CarlesonHunt.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

7 declarations, 0 contain sorry, 0 declared as axioms

This file contains the Carleson-Hunt theorem, a generalization of classical_carleson.

Imports (3)

Imported by 1 file

Carleson.lean

theorem exceptional_set_carleson Theorem 1.1 (Classical Carleson) line 23 source
Source comment
Theorem 1.1 (Classical Carleson)

depends on sorry what this means
Statement uses: Real, Complex, Function.Periodic, HMul.hMul, OfNat.ofNat, Nat, Real.pi, ENNReal, LT.lt, MeasureTheory.MemLp, MeasureTheory.Measure.restrict, MeasureTheory.MeasureSpace.volume, Set.Ioc, NNReal, Exists, LE.le, MeasureTheory.distribution, iSup, GT.gt, ENorm.enorm and 2 more
Proof uses: ENNReal.toNNReal, Min.min, HDiv.hDiv, Eq, rfl, Ne, Top.top, ne_of_lt, Eq.mpr, Or, id, congrArg, propext, min_lt_iff, Or.inl, ENNReal.div_lt_top, of_eq_true, Not, Eq.trans, False and 181 more
304 instances, coercions and auxiliary constants left out.

Tactic trace: 111 steps
Open to load the checked before-and-after goal states.
```
theorem exceptional_set_carleson {f : ℝ → ℂ} (periodic_f : f.Periodic (2 * π))
  {q : ℝ≥0∞} (hq : 1 < q) (hf : MemLp f q (volume.restrict (Set.Ioc 0 (2 * π))))
  {δ ε : NNReal} (δpos : 0 < δ) (εpos : 0 < ε) :
    ∃ N₀, distribution (fun x ↦ ⨆ N > N₀, ‖f x - S_ N f x‖ₑ) δ (volume.restrict (Set.Ioc 0 (2 * π))) ≤ ε := by
…
```
theorem carleson_hunt_interval line 136 source
depends on sorry what this means
Statement uses: Real, Complex, Function.Periodic, HMul.hMul, OfNat.ofNat, Nat, Real.pi, ENNReal, LT.lt, MeasureTheory.MemLp, MeasureTheory.Measure.restrict, MeasureTheory.MeasureSpace.volume, Set.Ioc, Filter.Eventually, Filter.Tendsto, partialFourierSum, Filter.atTop, nhds, MeasureTheory.ae, MeasureTheory.Measure
Proof uses: ae_tendsto_zero_of_distribution_le, NNReal, GT.gt, exceptional_set_carleson
69 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
theorem carleson_hunt_real Carleson's theorem asserting a.e. point-wise convergence of the partial Fourier sums for periodic L^p functions, p > 1 line 146 source
Source comment
**Carleson's theorem** asserting a.e. point-wise convergence of the partial Fourier sums for periodic L^p functions, p > 1.

depends on sorry what this means
Statement uses: Real, Complex, Function.Periodic, HMul.hMul, OfNat.ofNat, Nat, Real.pi, ENNReal, LT.lt, MeasureTheory.MemLp, MeasureTheory.Measure.restrict, MeasureTheory.MeasureSpace.volume, Set.Ioc, Filter.Eventually, Filter.Tendsto, partialFourierSum, Filter.atTop, nhds, MeasureTheory.ae, MeasureTheory.Measure
Proof uses: Function.Periodic.ae_of_ae_restrict, Real.two_pi_pos, Eq.mpr, Eq, HAdd.hAdd, id, congrArg, Function.Periodic.eq_1, congrFun', Filter, funext, Eq.trans, partialFourierSum_periodic, Eq.refl, MeasureTheory.ae_restrict_of_ae_eq_of_ae_restrict, Set.Icc, Set.Ico, Filter.EventuallyEq.symm, MeasureTheory.Ico_ae_eq_Icc, zero_add and 3 more
78 instances, coercions and auxiliary constants left out.

Tactic trace: 10 steps
Open to load the checked before-and-after goal states.
```
theorem carleson_hunt_real {f : ℝ → ℂ} (periodic_f : f.Periodic (2 * π))
  {p : ℝ≥0∞} (hp : 1 < p) (hf : MemLp f p (volume.restrict (Set.Ioc 0 (2 * π)))) :
    ∀ᵐ x, Tendsto (S_ · f x) atTop (𝓝 (f x)) := by
…
```
theorem classical_carleson_from_carleson_hunt_real Classical Carleson from the Carleson-Hunt theorem line 162 source
Source comment
Proves that the old, weak version of classical Carleson is implied by the L^p version.
L4DEL explanation
A continuous periodic function is bounded and hence lies in every L^p, so the Carleson-Hunt a.e. convergence result for L^p functions applies.
1. Unfold ClassicalCarleson and take f continuous and periodic.
2. Apply carleson_hunt_real with p equal to infinity.
3. Boundedness of f on [0, 2*pi] from continuity gives membership in L-infinity on the period.
Prerequisites: carleson_hunt_real
See also: ClassicalCarleson, classical_carleson
Sources and attribution: Pinned Lean source (8972acdfd953)
depends on sorry what this means
Statement uses: ClassicalCarleson
Proof uses: id, Real, Complex, Continuous, Function.Periodic, HMul.hMul, OfNat.ofNat, Real.pi, carleson_hunt_real, Top.top, ENNReal, of_eq_true, LT.lt, Membership.mem, Set, Set.Icc, Nat, LE.le, Norm.norm, Exists and 21 more
66 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
```
theorem classical_carleson_from_carleson_hunt_real : ClassicalCarleson := by
…
```
Mentions: ClassicalCarleson
theorem carleson_hunt_two_pi line 181 source
depends on sorry what this means
Statement uses: AddCircle, Real, HMul.hMul, OfNat.ofNat, Nat, Real.pi, Complex, ENNReal, LT.lt, MeasureTheory.MemLp, instFactLtRealOfNatHMulPi_1, MeasureTheory.MeasureSpace.volume, Filter.Eventually, Filter.Tendsto, ContinuousMap, AddSubgroup.zmultiples, partialFourierSum', Filter.atTop, nhds, MeasureTheory.ae and 1 more
Proof uses: Measurable, Not, Or, Classical.em, Function.Periodic, id, of_eq_true, Eq, HAdd.hAdd, Eq.trans, True, congrFun', congrArg, HasQuotient.Quotient, AddSubgroup, QuotientAddGroup.mk_add_of_mem, Membership.mem, eq_self, MeasureTheory.Measure.restrict, Set.Ioc and 91 more
193 instances, coercions and auxiliary constants left out.

Tactic trace: 55 steps
Open to load the checked before-and-after goal states.
theorem carleson_hunt' Version of carleson_hunt with volume instead of haarAddMeasure in the assumption line 236 source
Source comment
Version of carleson_hunt with `volume` instead of `haarAddMeasure` in the assumption. TODO: decide whether we really want to keep the other (equivalent) version

depends on sorry what this means
Statement uses: Real, Fact, LT.lt, OfNat.ofNat, AddCircle, Complex, ENNReal, MeasureTheory.MemLp, MeasureTheory.MeasureSpace.volume, Filter.Eventually, Filter.Tendsto, Nat, ContinuousMap, AddSubgroup.zmultiples, partialFourierSum', Filter.atTop, nhds, MeasureTheory.ae, MeasureTheory.Measure
Proof uses: HMul.hMul, Real.pi, AddEquiv, AddCircle.equivAddCircle, LT.lt.ne', Real.two_pi_pos, Fact.out, instFactLtRealOfNatHMulPi_1, id, Eq.mpr, AddCircle.haarAddCircle, Eq, congrArg, Eq.symm, propext, MeasureTheory.memLp_haarAddCircle_iff, MeasureTheory.MemLp.comp_measurePreserving, Eq.mp, AddCircle.measurePreserving_equivAddCircle, carleson_hunt_two_pi and 24 more
100 instances, coercions and auxiliary constants left out.

Tactic trace: 16 steps
Open to load the checked before-and-after goal states.
```
theorem carleson_hunt' {T : ℝ} [hT : Fact (0 < T)] {f : AddCircle T → ℂ} {p : ℝ≥0∞} (hp : 1 < p)
  (hf : MemLp f p) :
    ∀ᵐ x, Tendsto (partialFourierSum' · f x) atTop (𝓝 (f x)) := by
…
```
Mentions: T, partialFourierSum'
theorem carleson_hunt Classical theorem of Carleson and Hunt asserting a.e. convergence of the partial Fourier sums for L^p functions for p > 1 line 261 source
Source documentation
Classical theorem of Carleson and Hunt asserting a.e. convergence of the partial Fourier sums for L^p functions for p > 1. This is a strengthening of classical_carleson, and not officially part of the blueprint.

depends on sorry what this means
Statement uses: Real, Fact, LT.lt, OfNat.ofNat, AddCircle, Complex, ENNReal, MeasureTheory.MemLp, AddSubgroup.zmultiples, AddCircle.haarAddCircle, Filter.Eventually, Filter.Tendsto, Nat, ContinuousMap, partialFourierSum', Filter.atTop, nhds, MeasureTheory.ae, MeasureTheory.Measure, MeasureTheory.MeasureSpace.volume
Proof uses: carleson_hunt', MeasureTheory.MemLp.of_haarAddCircle
61 instances, coercions and auxiliary constants left out.
```
theorem carleson_hunt {T : ℝ} [hT : Fact (0 < T)] {f : AddCircle T → ℂ} {p : ℝ≥0∞} (hp : 1 < p)
  (hf : MemLp f p AddCircle.haarAddCircle) :
    ∀ᵐ x, Tendsto (partialFourierSum' · f x) atTop (𝓝 (f x)) :=
…
```
Mentions: T, partialFourierSum'