BCAM Working Seminar APDE: Vector-valued extensions of multilinear operators and a multilinear UMD condition

Abstract
Vector-valued extensions of important operators in harmonic analysis have been actively studied in the past decades. A centerpoint of the theory is the result of Burkholder and Bourgain that the Hilbert transform extends to a bounded operator on $L^p(mathbb{R};X)$ if and only if the Banach space $X$ has the so-called UMD property. In the specific case where $X$ is a Banach function space, it is a deep result of Bourgain and Rubio de Francia that this UMD property is equivalent to having bounded vector-valued extensions of the Hardy-Littlewood maximal operator to both $X$ and to its dual $X^ast$. In this talk I will place these ideas in the context of the modern technique of domination by sparse forms. These forms are intimately related to Muckenhoupt weight classes and the multilsubinear Hardy-Littlewood maximal operator. 

Moreover, I will discuss the current progress in extending the UMD property to a multilinear setting. In joint work with Emiel Lorist, we first considered such a condition for tuples of Banach function spaces. Using this notion, we developed a way of obtaining vector-valued sparse domination from scalar-valued sparse domination. I will talk about how this technique provides new quantitative weighted vector-valued bounds for a class of operators including multilinear Calderón-Zygmund operators and the bilinear Hilbert transform.

Link to the session: https://zoom.us/j/93027483205?pwd=Z2pyQTdJcmM3M2RIMkZ1bHZOYXhDQT09