In this talk I will give a brief overview of the theory of maximal directional averages and singular integrals in 2 and higher dimensions. We will see the main obstructions to the boundedness of these objects which will naturally lead us to the discussion of the connections with the Kakeya conjecture and the Stein and Zygmund conjectures. Finally, I will present a sharp estimate for maximal directional singular integrals in codimension one and general ambient dimension. This reports on joint work with O. Bakas (U. of Patras), F. Di Plinio (Napoli, Federico II) and L. Roncal (BCAM).

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