Year: 2026
Type: preprint
Abstract: We completely characterize a two-depth version of Journé's covering lemma in the bi-parameter setting. Let $\mathcal U$ be an incomparable family of bi-parameter dyadic rectangles contained in $Ω$, and write $e_i(I):=e_i(I;Ω)$ for the two embeddedness depths. For every coordinatewise nonincreasing $w\colon\mathbb N^2\to[0,\infty)$, the estimate \[ \sum_{I\in\mathcal U}w(e_1(I),e_2(I))|I|\lesssim_w|Ω| \] holds uniformly in $(\mathcal U,Ω)$ if and only if the following fixed-total condition holds: \[ \|w\|_{\mathrm{ft}}:=\sup_{N\geq0}\sum_{a=0}^Nw(a,N-a)<\infty. \] For instance, $\|(a+1)^{-s}(b+1)^{-(1-s)}\|_{\mathrm{ft}}\sim[s(1-s)]^{-1}$ for $0
Source: arXiv (Cornell University)
Author: Martikainen, Henri
Institution: Washington University in St. Louis
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204204431