"Maximal averages and singular integrals along vector fields"

Abstract: The averages of a locally integrable function over families of shrinking Euclidean balls converge almost everywhere: this is the Lebesgue differentiation theorem. A fundamental conjecture, attributed to Antoni Zygmund, is whether a square integrable function on the plane may be differentiated by Lipschitz families of lines: this is equivalent to L^2 bounds for the maximal averaging operator along a Lipschitz vector field. The operator obtained by replacing the maximal average by a directional singular integral is the object of a companion conjecture due to Elias Stein. Directional singular integrals share deep ties to other central objects in Analysis: most notably, the pointwise convergence of Fourier series and the polygonal summation problem. I will discuss recent progress on Stein’s conjecture and on finitary models of both problems. In particular, I will focus on the L^2 sharp bound for directional maximal averages along finite subsets of algebraic varieties of arbitrary codimension, and on its proof involving polynomial partitioning techniques. This result is in collaboration with Ioannis Parissis (University of Basque Country).


Host: Brett Wick

Tea will be served @ 3:45 in room 200