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Source. Lemma 4.35, p. 22, of Cabrelli, Lacey, Molter and Pipher, Variations on the theme of Journé's lemma, in the edition named on the source card.

Statement

Lemma 4.35 (p. 22). For every $\epsilon>0$ and every collection $\mathcal U$ of rectangles whose shadow has finite measure there are a set $V\supset\operatorname{sh}(\mathcal U)$ with $\lvert V\rvert\lesssim\lvert\operatorname{sh}(\mathcal U)\rvert$, a map $\operatorname{emb}:\mathcal U\to[1,\infty)$ and a map $\imath:\mathcal U\to\{1,2,\ldots,d\}$ such that $\operatorname{emb}(R)\cdot R\subset V$ for every $R\in\mathcal U$, and for every collection $\mathcal U'\subset\mathcal U$,

$$\sum_{j=1}^d\sum_{v=0}^\infty\sum_{I\in\mathcal D}2^{-(d+\epsilon)v}\lvert F(I,j,v,\mathcal U')\rvert \lesssim\lvert\operatorname{sh}(\mathcal U')\rvert,$$

where

$$F(I,j,v,\mathcal U')=\bigcup\{R\in\mathcal U':2^v<\operatorname{emb}(R)\le2^{v+1},\ R_{(j)}=I,\ \imath(R)=j\}.$$

Here $\operatorname{emb}(R)\cdot R$ dilates every side of $R$ by $\operatorname{emb}(R)$ about its center, so the embeddedness is uniform across coordinates, while the coordinate $\imath(R)$ selects which side indexes the sum. The paper remarks (p. 22) that the price is a worse power: the weight decays like $\operatorname{emb}^{-(d+\epsilon)}$, a power strictly below $-d$.

Proof pointer

Pp. 22--24. Apply Lemma 4.34 inductively, one coordinate at a time, on products of shifted dyadic grids: each stage enlarges the current rectangles in one coordinate to the extent of their embeddedness and builds a new set $V^m$, and $V=V^n$. The map $\operatorname{emb}$ is $\frac1{16}$ of the infimum over $m$ of inductively defined embeddedness quantities $\beta^m(R)$, and $\imath(R)$ is the coordinate attaining it. The final step bounds the shadow of the enlarged rectangles by $2^{dk}$ times the shadow of $\mathcal U'$, using the one-dimensional weak $L^1$ bound in each coordinate; this is where the power $d$ is lost.

Dependencies

Lemma 4.34. Read depth: claims checked; the statement was read clause by clause on p. 22, the proof for structure only. Nothing here is independently reviewed.

Bears on

The paper names no Erdős problem.