LorentzType.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

11 declarations, 0 contain sorry, 0 declared as axioms

Imports (4)

Imported by 2 files

def MeasureTheory.HasLorentzType An operator has Lorentz type (p, r, q, s) if it is bounded as a map from L^{q, s} to L^{p, r} line 24 source
Source documentation
An operator has Lorentz type (p, r, q, s) if it is bounded as a map from L^{q, s} to L^{p, r}. HasLorentzType T p r q s μ ν c means that T has Lorentz type (p, r, q, s) w.r.t. measures μ, ν and constant c.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, TopologicalSpace, ENorm, ENNReal, MeasureTheory.Measure
Proof uses: MeasureTheory.MemLorentz, And, MeasureTheory.AEStronglyMeasurable, LE.le, MeasureTheory.eLorentzNorm, HMul.hMul
6 instances, coercions and auxiliary constants left out.
```
def HasLorentzType [ENorm ε₁] [ENorm ε₂] (T : (α → ε₁) → (α' → ε₂))
    (p r q s : ℝ≥0∞) (μ : Measure α) (ν : Measure α') (c : ℝ≥0∞) : Prop :=
…
```
Mentions: T, α', s, c
lemma MeasureTheory.hasStrongType_iff_hasLorentzType line 29 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, MeasureTheory.Measure, TopologicalSpace, ENNReal, ESeminormedAddMonoid, Iff, MeasureTheory.HasStrongType, MeasureTheory.HasLorentzType
Proof uses: id, MeasureTheory.MemLp, And, MeasureTheory.AEStronglyMeasurable, LE.le, MeasureTheory.eLpNorm, HMul.hMul, Iff.intro, MeasureTheory.MemLorentz, MeasureTheory.eLorentzNorm, Eq.mpr, Eq, congrArg, MeasureTheory.eLorentzNorm_eq_eLpNorm, And.left, LT.lt, Top.top, Eq.mp, Eq.symm
14 instances, coercions and auxiliary constants left out.

Tactic trace: 14 steps
Open to load the checked before-and-after goal states.
def MeasureTheory.HasRestrictedWeakType Defines when an operator "has restricted weak type" line 50 source
Source documentation
Defines when an operator "has restricted weak type". This is an even weaker version of HasBoundedWeakType.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, TopologicalSpace, Zero, One, ENorm, ENNReal, MeasureTheory.Measure
Proof uses: Set, MeasurableSet, LT.lt, Top.top, And, MeasureTheory.AEStronglyMeasurable, Set.indicator, OfNat.ofNat, LE.le, MeasureTheory.eLpNorm, MeasureTheory.Measure.restrict, HMul.hMul, HPow.hPow, Real, ENNReal.toReal, Inv.inv
17 instances, coercions and auxiliary constants left out.
```
def HasRestrictedWeakType [ENorm ε₂] (T : (α → β) → (α' → ε₂)) (p q : ℝ≥0∞)
  (μ : Measure α) (ν : Measure α')
    (c : ℝ≥0∞) : Prop :=
…
```
Mentions: T, α', c
lemma MeasureTheory.HasRestrictedWeakType.without_finiteness line 58 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, MeasureTheory.Measure, TopologicalSpace, ENNReal, Zero, One, ESeminormedAddMonoid, Ne, OfNat.ofNat, Top.top, NNReal, LT.lt, MeasureTheory.HasRestrictedWeakType, Filter.EventuallyEq, MeasureTheory.ae, Function.comp, ENorm.enorm, Set, MeasurableSet, LE.le and 8 more
Proof uses: of_eq_true, Eq.trans, True, congrArg, ENNReal.toReal_inv, eq_true, ENNReal.toReal_pos, dite, And, And.right, MeasureTheory.AEStronglyMeasurable, And.left, Not, Or, Eq.mp, propext, not_and_or, Eq, Eq.mpr, id and 35 more
106 instances, coercions and auxiliary constants left out.

Tactic trace: 54 steps
Open to load the checked before-and-after goal states.
def MeasureTheory.HasRestrictedWeakType' An enhanced version of HasRestrictedWeakType line 113 source
Source documentation
An enhanced version of HasRestrictedWeakType

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, TopologicalSpace, ENorm, ENNReal, MeasureTheory.Measure
Proof uses: MeasureTheory.MemLorentz, OfNat.ofNat, Set, MeasurableSet, And, MeasureTheory.AEStronglyMeasurable, LE.le, MeasureTheory.eLpNorm, MeasureTheory.Measure.restrict, HMul.hMul, MeasureTheory.eLorentzNorm, HPow.hPow, Real, ENNReal.toReal, Inv.inv
13 instances, coercions and auxiliary constants left out.
```
def HasRestrictedWeakType' [TopologicalSpace β] [ENorm β] [ENorm ε₂] (T : (α → β) → (α' → ε₂)) (p q : ℝ≥0∞)
  (μ : Measure α) (ν : Measure α')
    (c : ℝ≥0∞) : Prop :=
…
```
Mentions: T, α', c
def MeasureTheory.WeaklyContinuous The weak continuity assumption needed for HasRestrictedWeakType.hasLorentzType_helper line 124 source
Source documentation
The weak continuity assumption needed for HasRestrictedWeakType.hasLorentzType_helper.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, TopologicalSpace, ENorm, SupSet, Preorder, ENNReal, MeasureTheory.Measure
Proof uses: Nat, MeasureTheory.SimpleFunc, Monotone, BddAbove, Set.range, iSup, MeasureTheory.MemLorentz, OfNat.ofNat, Set, LE.le, MeasureTheory.eLpNorm, MeasureTheory.Measure.restrict, Filter.limsup, Filter.atTop
13 instances, coercions and auxiliary constants left out.
```
def WeaklyContinuous [TopologicalSpace ε] [ENorm ε] [SupSet ε]
  [Preorder ε] [ENorm ε'] (T : (α → ε) → (α' → ε')) (p : ℝ≥0∞) (μ : Measure α) (ν : Measure α') : Prop :=
…
```
Mentions: T, α'
theorem MeasureTheory.HasRestrictedWeakType.hasRestrictedWeakType'_nnreal line 131 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, MeasureTheory.Measure, ENNReal, TopologicalSpace, ENormedAddCommMonoid, Module, NNReal, ENormSMulClass, LT.lt, OfNat.ofNat, Ne, Top.top, ENNReal.HolderConjugate, MeasureTheory.MemLorentz, MeasureTheory.AEStronglyMeasurable, Filter.Eventually, LE.le, ENorm.enorm, HAdd.hAdd, MeasureTheory.ae and 7 more
Proof uses: Function.comp, Filter.mp_mem, Set.ofPred, Eq, Membership.mem, Set, Filter.univ_mem', id, Eq.mpr, congrArg, And, Not, Eq.trans, congrFun', And.intro, LT.lt.ne', MeasurableSet, MeasureTheory.eLpNorm, MeasureTheory.Measure.restrict, MeasureTheory.eLorentzNorm and 168 more
254 instances, coercions and auxiliary constants left out.

Tactic trace: 181 steps
Open to load the checked before-and-after goal states.
lemma MeasureTheory.HasRestrictedWeakType'.hasLorentzType line 312 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, MeasureTheory.Measure, ENNReal, MeasureTheory.SigmaFinite, RCLike, TopologicalSpace, ContinuousENorm, ENNReal.HolderConjugate, Ne, Top.top, MeasureTheory.HasRestrictedWeakType', MeasureTheory.HasLorentzType, OfNat.ofNat
Proof uses: MeasureTheory.MemLorentz, MeasureTheory.AEStronglyMeasurable, And.left, LE.le, MeasureTheory.eLpNorm, MeasureTheory.Measure.restrict, EmptyCollection.emptyCollection, Set, HMul.hMul, MeasureTheory.eLorentzNorm, HPow.hPow, Real, ENNReal.toReal, Inv.inv, MeasurableSet.empty, And.intro, Eq.mpr, MeasureTheory.wnorm, id, Eq and 125 more
177 instances, coercions and auxiliary constants left out.

Tactic trace: 127 steps
Open to load the checked before-and-after goal states.
theorem MeasureTheory.memLorentz_iff_memLorentz_embedRCLike line 429 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, MeasureTheory.Measure, ENNReal, RCLike, NNReal, Iff, MeasureTheory.MemLorentz, Function.comp, Real, RingHom, Algebra.algebraMap
Proof uses: Iff.intro, And.intro, MeasureTheory.AEStronglyMeasurable, LT.lt, MeasureTheory.eLorentzNorm, Top.top, And.left, Eq.mp, congrArg, propext, RCLike.aestronglyMeasurable_iff_aestronglyMeasurable_embedRCLike, Eq.mpr, eq_of_heq, LT, Eq, HEq, Eq.refl, HEq.refl, Eq.symm, MeasureTheory.eLorentzNorm_congr_enorm_ae and 6 more
68 instances, coercions and auxiliary constants left out.

Tactic trace: 24 steps
Open to load the checked before-and-after goal states.
lemma MeasureTheory.HasRestrictedWeakType'.of_hasRestrictedWeakType'_nnreal line 450 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, MeasureTheory.Measure, ENNReal, MeasureTheory.SigmaFinite, RCLike, TopologicalSpace, ENormedAddMonoid, MeasureTheory.MemLorentz, OfNat.ofNat, MeasureTheory.AEStronglyMeasurable, Filter.EventuallyEq, MeasureTheory.ae, Filter.Eventually, LE.le, ENorm.enorm, HAdd.hAdd, HSMul.hSMul, HMul.hMul, MeasureTheory.HasRestrictedWeakType', NNReal and 5 more
Proof uses: Set, MeasurableSet, And.intro, MeasureTheory.eLpNorm, MeasureTheory.Measure.restrict, MeasureTheory.eLorentzNorm, HPow.hPow, ENNReal.toReal, Inv.inv, Eq, Mathlib.Meta.NormNum.isNat_eq_true, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Mathlib.Meta.NormNum.instAtLeastTwo, Mathlib.Meta.NormNum.isNat_mul, Nat.cast_one, Eq.mpr, id, congrArg, Membership.mem and 90 more
249 instances, coercions and auxiliary constants left out.

Tactic trace: 108 steps
Open to load the checked before-and-after goal states.
lemma MeasureTheory.HasRestrictedWeakType.hasLorentzType line 552 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: MeasurableSpace, MeasureTheory.Measure, MeasureTheory.SigmaFinite, RCLike, TopologicalSpace, ENormedAddCommMonoid, Module, NNReal, ENormSMulClass, ENNReal, ENNReal.HolderConjugate, Ne, Top.top, LT.lt, OfNat.ofNat, MeasureTheory.HasRestrictedWeakType, MeasureTheory.MemLorentz, MeasureTheory.AEStronglyMeasurable, Filter.Eventually, LE.le and 19 more
Proof uses: Eq.mpr, id, Eq, congrArg, mul_div_assoc, MeasureTheory.HasRestrictedWeakType'.hasLorentzType, ENNReal.HolderConjugate.ne_zero, ENNReal.mul_ne_top, ENNReal.ofNat_ne_top, ENNReal.div_ne_top, ENNReal.coe_ne_top, MeasureTheory.HasRestrictedWeakType'.of_hasRestrictedWeakType'_nnreal, MeasureTheory.HasRestrictedWeakType.hasRestrictedWeakType'_nnreal, Function.comp, RingHom, Algebra.algebraMap, propext, MeasureTheory.memLorentz_iff_memLorentz_embedRCLike, Filter.mp_mem, Set.ofPred and 86 more
317 instances, coercions and auxiliary constants left out.

Tactic trace: 103 steps
Open to load the checked before-and-after goal states.