Calculus of Variations and Geometric Measure Theory

Kakeya singular integral operators and quantitative estimates for Lagrangian flows with $BV$ vector fields


created by ambrosio on 27 Apr 2018
modified by paolini on 01 May 2018

3 may 2018 -- 16:00   [open in google calendar]

Aula Tonelli, Scuola Normale Superiore


Abstract: In this talk, we introduce a Kakeya singular integral operator and establish a weak type $(1,1)$ bound for this operator. We then apply it to solve a main open problem mentioned in L. Ambrosio and G. Crippa, 2014. Specifically, we prove the well posedness of regular Lagrangian flows associated to vector fields \[ B=(B^1,\ldots,B^d)\in L^1((0,T);L^1\cap L^\infty({\bf R}^d)) \] representable as \[ B^i=\sum_{j=1}^{m}{\bf K}_j^i*b_j,\qquad b_j\in L^1((0,T),BV({\bf R}^d)) \] with ${\rm div}(B)\in L^1((0,T);L^\infty({\bf R}^d))$, where $(K_j^i)_{i,j}$ are singular kernels in ${\bf R}^d$.