A characterization of two-depth Journ\'{e} packing for bi-parameter and Zygmund rectangles

Abstract

We completely characterize a two-depth version of Journ\'{e}'s covering lemma in the bi-parameter setting. Let $\mathcal U$ be an incomparable family of bi-parameter dyadic rectangles contained in $\Omega$, and write $e_i(I):=e_i(I;\Omega)$ for the two embeddedness depths. For every coordinatewise nonincreasing $w\colon\mathbb N^2\to[0,\infty)$, the estimate \[ \sum_{I\in\mathcal U}w(e_1(I),e_2(I))|I|\lesssim_w|\Omega| \] holds uniformly in $(\mathcal U,\Omega)$ 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 $0<s<1$, so this allows weights that are not summable in either variable, while one-depth theory holds precisely for summable weights. Exactly the same characterization holds for dyadic Zygmund rectangles $I=I^1\times I^2\times I^3$ with $\ell(I^3)=\ell(I^1)\ell(I^2)$. We use probabilistic packing methods to prove the sufficiency, and these techniques also yield sparse refinements of the preceding estimates, which self-improve to various forms of exponential integrability.