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 𝒰 be an incomparable family of bi-parameter dyadic rectangles contained in Ω, and write ei(I):=ei(I;Ω) for the two embeddedness depths. For every coordinatewise nonincreasing w:ℕ2→[0,∞), the estimate ∑I∈𝒰w(e1(I),e2(I))|I|≲w|Ω| holds uniformly in (𝒰,Ω) if and only if the following fixed-total condition holds: |w|ft:=supN≥0∑a=0Nw(a,N−a)<∞. For instance, |(a+1)−s(b+1)−(1−s)|ft∼[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=I1× I2× I3 with ℓ(I3)=ℓ(I1)ℓ(I2). 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.
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BibTeX
@misc{martikainen2026characterization,
title = {{A characterization of two-depth Journé packing for bi-parameter and Zygmund rectangles}},
author = {Henri Martikainen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.22628},
eprint = {2608.22628},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.22628},
}
RIS
TY - UNPB
TI - A characterization of two-depth Journé packing for bi-parameter and Zygmund rectangles
AU - Henri Martikainen
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2608.22628
UR - https://arxiv.org/abs/2608.22628
AB - We completely characterize a two-depth version of Journé's covering lemma in the bi-parameter setting. Let 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:ℕ^2→[0,∞), the estimate ∑_I∈𝒰w(e_1(I),e_2(I))|I|≲_w|Ω| holds uniformly in (𝒰,Ω) if and only if the following fixed-total condition holds: |w|_ft:=sup_N≥0∑_a=0^Nw(a,N−a)<∞. For instance, |(a+1)^−s(b+1)^−(1−s)|_ft∼[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× I^2× I^3 with ℓ(I^3)=ℓ(I^1)ℓ(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.
ER -