Let us denote by Rubio de Francia square function the square function formed by frequency projections on a collection of disjoint intervals of the real line. J. L. Rubio de Francia established in 1985 that this operator is bounded on $L^p$ for $p\ge 2$ and on $L^p(w)$, for $p>2$, with weights $w$ in the Muckenhoupt class $A_{p/2}$. What happens in the endpoint $L^1(w)$ for $w \in A_1$ was left open, and Rubio de Francia conjectured the validity of the estimate in this endpoint.

In this talk we will show a new pointwise sparse estimate for the Rubio de Francia square function. Such a bound implies quantitative weighted estimates which, in some cases, improve the available results. We will also confirm that the $L^2(w)$ conjecture is verified for radially decreasing even $A_1$ weights, and in full generality for the Walsh group analogue to the Rubio de Francia square function. In general, the $L^2$ weighted inequality is still an open problem.

Joint work with Francesco Di Plinio, Mikel Flórez-Amatriain, and Ioannis Parissis.

The video of this talk is available on the IISc Math Department channel.