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DESCRIPTION:✒️ TITLE / TITRE\n\nEvery weak type $L^1$ bound for the maximal
  function has an underlying covering lemma.\n\n \n\n📄 ABSTRACT / RÉSUMÉ\n
 \nLet $X$ be a separable metric space\, $\mu$ a Borel measure on $X$\, and
  $B$ some family of open sets of finite positive measure. Define the maxim
 al function of a non-negative integrable function $f$ by $$ Mf(x)=\sup_{R\
 in B\, x\in R}\mu(R)^{-1}\int_R f\\,d\mu\\,. $$ 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 foll
 ows:\n	\n	There exist constants $c\,C\in(0\,+\infty)$ such that for every fi
 nite family $B_0\subset B$\, there is a subfamily $B_1\subset B_0$ satisfy
 ing $\mu(\cup_{R\in B_1}R)\ge c\mu(\cup_{R\in B_0}R)$ and $\sum_{R\in B_1}
 \chi_R\le C$ $\mu$-almost everywhere.\n	\n	We shall show that there cannot b
 e 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 $M$ associate
 d with the family $B$\, then $B$ necessarily has this covering selection p
 roperty. Time permitting\, we'll discuss analogues of this theorem for the
  similar bounds for $M$ involving $L\log L$ and other Orlich type expressi
 ons on the right-hand side.\n	\n	This is a joint work with Paul Hagelstein a
 nd Blanca Radillo-Murguia.\n\n📍 PLACE / LIEU \n	Hybride - CRM\, Salle / Roo
 m 1177\, Pavillon André Aisenstadt\n
DTSTART:20260501T193000Z
DTEND:20260501T193000Z
SUMMARY:Fedor Nazarov (Kent State University)
URL:/mathstat/channels/event/fedor-nazarov-kent-state-
 university-372757
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