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Source. Lemma 1.1, p. 1, of Carlos Cabrelli, Michael T. Lacey, Ursula Molter and Jill C. Pipher, Variations on the theme of Journé's lemma, in the arXiv version math/0412174v2 (9 March 2005) named on the source card. The paper attributes the lemma to J.-L. Journé, A covering lemma for product spaces, Proc. Amer. Math. Soc. 96 (1986), 593--598.

Statement

Setting (pp. 1, 3--4). $M$ is the strong maximal function in the plane, the supremum of averages of $\lvert f\rvert$ over all rectangles, dyadic or not, containing the point. $\mathcal U$ is a collection of dyadic rectangles of the plane whose union, the shadow $\operatorname{sh}(\mathcal U)$, has finite measure, and

$$\operatorname{Enl}(\mathcal U)=\{M\mathbf 1_{\operatorname{sh}(\mathcal U)}>\tfrac12\}.$$

For a dyadic rectangle $R=R_{(1)}\times R_{(2)}\in\mathcal U$,

$$\operatorname{emb}(R;\mathcal U)=\sup\{\mu>1:(\mu R_{(1)})\times R_{(2)}\subset\operatorname{Enl}(\mathcal U)\},$$

where $\lambda R$ is the set with the same center as $R$ dilated by $\lambda$. So only the first side of $R$ is stretched. Section 3.1.1 (p. 13, (3.17)) restates the same embeddedness with $\mu\ge1$, as $\sup\{\mu\ge1:\operatorname{Dil}_{(\mu,1)}R\subset\operatorname{Enl}(\mathcal U)\}$.

Lemma 1.1 (p. 1). For every $\epsilon>0$ and every subcollection $\mathcal U'\subset\mathcal U$ of pairwise incomparable dyadic rectangles,

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

with an implied constant depending only on $\epsilon$. The paper stresses (p. 1) that the bound is uniform over all subcollections $\mathcal U'$.

The paper notes (p. 3) that, by the product John--Nirenberg inequality (Lemma 2.12, p. 8), the conclusion yields $\bigl\lVert\sum_{R\in\mathcal U}\operatorname{emb}(R,\mathcal U)^{-\epsilon}\mathbf 1_R\bigr\rVert_p\lesssim\lvert\operatorname{sh}(\mathcal U)\rvert^{1/p}$ for $1<p<\infty$.

Proof pointer

Section 3.1, pp. 13--14, gives two proofs. Both pass to the paper's standard reduction (1.8), p. 6: it suffices to bound the total area of rectangles with $\mu\le\operatorname{emb}\le2\mu$ and widely separated scales by their shadow, which holds when the rectangles are essentially disjoint. The first proof shows that no rectangle can be $7/8$ covered by rectangles longer in the first coordinate without having embeddedness at least $10\mu$. The second counts, for each dyadic $I$ and $k\ge0$, the rectangles whose first side dilated by $2^k$ stays in the shadow.

Dependencies

None in the corpus. Read depth: claims checked; the setting and statement were read clause by clause on pp. 1, 3--4 and 13 of the print, the proofs for structure only. Nothing here is independently reviewed.

Bears on

The paper names no Erdős problem.