LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6.1-sol)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Calderón–Zygmund decomposition at height λ

Statement

Assume Countable Choice. Let $f \in L^{1} (R^{n})$ and $λ > 0$ . Then $f = g + \sum_{j} b_{j}$ almost everywhere, where the cubes $Q_{j}$ are the maximal all-generation dyadic cubes of Maximal dyadic cubes above a level, $b_{j} = (f - ∣ Q_{j} ∣^{- 1} \int_{Q_{j}} f) 1_{Q_{j}}$ satisfies $\int b_{j} = 0$ and $∥ b_{j} ∥_{1} \leq 2^{n + 1} λ ∣ Q_{j} ∣$ , and $g = f outside ⋃_{j} Q_{j}, g = ∣ Q_{j} ∣^{- 1} \int_{Q_{j}} f on Q_{j}$ satisfies $∥ g ∥_{1} \leq ∥ f ∥_{1}$ , $∣ g ∣ \leq 2^{n} λ$ almost everywhere, and $∥ g ∥_{2}^{2} \leq 2^{n} λ ∥ f ∥_{1}$ ; moreover $\sum_{j} ∣ Q_{j} ∣ \leq λ^{- 1} ∥ f ∥_{1}$ .

Facts & Assumptions

Given: $f \in L^{1} (R^{n})$ and $λ > 0$ ; the maximal bad dyadic cubes $Q_{j}$ of the previous lemma, pairwise disjoint with $\sum_{j} ∣ Q_{j} ∣ \leq λ^{- 1} ∥ f ∥_{1}$ and $∣ Q_{j} ∣^{- 1} \int_{Q_{j}} ∣ f ∣ \leq 2^{n} λ$ ; the functions $g$ and $b_{j}$ defined above.

[F1]

The maximal bad cubes $Q_{j}$ (cubes with $∣ Q ∣^{- 1} \int_{Q} ∣ f ∣ > λ$ , maximal under inclusion) are countable, pairwise disjoint, have union ${M_{d} f > λ}$ , and satisfy $∣ Q_{j} ∣^{- 1} \int_{Q_{j}} ∣ f ∣ \leq 2^{n} λ$ and $\sum_{j} ∣ Q_{j} ∣ \leq λ^{- 1} ∥ f ∥_{1}$ (Maximal dyadic cubes above a level).

[F2]

A family $(E_{r})_{r > 0}$ shrinks nicely to $x$ with constant $α$ when $E_{r} \subseteq B (x, r)$ and $λ (E_{r}) \geq α λ (B (x, r))$ ; if for each $x$ in a set $A$ such a family is given, then for almost every $x \in A$ the averages of an $L_{l o c}^{1}$ function over $E_{r}$ converge to the function value as $r \to 0^{+}$ (Differentiation holds along families shrinking nicely, Almost every point is a Lebesgue point of a locally integrable function).

Proof

technique · direct

1.1F1givenalgebra

The functions $g$ and $b_{j}$ are measurable, $b_{j}$ is supported in $Q_{j}$ , and $f = g + \sum_{j} b_{j}$ everywhere: on $Q_{j}$ the sum $g + \sum_{k} b_{k}$ has the single nonzero term $g + b_{j} = ∣ Q_{j} ∣^{- 1} \int_{Q_{j}} f + f - ∣ Q_{j} ∣^{- 1} \int_{Q_{j}} f = f$ by disjointness of the $Q_{k}$ , while off $⋃_{k} Q_{k}$ one has $g = f$ and every $b_{k} = 0$ . Moreover $\int b_{j} = \int_{Q_{j}} f - ∣ Q_{j} ∣^{- 1} \int_{Q_{j}} f ∣ Q_{j} ∣ = 0$ and $∥ b_{j} ∥_{1} \leq \int_{Q_{j}} ∣ f ∣ + ∣ Q_{j} ∣^{- 1} ∣ \int_{Q_{j}} f ∣ ∣ Q_{j} ∣ \leq 2 \int_{Q_{j}} ∣ f ∣ \leq 2 \cdot 2^{n} λ ∣ Q_{j} ∣ = 2^{n + 1} λ ∣ Q_{j} ∣$ .

2.1F1F2step 1.1algebra

On each bad cube, $∣ g ∣ = ∣ Q_{j} ∣^{- 1} ∣ \int_{Q_{j}} f ∣ \leq ∣ Q_{j} ∣^{- 1} \int_{Q_{j}} ∣ f ∣ \leq 2^{n} λ$ , so $∣ g ∣ \leq 2^{n} λ$ on $⋃_{j} Q_{j}$ ; off $⋃_{j} Q_{j}$ one has $g = f$ and all dyadic cubes through $x$ are good, so the averages of $∣ f ∣$ over those cubes are at most $λ$ . These cubes, indexed by generation and assigned to the parameter $r = 2 \sqrt{n} 2^{- k}$ for the generation- $k$ cube through $x$ and extended constantly on $[2 \sqrt{n} 2^{- k}, 2 \sqrt{n} 2^{- k + 1})$ , shrink nicely to $x$ with a dimensional constant: each lies in $B (x, r)$ and has measure $2^{- k n} = c_{n} λ (B (x, 2 \sqrt{n} 2^{- k}))$ . Hence [F2] gives $∣ f (x) ∣ \leq λ$ for almost every $x \notin ⋃_{j} Q_{j}$ , and since $g = f$ there, $∣ g ∣ \leq \max (2^{n} λ, λ) = 2^{n} λ$ almost everywhere.

3.1F1step 1.1step 2.1algebra∎

From step 2.1, $∥ g ∥_{1} \leq \int_{⋃ Q_{j}} ∣ g ∣ + \int_{R^{n} ∖ ⋃ Q_{j}} ∣ f ∣ \leq \sum_{j} \int_{Q_{j}} ∣ f ∣ + \int_{R^{n} ∖ ⋃ Q_{j}} ∣ f ∣ = ∥ f ∥_{1}$ , and $∥ g ∥_{2}^{2} = \int ∣ g ∣^{2} \leq ∥ g ∥_{\infty} \int ∣ g ∣ \leq 2^{n} λ ∥ f ∥_{1}$ . Together with step 1.1 and the bounds $\sum_{j} ∣ Q_{j} ∣ \leq λ^{- 1} ∥ f ∥_{1}$ and $∣ Q_{j} ∣^{- 1} \int_{Q_{j}} ∣ f ∣ \leq 2^{n} λ$ from [F1], this is the asserted decomposition.

Depends on

Used by

Dependency tree · two levels

22 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Loukas Grafakos, Classical Fourier Analysis, third edition (standard reference, not scraped)
Terence Tao, Math 247A Lecture Notes 3 (standard reference, not scraped)
Juha Kinnunen, Harmonic Analysis (standard reference, not scraped)