Sharp weak type estimates for a family of Soria bases
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Publication:6358747
DOI10.1007/S12220-022-00903-5zbMath1485.42034arXiv2101.08736MaRDI QIDQ6358747
Dmitriy Dmitrishin, Alex Stokolos, Paul Alton Hagelstein
Publication date: 21 January 2021
Abstract: Let $mathcal{B}$ be a collection of rectangular parallelepipeds in $mathbb{R}^3$ whose sides are parallel to the coordinate axes and such that $mathcal{B}$ contains parallelepipeds with side lengths of the form $s, frac{2^N}{s} , t $, where $s, t > 0$ and $N$ lies in a nonempty subset $S$ of the natural numbers. We show that if $S$ is an infinite set, then the associated geometric maximal operator $M_mathcal{B}$ satisfies the weak type estimate $$left|left{x in mathbb{R}^3 : M_{mathcal{B}}f(x) > alpha
ight}
ight| leq C int_{mathbb{R}^3} frac{|f|}{alpha} left(1 + log^+ frac{|f|}{alpha}
ight)^{2}$$ but does not satisfy an estimate of the form $$left|left{x in mathbb{R}^3 : M_{mathcal{B}}f(x) > alpha
ight}
ight| leq C int_{mathbb{R}^3} phileft(frac{|f|}{alpha}
ight)$$ for any convex increasing function $phi: mathbb[0, infty)
ightarrow [0, infty)$ satisfying the condition $$lim_{x
ightarrow infty}frac{phi(x)}{x (log(1 + x))^2} = 0;.$$
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