Joint BCAM-UPV/EHU Analysis and PDE seminar: Pointwise localization and sharp weighted bounds for Rubio de Francia square functions

The Rubio de Francia square function is the square function formed by frequency projections over a collection of disjoint intervals of the real line. J. L. Rubio de Francia proved in 1985 that this operator is bounded in L^p for p\ge 2 and in L^p(w) for p > 2 with weights w in the Muckenhoupt class A_{p/2}. What happens in the endpoint L^2(w) for w\in A_1 was left open, and Rubio de Francia conjectured the validity of the boundedness.

In this talk we will show a new pointwise sparse bound for the Rubio de Francia square function. This sparse bound leads to quantified weighted norm inequalities. We will also show that the weighted L^2-conjecture holds for radially decreasing even weights and in full generality for the Walsh group analogue of the Rubio de Francia square function; in general the weighted L^2 inequality is at this point still an open problem. 

In the first part of the talk, we will give the background of the problem while in the second part we will explain the new results mentioned above.

This talk is based on a joint work with F. Di Plinio, I. Parissis and L. Roncal.