2021/01/21 by Dmitriy Dmitrishin, Dmitrishin, Dmitriy, Paul Hagelstein +3
Mathematics · #Advanced Harmonic Analysis Research #Nonlinear Partial Differential Equations #Advanced Banach Space Theory
paper · pdf · doi:10.48550/arxiv.2101.08736
Let B be a collection of rectangular parallelepipeds in ℝ3 whose sides are parallel to the coordinate axes and such that B contains parallelepipeds with side lengths of the form s, (2N)/(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 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 .