Cabrelli lacey molter pipher 2005 journe lemma variations

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lemma_1_1: Journé's lemma as the paper states it: for every eps > 0 and every subcollection of pairwise incomparable dyadic rectangles of the plane, the sum of |R| times emb(R,U)^{-eps} is at most a constant depending only on eps times the area of the subcollection's shadow.

lemma_3_23: The two-parameter Journé lemma with embeddedness measured by dilating all sides of R equally inside Enl_2(U) = {M 1_sh(U) > 1/16}: for every eps > 0 the emb^{-eps}-weighted area of any subcollection is at most a constant depending only on eps times its shadow.

lemma_3_30: The two-parameter Journé lemma with embeddedness the largest product mu_1 mu_2 of independent side dilations keeping R inside Enl_2(U): for every eps > 0 the weighted area of a subcollection of maximal rectangles is at most a constant depending only on eps times its shadow; the proof is only sketched.

lemma_4_33: Pipher's d-parameter variant of Journé's lemma: with embeddedness measured in the first coordinate only, the sets F(I,j,U') of rectangles over a fixed first side I with embeddedness about 2^j satisfy a 2^{-eps j}-weighted shadow bound, and an L^p bound for powers of their maximal functions.

lemma_4_34: The small-enlargement form of Lemma 4.33: for all delta, eps > 0 one can choose V containing the shadow with |V| <= (1+delta)|sh(U)| so that the 2^{-eps j}-weighted sets F(I,j,U'), with first-coordinate embeddedness in V, satisfy the shadow bound and the L^p bound for every subcollection.

lemma_4_35: For every eps > 0 and every collection of rectangles in R^d with finite-measure shadow there are a set V of measure comparable to the shadow, an embeddedness map with emb(R) R inside V and a coordinate map, such that the 2^{-(d+eps)v}-weighted sets F(I,j,v,U') are bounded by the shadow of every subcollection.

lemma_4_36: The small-enlargement form of Lemma 4.35: for all delta, eps > 0 there is K_{delta,eps} such that every collection of rectangles with finite-measure shadow has V with |V| <= (1+delta)|sh(U)|, an embeddedness map and a coordinate map giving the 2^{-(d+eps)v}-weighted bound with constant K_{delta,eps}; the paper refers to Lacey and Terwilleger for the proof.

lemma_5_38: Pipher's rectangle form of Journé's lemma in d >= 3 parameters: for 0 < eps < 1 and a collection of pairwise incomparable dyadic rectangles, the sum of |R| times the product over j < d of emb(j,R)^{-eps} is at most a constant times the shadow, uniformly over subcollections.

proposition_3_26: For each 0 < delta, eps < 1, every collection of rectangles in the plane with finite-measure shadow has a set V containing the shadow with |V| < (1+delta)|sh(U)| for which the emb(R,V)^{-eps}-weighted area of any subcollection is at most a constant depending on eps and delta times its shadow.

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Carlos Cabrelli, Michael T. Lacey, Ursula M. Molter, and Jill C. Pipher, “Variations on the theme of Journé's lemma,” Houston Journal of Mathematics 32 (2006), no. 3, 833–861. The arXiv version is math/0412174, version 2 dated 9 March 2005; the labels and page numbers on this card and its result pages are those of that version's 27 numbered pages. The arXiv record carries no license field, so arXiv's assumed license applies (arXiv:math/0412174), every other right reserved.

The paper collects forms of Journé's covering lemma, some known, some implicit in the literature and some new. In the plane, $M$ is the strong maximal function (averages over all rectangles, dyadic or not). For a collection $\mathcal U$ of dyadic rectangles whose union, the shadow $\operatorname{sh}(\mathcal U)$, has finite measure, the paper sets $\operatorname{Enl}(\mathcal U)=\{M\mathbf 1_{\operatorname{sh}(\mathcal U)}>\frac12\}$ and, for $R=R_{(1)}\times R_{(2)}\in\mathcal U$,

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

where $\mu R_{(1)}$ is dilated about its center (p. 1). Lemma 1.1 (p. 1), Journé's lemma, says that 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 later sections vary three things: how embeddedness is measured (one side, all sides equally, or independent side dilations), how large the enlarged set may be (comparable to the shadow, or at most $(1+\delta)$ times it), and the number of parameters. Section 2 records the consequences for product Carleson measures and product BMO (Corollaries 2.10 and 2.16, Proposition 2.11, the John--Nirenberg inequality Lemma 2.12, Theorem 2.14), which are not extracted here.

Read status: claims checked, clause by clause on the page images of the arXiv version, for the defining displays and the statements of Lemma 1.1, Lemma 3.23, Proposition 3.26, Lemma 3.30, Lemma 4.33, Lemma 4.34, Lemma 4.35, Lemma 4.36 and Lemma 5.38; the proofs were followed for structure only. Lemma 3.30 has only a sketched proof in the paper, and Lemma 4.36 is proved elsewhere (Lacey and Terwilleger). Nothing here is independently reviewed.

Results.

• Lemma 1.1 (p. 1): Journé's lemma in the plane, embeddedness in the first coordinate, for pairwise incomparable dyadic rectangles.
• Lemma 3.23 (p. 14): the planar lemma with all sides dilated equally inside $\operatorname{Enl}_2(\mathcal U)=\{M\mathbf 1_{\operatorname{sh}(\mathcal U)}>\frac1{16}\}$, for every subcollection.
• Proposition 3.26 (p. 17): the planar uniform-embeddedness lemma with an enlarged set of measure below $(1+\delta)\lvert\operatorname{sh}(\mathcal U)\rvert$.
• Lemma 3.30 (p. 19): the planar lemma with embeddedness $\mu_1\mu_2$ from independent side dilations, proof sketched.
• Lemma 4.33 (p. 20): Pipher's $d$-parameter form, one-coordinate embeddedness, sums over unions $F(I,j,\mathcal U')$ of rectangles, with an $L^p$ companion.
• Lemma 4.34 (p. 21): Lemma 4.33 with an enlarged set of measure at most $(1+\delta)\lvert\operatorname{sh}(\mathcal U)\rvert$.
• Lemma 4.35 (p. 22): $d$ parameters with uniform embeddedness, at the cost of the weight $2^{-(d+\epsilon)v}$.
• Lemma 4.36 (p. 24): Lemma 4.35 with an enlarged set of measure at most $(1+\delta)\lvert\operatorname{sh}(\mathcal U)\rvert$, proof referred to Lacey and Terwilleger.
• Lemma 5.38 (p. 24): for $d\ge3$, pairwise incomparable dyadic rectangles weighted by $\prod_{j<d}\operatorname{emb}(j,R)^{-\epsilon}$.

Bears on. None: the paper names no Erdős problem, and this is an analysis method source.

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