涩里番

鉁掞笍 TITLE / TITRE

Every weak type $L^1$ bound for the maximal function has an underlying covering lemma.

听

馃搫 ABSTRACT / R脡SUM脡

Let $X$ be a separable metric space, $\mu$ a Borel measure on $X$, and $B$ some family of open sets of finite positive measure. Define the maximal function of a non-negative integrable function $f$ by听$$听Mf(x)=\sup_{R\in B, x\in R}\mu(R)^{-1}\int_R f\,d\mu\,. $$听The weak type $L^1$ bound for $M$, i.e., the inequality $$ \mu(\{x\in X:Mf(x)>t\})\le Ct^{-1}\int_X f\,d\mu\,$$ in various settings is usually derived from some covering selection property, the most general version of which seems to be as follows:

There exist constants $c,C\in(0,+\infty)$ such that for every finite family $B_0\subset B$, there is a subfamily $B_1\subset B_0$ satisfying $\mu(\cup_{R\in B_1}R)\ge c\mu(\cup_{R\in B_0}R)$ and $\sum_{R\in B_1}\chi_R\le C$ $\mu$-almost everywhere.

We shall show that there cannot be any other reason for a weak type $L^1$ bound of the above type, namely, that if the weak type bound holds for the maximal function $M$ associated with the family $B$, then $B$ necessarily has this covering selection property. Time permitting, we'll discuss analogues of this theorem for the similar bounds for $M$ involving $L\log L$ and other Orlich type expressions on the right-hand side.

This is a joint work with Paul Hagelstein and Blanca Radillo-Murguia.

馃搷 PLACE / LIEU
Hybride - CRM, Salle / Room 1177, Pavillon Andr茅 Aisenstadt