Calder\'on-Zygmund operators cannot be bounded on $L^2$ with totally irregular measures

Series
Analysis Seminar
Time
Wednesday, March 14, 2018 - 1:55pm for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Jose Conde Alonso – Brown University – jose_conde_alonso@brown.eduhttps://sites.google.com/view/josecondealonso
Organizer
Galyna Livshyts
We consider totally irregular measures μ in Rn+1, that is, lim supr0μ(B(x,r))(2r)n>0&lim infr0μ(B(x,r))(2r)n=0for μ almost every x. We will show that if Tμf(x)=K(x,y)f(y)dμ(y) is an operator whose kernel K(,) is the gradient of the fundamental solution for a uniformly elliptic operator in divergence form associated with a matrix with H\"older continuous coefficients, then Tμ is not bounded in L2(μ).This extends a celebrated result proved previously by Eiderman, Nazarov and Volberg for the n-dimensional Riesz transform and is part of the program to clarify the connection between rectifiability of sets/measures on Rn+1 and boundedness of singular integrals there. Based on joint work with Mihalis Mourgoglou and Xavier Tolsa.