LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generated

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.

The good part has controlled L2 image

Statement

Assume Countable Choice. Let $T$ be a Calderón–Zygmund operator with kernel constants $A_{1}, A_{2}$ and $L^{2}$ norm $B$ , and let $f = g + \sum_{j} b_{j}$ be the Calderón–Zygmund decomposition of $f \in L^{1} (R^{n})$ at height $λ > 0$ (Calderón–Zygmund decomposition at height λ). Then $∣ {∣ T g ∣ > λ / 2} ∣ \leq 4 B^{2} 2^{n} λ^{- 1} ∥ f ∥_{1} .$

Facts & Assumptions

Given: $f \in L^{1}$ , $λ > 0$ , its Calderón–Zygmund decomposition with good part $g$ at height $λ$ ; a Calderón–Zygmund operator $T$ with $L^{2}$ norm bound $B$ .

[F1]

The good part satisfies $g \in L^{2} (R^{n})$ with $∥ g ∥_{2}^{2} \leq 2^{n} λ ∥ f ∥_{1}$ (Calderón–Zygmund decomposition at height λ).

[F2]

$T : L^{2} \to L^{2}$ is linear with $∥ T h ∥_{2} \leq B ∥ h ∥_{2}$ for every $h \in L^{2}$ (Calderón–Zygmund kernels and their associated operators).

[F3]

For a measurable $u$ and $μ > 0$ , $μ ({∣ u ∣ \geq μ}) \leq μ^{- 2} \int ∣ u ∣^{2} d μ$ ; more precisely $λ ({∣ u ∣ > μ}) \leq μ^{- 2} \int ∣ u ∣^{2} d λ$ by Chebyshev's inequality applied to $∣ u ∣^{2}$ at level $μ^{2}$ (Chebyshev-Markov inequality for the integral).

Proof

technique · direct

1.1F1F2given

Since $g \in L^{2}$ by [F1], linearity of $T$ gives $T g \in L^{2}$ with $∥ T g ∥_{2} \leq B ∥ g ∥_{2}$ .

2.1F1F3step 1.1algebra∎

Chebyshev's inequality at level $(λ / 2)^{2}$ applied to $∣ T g ∣^{2}$ , followed by step 1.1 and the bound of [F1], gives $∣ {∣ T g ∣ > λ / 2} ∣ \leq \frac{4}{λ^{2}} ∥ T g ∥_{2}^{2} \leq \frac{4 B^{2}}{λ^{2}} ∥ g ∥_{2}^{2} \leq \frac{4 B^{2}}{λ^{2}} 2^{n} λ ∥ f ∥_{1} = 4 B^{2} 2^{n} λ^{- 1} ∥ f ∥_{1},$ which is the asserted estimate.

Depends on

Used by

Calderón–Zygmund operators are of weak type (1,1) Theorem

Dependency tree · two levels

21 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 4 (standard reference, not scraped)