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Source. Lemma 5.38, p. 24, of Cabrelli, Lacey, Molter and Pipher, Variations on the theme of Journé's lemma, in the edition named on the source card. The paper describes it as the version of Journé's lemma given by J. Pipher, Journé's covering lemma and its extension to higher dimensions, Duke Math. J. 53 (1986), 683--690.

Statement

Setting (p. 24). In $\mathbb R^d$, $\operatorname{Enl}(\mathcal U)=\{M\mathbf 1_{\operatorname{sh}(\mathcal U)}>\frac1{2d}\}$, and for $1\le j\le d$ the $j$-th embeddedness stretches only the $j$-th side:

$$\operatorname{emb}(j,R)=\sup\{\mu\ge1:R_{(1)}\times\cdots\times\mu R_{(j)}\times\cdots\times R_{(d)}\subset\operatorname{Enl}(\mathcal U)\}\quad(5.37).$$

The paper uses it only for $1\le j<d$.

Lemma 5.38 (p. 24). For each $d\ge3$ and $0<\epsilon<1$, every subset $U$ of $\mathbb R^d$ of finite measure, and every collection $\mathcal U$ of pairwise incomparable dyadic rectangles,

$$\sum_{R\in\mathcal U'}\lvert R\rvert\prod_{j=1}^{d-1}\operatorname{emb}(j,R)^{-\epsilon} \lesssim\lvert\operatorname{sh}(\mathcal U')\rvert,$$

uniformly over all subsets $\mathcal U'$ of $\mathcal U$. The set $U$ named in the hypotheses plays no visible role in the printed statement.

The paper remarks (p. 25) that the sum runs over rectangles, simpler objects than the unions of Lemma 4.33, at the cost of a product of embeddedness terms, so that $\lvert R\rvert$ is essentially weighted by the largest one.

Proof pointer

Pp. 25--26, two proofs. The first uses partial orders $<_j$ (intersecting, with the $j$-th side strictly contained) and a standard reduction with each $\operatorname{emb}(j,R)$ between $\mu_j$ and $2\mu_j$ and scales separated by $10\max_j\mu_j$; a rectangle covered to $7/8$ by others would be covered to $7/(8d)$ along one order, contradicting $\operatorname{emb}(j,R)\simeq\mu_j$, so the rectangles are essentially disjoint. The second, Pipher's, is given for three parameters only: fixing the first side reduces to Lemma 1.1, which brings in Lemma 4.33; the paper omits the details in higher parameters.

Dependencies

Lemma 1.1 and Lemma 4.33, in the second proof. Read depth: claims checked; the setting and statement were read clause by clause on p. 24, the proofs for structure only. Nothing here is independently reviewed.

Bears on

The paper names no Erdős problem.