Hanninen 2018 sparse carleson coefficients general sets

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proposition_1_4: Hänninen's proposition that for a locally finite Borel measure on R^d and a countable collection of Borel sets, non-negative coefficients are Carleson with constant C exactly when the sum of lambda_S a_S is at most C times the integral of sup_S a_S 1_S for all non-negative families a.

theorem_1_3: Hänninen's theorem that for a locally finite Borel measure on R^d without point masses and a countable collection of Borel sets, a family of non-negative coefficients is Carleson if and only if it is sparse, with the same constant in both conditions.

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Timo S. Hänninen, “Equivalence of sparse and Carleson coefficients for general sets,” Arkiv för Matematik 56 (2018), 333–339, DOI 10.4310/ARKIV.2018.v56.n2.a8; see also arXiv:1709.10457. The published article prints "© 2018 by Institut Mittag-Leffler. All rights reserved" on its first page, every other right reserved.

Read status: claims checked for the results with pages below, each read clause by clause on the page images of the print (pp. 333--339), with their proofs followed. Result pages: Theorem 1.3 (p. 335), Carleson coefficients are sparse when $\mu$ has no point masses; Proposition 1.4 (p. 336), the dual reformulation of the Carleson condition.

Let $\mu$ be a locally finite Borel measure on $\mathbb R^d$, let $\mathcal S$ be a countable collection of Borel sets, and let $\{\lambda_S\}_{S\in\mathcal S}$ be non-negative reals. The family is Carleson with constant $C\ge1$ (Definition 1.1, p. 333) when

$$\sum_{S\in\mathcal S:\ S\subseteq\Omega}\lambda_S\le C\mu(\Omega)$$

for every union $\Omega$ of sets of $\mathcal S$, and sparse with constant $C\ge1$ (Definition 1.2, p. 334) when each $S$ has a subset $E_S\subseteq S$ with $\lambda_S\le C\mu(E_S)$, the sets $E_S$ being pairwise disjoint. Sparse families are always Carleson with the same constant (p. 334).

Theorem 1.3 (p. 335) states that if $\mu$ has no point masses, a non-negative family on $\mathcal S$ is Carleson if and only if it is sparse, with the same constant in both conditions; this covers in particular the dyadic rectangles of bi-parameter theory, where Barron and Pipher had raised the converse as an open problem. The proof runs the Dor–Verbitsky argument through Proposition 1.4 (p. 336), which extends to general $\mathcal S$, with no point-mass hypothesis, Verbitsky's dual reformulation for dyadic cubes: the family is Carleson with constant $C$ exactly when

$$\sum_{S\in\mathcal S}\lambda_Sa_S\le C\int\sup_{S\in\mathcal S}a_S1_S\,d\mu$$

for every family $\{a_S\}$ of non-negative reals. Dor's characterization (Proposition 1.5, p. 338) then supplies the disjoint sets. A Remark on p. 334 shows that point masses can obstruct the converse.

Bears on. None: the paper is a harmonic-analysis source on sparse domination and Carleson embeddings, and it mentions no Erdős problem.

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