On Weak Type Inequalities for Rare Maximal Functions in ℝⁿ

<div class="line" id="line-5"> A weak L(log+ L)n&minus;1 type inequality |{Mf(x) &gt; &lambda;}| &le; C &int; |f|&lambda; (1 + log+ |f|&lambda;)n-1 is a quantitative version of Jessen-Marcinkiewich-Zygmund theorem. It was established by Miguel de Guzman in 1972. It is easy to see that this is the best possible estimate.</div><div class="line" id="line-7"> <br/></div><div class="line" id="line-9"> By Rare Maximal Function we understand the maximal function with respect to rectangles whose side length could be any number from a given in&filig;nite sparse set of positive real numbers. If this set is dense enough then the Rare Maximal Function is pointwise comparable with the Strong Maximal Function, and thus is of a weak type L(log+ L)n&minus;1. Generally, a rarefaction of the set of rectangles could improve the estimate for the corresponding maximal function. In the talk we prove that this is not true for the Rare Maximal Functions in Rn for any n &gt; 1, i.e. that the rarefaction of the side-length of the rectangles does not improve the properties of the corresponding maximal functions. Thus, we extend some results known for R2 only.</div>