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Source. Proposition 1.4, p. 336, of 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; the edition read is named on the source card.

Read depth. Claims checked: the statement was read clause by clause on the page images of the print, and the proof on pp. 336--337 was followed. Nothing here is independently reviewed.

Statement

Proposition 1.4 (p. 336). 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 a family of non-negative reals. The following are equivalent.

(i) The family is Carleson, in the form

$$\sum_{S\in\mathcal S'}\lambda_S\le C\mu\Bigl(\bigcup_{S\in\mathcal S'}S\Bigr) \qquad\text{for every subcollection }\mathcal S'\text{ of }\mathcal S.$$

(ii) For every family $a=\{a_S\}_{S\in\mathcal S}$ of non-negative reals,

$$\sum_{S\in\mathcal S}\lambda_Sa_S\le C\int\sup_{S\in\mathcal S}a_S1_S\,d\mu. \qquad(1.5)$$

The same constant $C$ appears in (i) and (ii). Unlike Theorem 1.3, the proposition makes no assumption that $\mu$ has no point masses. For the collection $\mathcal D$ of dyadic cubes it is Verbitsky's dual reformulation (inequality (1.3), p. 335), which the paper reads as the dual norm formula for the discrete Littlewood--Paley spaces $f^{\infty,1}(\mu)$ and $f^{1,\infty}(\mu)$ (pp. 335--336).

Proof pointer

Pp. 336--337. For (ii) implies (i), take $a_S$ to be the indicator of membership in $\mathcal S'$, so that $\sup_Sa_S1_S$ is the indicator of $\bigcup_{S\in\mathcal S'}S$. For (i) implies (ii), write both sides through the layer-cake formula $\int f\,d\nu=\int_0^\infty\nu(f>t)\,dt$: the level set $\{\sup_Sa_S1_S>t\}$ is the union of the $S$ with $a_S>t$, and (i) applied to that subcollection bounds the integrand on the left by $C$ times the integrand on the right. The paper describes this as a slight variant of the standard proof of the dyadic Carleson embedding theorem.

Dependencies

None beyond the layer-cake formula.

Bears on

The proposition is a step toward Theorem 1.3 and bears on no Erdős problem; the paper mentions none.