刊名:Journal of Mathematical Analysis and Applications
出版年:2017
出版时间:15 March 2017
年:2017
卷:447
期:2
页码:951-956
全文大小:291 K
文摘
We consider Ramsey-type problems associated to collections of sets in Rn satisfying a standard geometric regularity condition. In particular, let be a collection of measurable sets in Rn such that every Rj is contained in a cube Qj whose sides are parallel to the axes and such that |Rj|/|Qj|≥ρ>0. Moreover, suppose that there exists 0<γ<∞ such that |Rj|/|Rk|≤γ for every j,k. We prove that there exists a subcollection of consisting of at least R(N) sets that either have a point of common intersection or that are pairwise disjoint, where . If the sets in the collection {Rj} are convex, we obtain the improved Ramsey estimate R(N)≥(3−nρN)1/2. Applications of these results to weak type bounds of geometric maximal operators are provided.