about what I will call Calderon-Hankel operators (see Rydhe, Geom. Funct. Anal. 2017), Schur multipliers, and factorization theorems. These notions have been central in my work on harmonic analysis of vector-valued functions.

Multi-parameter harmonic analysis forms a natural middle ground between the classical and vectorial settings.The fourth subproject investigates applications to weighted Fourier estimates and Fourier restriction theory.

The fifth relates vectorial Hankel operators to bounded functional calculi and questions from control theory and C_0-semigroups.