CarlesonOnTheRealLineContinuous.lean`

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Annotation provenance. This overlay contains 10 curated AI-written annotations and 981 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned project documentation, Pinned Carleson blueprint

2 declarations, 0 contain sorry, 0 declared as axioms

Imports (2)

Imported by 2 files

lemma rcarleson_general Version of Lemma 11.1.5 for general exponents line 17 source
Source comment
Version of Lemma 11.1.5 for general exponents

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: NNReal, Membership.mem, Set, Set.Ioc, OfNat.ofNat, Nat, NNReal.HolderConjugate, Real, MeasurableSet, Complex, Measurable, LE.le, Norm.norm, Set.indicator, ENNReal, MeasureTheory.lintegral, MeasureTheory.Measure.restrict, MeasureTheory.MeasureSpace.volume, carlesonOperatorReal, K and 5 more
Proof uses: Trans.trans, carlesonOperator, Eq, rfl, MeasureTheory.lintegral_mono, carlesonOperatorReal_le_carlesonOperator, two_sided_metric_carleson, Mathlib.Meta.NormNum.isNat_le_true, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Bool, Bool.true, Hilbert_strong_2_2
93 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
lemma rcarleson_general {q q' : ℝ≥0} (hq : q ∈ Set.Ioc 1 2) (hqq' : q.HolderConjugate q')
    {F G : Set ℝ} (hF : MeasurableSet F) (hG : MeasurableSet G)
    (f : ℝ → ℂ) (hmf : Measurable f) (hf : ∀ x, ‖f x‖ ≤ F.indicator 1 x) :
    ∫⁻ x in G, T f x ≤ C10_0_1 4 q * (volume G) ^ (q' : ℝ)⁻¹ * (volume F) ^ (q : ℝ)⁻¹ := by
…
```
Mentions: T, C10_0_1
lemma rcarleson Lemma 11.1.5 line 29 source
Source comment
Lemma 11.1.5

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Real, MeasurableSet, Complex, Measurable, LE.le, Norm.norm, Set.indicator, OfNat.ofNat, ENNReal, MeasureTheory.lintegral, MeasureTheory.Measure.restrict, MeasureTheory.MeasureSpace.volume, carlesonOperatorReal, K, HMul.hMul, C10_0_1, Nat, NNReal, HPow.hPow and 2 more
Proof uses: NNReal.HolderConjugate, Eq.mpr, Eq, HDiv.hDiv, HSub.hSub, id, congrArg, propext, NNReal.holderConjugate_iff_eq_conjExponent, Mathlib.Meta.NormNum.isNat_lt_true, Mathlib.Meta.NormNum.isNat_ofNat, Nat.cast_one, Eq.refl, Mathlib.Meta.NormNum.instAtLeastTwo, Bool, Bool.false, NNReal.eq, of_eq_true, Eq.trans, True and 21 more
86 instances, coercions and auxiliary constants left out.

Tactic trace: 8 steps
Open to load the checked before-and-after goal states.
```
lemma rcarleson {F G : Set ℝ} (hF : MeasurableSet F) (hG : MeasurableSet G)
    (f : ℝ → ℂ) (hmf : Measurable f) (hf : ∀ x, ‖f x‖ ≤ F.indicator 1 x) :
    ∫⁻ x in G, T f x ≤ C10_0_1 4 2 * (volume G) ^ (2 : ℝ)⁻¹ * (volume F) ^ (2 : ℝ)⁻¹ := by
…
```
Mentions: T, C10_0_1