Directional operators and the multiplier problem for the polygon

Abstract: I will discuss two recent results obtained in collaboration with I. Parissis (U Basque Country). The first is a sharp L^2 estimate for the maximal averaging operator associated to sets of directions from algebraic sets in R^n of arbitrary codimension. The proof uses a new scheme of polynomial partitioning on manifolds which extends ideas by Larry Guth. The second result is a sharp L^4 estimate for the Fourier multiplier associated to a polygon of N sides in R^2, and a sharp form of the two parameter Meyer’s lemma. These results improve on the usual ones obtained via weighted norm inequalities and rely on a novel Carleson measure estimate for directional square functions of time-frequency nature.