LorentzInterpolation.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL: Lean4 Domain Explanation Language

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

3 declarations, 3 contain sorry, 0 declared as axioms

The Marcinkiewisz interpolation theorem for Lorentz spaces

Upstreaming status: this file is under construction and being worked on. Don't upstream yet!

Imports (2)

Imported by 2 files

def C_LorentzInterpolation The constant occurring in the general real interpolation theorem (for Lorentz spaces) sorry line 22 source
Source documentation
The constant occurring in the general real interpolation theorem (for Lorentz spaces)

depends on sorry what this means
Statement uses: ENNReal, NNReal
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
1 instances, coercions and auxiliary constants left out.
```
def C_LorentzInterpolation (p₀ p₁ q₀ q₁ q : ℝ≥0∞) (C₀ C₁ A : ℝ≥0) (t : ℝ≥0∞) : ℝ≥0 :=
…
```
Mentions: A
lemma C_LorentzInterpolation_pos sorry line 25 source
depends on sorry what this means
Statement uses: ENNReal, NNReal, LT.lt, OfNat.ofNat, C_LorentzInterpolation
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, Nat
11 instances, coercions and auxiliary constants left out.

theorem exists_hasLorentzType_real_interpolation General Marcinkiewicz real interpolation theorem sorry line 31 source

Source documentation

General Marcinkiewicz real interpolation theorem

depends on sorry what this means

Statement uses: MeasurableSpace, MeasureTheory.Measure, ENNReal, TopologicalSpace, ENormedAddCommMonoid, LE.le, OfNat.ofNat, Ne, NNReal, LT.lt, Membership.mem, Set, Set.Ioo, Eq, Inv.inv, HAdd.hAdd, HDiv.hDiv, HSub.hSub, MeasureTheory.MemLorentz, MeasureTheory.AEStronglyMeasurable and 4 more

Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, Nat

51 instances, coercions and auxiliary constants left out.

theorem exists_hasLorentzType_real_interpolation {p₀ p₁ r₀ r₁ q₀ q₁ s₀ s₁ p q : ℝ≥0∞}
    [MeasurableSpace E₁] [TopologicalSpace E₁] [ENormedAddCommMonoid E₁]
    [MeasurableSpace E₂] [TopologicalSpace E₂] [ENormedAddCommMonoid E₂]
    -- TODO: find out which of the conditions `(1 ≤ ·)` are actually necessary.
    (hp₀ : 1 ≤ p₀) (hp₁ : 1 ≤ p₁) (hr₀ : 1 ≤ r₀) (hr₁ : 1 ≤ r₁)
    (hq₀ : 1 ≤ q₀) (hq₁ : 1 ≤ q₁) (hs₀ : 1 ≤ s₀) (hs₁ : 1 ≤ s₁)
    (hp₀p₁ : p₀ ≠ p₁) (hq₀q₁ : q₀ ≠ q₁)
    {C₀ C₁ t A : ℝ≥0} (hA : 0 < A) (ht : t ∈ Ioo 0 1) (hC₀ : 0 < C₀) (hC₁ : 0 < C₁)
    (hp : p⁻¹ = (1 - t) / p₀ + t / p₁) (hq : q⁻¹ = (1 - t) / q₀ + t / q₁)
    (hmT : ∀ f, MemLorentz f p q μ → AEStronglyMeasurable (T f) ν)
    (hT : AESubadditiveOn T (fun f ↦ MemLorentz f p₀ r₀ μ ∨ MemLorentz f p₁ r₁ μ) A ν)
    (h₀T : HasLorentzType T p₀ r₀ q₀ s₀ μ ν C₀) (h₁T : HasLorentzType T p₁ r₁ q₁ s₁ μ ν C₁) :
…

Mentions: E₁, E₂, A, T