A characterization of two-depth Journé packing for bi-parameter and Zygmund rectangles

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 $\Omega$, and write $e_i(I):=e_i(I;\Omega )$ for the two embeddedness depths. For every coordinatewise nonincreasing $w:{\mathbb{N}}^2\to [0,\infty )$, the estimate \[ \sum_{I\in\mathcal U}w(e_1(I),e_2(I))|I|\lesssim_w|Ω| \] holds uniformly in $(\mathcal{U},\Omega )$ 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, $\Vert (a+1{)}^{-s}(b+1{)}^{-(1-s)}{\Vert }_{\mathrm{ft}}\sim [s(1-s){]}^{-1}$ 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.

arXiv commentv2: 49 pages; the main result has been strengthened to a complete characterization of two-depth packing