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Source. Lemma 4.36, p. 24, of Cabrelli, Lacey, Molter and Pipher, Variations on the theme of Journé's lemma, in the edition named on the source card.

Statement

Lemma 4.36 (p. 24). For all $\delta>0$ and $\epsilon>0$ there is a constant $K_{\delta,\epsilon}$ such that for every collection $\mathcal U$ of rectangles whose shadow has finite measure there are a set $V\supset\operatorname{sh}(\mathcal U)$ with $\lvert V\rvert\le(1+\delta)\lvert\operatorname{sh}(\mathcal U)\rvert$, 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 $\mathcal U'\subset\mathcal U$,

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

with $F(I,j,v,\mathcal U')=\bigcup\{R\in\mathcal U':2^v<\operatorname{emb}(R)\le2^{v+1},\ R_{(j)}=I,\ \imath(R)=j\}$ as in Lemma 4.35. It differs from Lemma 4.35 only in the size of $V$: at most $(1+\delta)\lvert\operatorname{sh}(\mathcal U)\rvert$ instead of comparable to $\lvert\operatorname{sh}(\mathcal U)\rvert$.

Proof pointer

None in this paper. The paper says (p. 24) that the lemma has been applied by M. T. Lacey and E. Terwilleger, Hankel operators in several complex variables and product BMO (2004), arXiv:math/0310348, and refers to that paper for the detailed proof.

Dependencies

None in the corpus. Read depth: claims checked; the statement was read clause by clause on p. 24. Its proof is not in this paper and was not read. Nothing here is independently reviewed.

Bears on

The paper names no Erdős problem.