Bounds for maximal operators arising from a geometric problem

ANDREA OLIVO

Rio de Janeiro

Jornada; World Meeting for Women in Mathematics; 2018

Instituto de matematica pura y aplicada (IMPA)

A problem recently studied is the relation between sizes of sets $B, S subseteq R^2$ when B contains the boundary of a square with center in every point of S and sides parallel to the axis. By size, we mean Lebesgue measure and some fractal dimensions.More generally, in higher dimensions, the sets consider are $B, S subseteq R^n$ when B contains the k-skeleton of an n-dimensional cube around every point of S.This type of problems have associated a maximal operator. In this work we studya possible maximal operator and present results about its behaviour from $L^p → L^q$, for $1 ≤ p ≤ infty$. Whit this bounds we recover several results for the sizes of sets B.