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Source. Lemma 3.23, p. 14, of Cabrelli, Lacey, Molter and Pipher, Variations on the theme of Journé's lemma, in the edition named on the source card. The paper says this form was first proved by S. H. Ferguson and M. T. Lacey, A characterization of product BMO by commutators, Acta Math. 189 (2002), 143--160.

Statement

Setting (p. 14). For a collection $\mathcal U$ of dyadic rectangles of the plane whose shadow has finite measure, the paper defines

$$\operatorname{Enl}_2(\mathcal U)=\{M\mathbf 1_{\operatorname{sh}(\mathcal U)}>\tfrac1{16}\}\quad(3.20),$$

iterates it by $\operatorname{Enl}_{j+1}=\operatorname{Enl}_2(\operatorname{Enl}_j)$ (3.21, printed for $j>2$), and measures the embeddedness of $R\in\mathcal U$ by dilating all sides of $R$ about its center by the same factor:

$$\operatorname{emb}(R,\operatorname{Enl}_j(\mathcal U))=\sup\{\mu\ge1:\mu R\subset\operatorname{Enl}_j(\mathcal U)\},\qquad j\ge2\quad(3.22).$$

The paper remarks that for many rectangles this can be essentially smaller than the one-coordinate embeddedness of Lemma 1.1.

Lemma 3.23 (p. 14). For every $\epsilon>0$ and every collection $\mathcal U$ of rectangles whose shadow has finite measure in the plane,

$$\sum_{R\in\mathcal U'}\operatorname{emb}(R,\operatorname{Enl}_2(\mathcal U))^{-\epsilon}\lvert R\rvert \lesssim\lvert\operatorname{sh}(\mathcal U')\rvert,$$

where the implied constant depends only on $\epsilon$ and the inequality holds uniformly over all collections $\mathcal U'\subset\mathcal U$. Unlike Lemma 1.1, the statement makes no incomparability assumption on $\mathcal U'$.

Proof pointer

Section 3.2, pp. 14--16, gives two proofs. The first applies Lemma 1.1 in the first coordinate on a shifted dyadic grid (1.6), replaces each first side by a maximal enlarged shifted interval, and applies Lemma 1.1 again in the second coordinate to $O(k)$ incomparable subcollections. The second runs the essentially-disjoint strategy, splitting the collection into a good part and two bad parts and showing that the bad decomposition terminates after three rounds.

Dependencies

Lemma 1.1, used in the first proof. Read depth: claims checked; the definitions and statement were read clause by clause on p. 14, the proofs for structure only. Nothing here is independently reviewed.

Bears on

The paper names no Erdős problem.