2021/12/03 by Hagelstein, Paul, Stokolos, Alex · 1 citation
#42B25 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2112.02038
Let B be a collection of rectangular parallelepipeds in ℝ3 whose sides are parallel to the coordinate axes and such that B consists of parallelepipeds with side lengths of the form s, 2j s, t , where s, t > 0 and j lies in a nonempty subset S of the integers. In this paper, we prove the following: If S is a finite set, then the associated geometric maximal operator MB satisfies the weak type estimate of the form |\x ∈ ℝ3 : MBf(x) gt; α\| ≤ C ∫ℝ3 \frac|f|α(1 + log+ \frac|f|α) but does not satisfy an estimate of the form |\x ∈ ℝ3 : MBf(x) gt; α\| ≤ C ∫ℝ3 ϕ(\frac|f|α) for any convex increasing function ϕ: \mathbb[0, ∞) → [0, ∞) satisfying the condition limx → ∞(ϕ(x))/(x (log(1 + x))) = 0 . On the other hand, if S is an infinite set, then the associated geometric maximal operator MB satisfies the weak type estimate |\x ∈ ℝ3 : MBf(x) gt; α\| ≤ C ∫ℝ3 \frac|f|α (1 + log+ \frac|f|α)2 but does not satisfy an estimate of the form |\x ∈ ℝ3 : MBf(x) gt; α\| ≤ C ∫ℝ3 ϕ(\frac|f|α) for any convex increasing function ϕ: \mathbb[0, ∞) → [0, ∞) satisfying the condition limx → ∞(ϕ(x))/(x (log(1 + x))2) = 0 .