[CvGmt News] [cvgmt seminar] Kakeya singular integral operators and quantitative estimates for Lagrangian flows with $BV$ vector fields

[ambrosio at sns.it]

Kakeya singular integral operators and quantitative estimates for Lagrangian flows with $BV$ vector fields
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http://cvgmt.sns.it/seminar/638/

Date: Thursday, May  3, 2018

Time: 16:00

Place: Aula Tonelli, Scuola Normale Superiore

Speaker: QUOC HUNG NGUYEN



Abstract. 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 ${\bf R}^d$.