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Source. Lemma 4.34, p. 21, of Cabrelli, Lacey, Molter and Pipher, Variations on the theme of Journé's lemma, in the edition named on the source card.

Statement

Setting (p. 21). For a collection $\mathcal U$ of rectangles in $\mathbb R^d$ whose shadow has finite measure, and a set $V$ containing the shadow,

$$\operatorname{emb}(R,V)=\sup\{\mu\ge1:\operatorname{Dil}_{(\mu,1,\ldots,1)}R\subset V\},$$ $$F(I,j,\mathcal U')=\bigcup\{I\times R':I\times R'\in\mathcal U',\ 2^{j-1}\le\operatorname{emb}(I\times R',V)<2^j\}.$$

This is the setting of Lemma 4.33 with the enlarged set replaced by $V$.

Lemma 4.34 (p. 21). For all $\delta,\epsilon>0$ and every such $\mathcal U$ one can choose $V\supset\operatorname{sh}(\mathcal U)$ with $\lvert V\rvert\le(1+\delta)\lvert\operatorname{sh}(\mathcal U)\rvert$ such that for every $\mathcal U'\subset\mathcal U$,

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

and moreover, for every integer $n>1$ and every $1<p<\infty$,

$$\Bigl\lVert\sum_{j=1}^\infty\sum_{I\in\mathcal D}2^{-\epsilon j}\bigl(M\mathbf 1_{F(I,j,\mathcal U')}\bigr)^n\Bigr\rVert_p \lesssim\lvert\operatorname{sh}(\mathcal U')\rvert^{1/p}.$$

The implied constants depend only on the dimension and on $\epsilon$ and $\delta$.

Proof pointer

Pp. 21--22. With $\delta=(1+2^{\mathsf d})^{-1}$, the set $V$ is the level set $\{M_1^{\mathcal D_{\mathsf d}}\mathbf 1_{\operatorname{Enl}_1(\mathcal U)}>1-\delta\}$ of the maximal function over Christ's shifted dyadic grids in the first coordinate, whose measure (1.7) controls. The rest repeats the disjointness argument of Lemma 4.33, with each set $H(I)$ now keeping a $\delta/2$ share of every rectangle, at a cost of $\delta^{-1}$.

Dependencies

The proof follows that of Lemma 4.33. Read depth: claims checked; the setting and statement were read clause by clause on p. 21, the proof for structure only. Nothing here is independently reviewed.

Bears on

The paper names no Erdős problem.