feat: Hölder triples and multiplication of MeasureTheory.Lp functions

[j-loreaux]

This introduces a class ENNReal.HolderTriple which takes arguments r p q : ℝ≥0∞ and simply requires 1 / r = 1 / p + 1 / q, and r is marked as a semiOutParam (not an outParam because this could cause issues for two values of r that are propositionally but not definitionally equal).

This class allows us to define an HMul instance for MeasureTheory.Lp, and then proves various facts about it. This culminates in the natural continuous linear map from MeasureTheory.Lp 𝕜 p μ to NormedSpace.Dual 𝕜 (MeasureTheory.Lp 𝕜 q μ). Of course, this can be upgraded to a linear isometry equivalence, but that requires significantly more work.

Note: this makes the second example (the first being matrices) of a true heterogeneous multiplication occurring in practice. Right now, none of our distributivity lemmas can be stated for heterogeneous multiplication. Moreover, we can't have IsScalarTower or SMulCommClass instances for these.

Maybe we should consider having heterogeneous variants of these classes?

---

• depends on: refactor: allow for bilinear maps of norm greater than one in Hölder's inequality and use HolderTriple #21792
• depends on: [Merged by Bors] - feat: Hölder triples / conjugates as a type class #21854

This PR can be split once the design decisions are approved.

[github-actions]

PR summary 495856e5d3

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.Tactic	1
294 files Mathlib.Algebra.Module.ZLattice.Basic Mathlib.Algebra.Module.ZLattice.Covolume Mathlib.Analysis.BoundedVariation Mathlib.Analysis.BoxIntegral.Integrability Mathlib.Analysis.BoxIntegral.UnitPartition Mathlib.Analysis.CStarAlgebra.ApproximateUnit Mathlib.Analysis.CStarAlgebra.CStarMatrix Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Integral Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order Mathlib.Analysis.CStarAlgebra.GelfandDuality Mathlib.Analysis.CStarAlgebra.Hom Mathlib.Analysis.CStarAlgebra.Module.Constructions Mathlib.Analysis.CStarAlgebra.Module.Defs Mathlib.Analysis.CStarAlgebra.SpecialFunctions.PosPart Mathlib.Analysis.CStarAlgebra.Spectrum Mathlib.Analysis.Calculus.BumpFunction.Convolution Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension Mathlib.Analysis.Calculus.BumpFunction.Normed Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts Mathlib.Analysis.Calculus.Monotone Mathlib.Analysis.Calculus.ParametricIntegral Mathlib.Analysis.Calculus.ParametricIntervalIntegral Mathlib.Analysis.Calculus.Rademacher Mathlib.Analysis.Complex.AbsMax Mathlib.Analysis.Complex.CauchyIntegral Mathlib.Analysis.Complex.Hadamard Mathlib.Analysis.Complex.Liouville Mathlib.Analysis.Complex.LocallyUniformLimit Mathlib.Analysis.Complex.OpenMapping Mathlib.Analysis.Complex.Periodic Mathlib.Analysis.Complex.PhragmenLindelof Mathlib.Analysis.Complex.Polynomial.Basic Mathlib.Analysis.Complex.Polynomial.UnitTrinomial Mathlib.Analysis.Complex.Positivity Mathlib.Analysis.Complex.RemovableSingularity Mathlib.Analysis.Complex.Schwarz Mathlib.Analysis.Complex.TaylorSeries Mathlib.Analysis.Complex.UpperHalfPlane.Exp Mathlib.Analysis.Convex.Integral Mathlib.Analysis.Convolution 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Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.ExpLog Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Isometric Mathlib.Analysis.SpecialFunctions.Gamma.Basic Mathlib.Analysis.SpecialFunctions.Gamma.Beta Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup Mathlib.Analysis.SpecialFunctions.Gamma.Deligne Mathlib.Analysis.SpecialFunctions.Gamma.Deriv Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral Mathlib.Analysis.SpecialFunctions.Gaussian.PoissonSummation Mathlib.Analysis.SpecialFunctions.ImproperIntegrals Mathlib.Analysis.SpecialFunctions.Integrals Mathlib.Analysis.SpecialFunctions.JapaneseBracket Mathlib.Analysis.SpecialFunctions.Log.Summable Mathlib.Analysis.SpecialFunctions.NonIntegrable Mathlib.Analysis.SpecialFunctions.PolarCoord Mathlib.Analysis.SpecialFunctions.Pow.Integral Mathlib.Analysis.SpecialFunctions.Stirling Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd Mathlib.Analysis.SumIntegralComparisons Mathlib.Data.Real.Pi.Irrational Mathlib.Data.Real.Pi.Leibniz Mathlib.Data.Real.Pi.Wallis Mathlib.Dynamics.Ergodic.Action.OfMinimal Mathlib.Dynamics.Ergodic.AddCircle Mathlib.Geometry.Manifold.BumpFunction Mathlib.Geometry.Manifold.Complex Mathlib.Geometry.Manifold.IntegralCurve.ExistUnique Mathlib.Geometry.Manifold.IntegralCurve.UniformTime Mathlib.Geometry.Manifold.PartitionOfUnity Mathlib.Geometry.Manifold.WhitneyEmbedding Mathlib.MeasureTheory.Constructions.HaarToSphere Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace Mathlib.MeasureTheory.Covering.Besicovitch Mathlib.MeasureTheory.Covering.DensityTheorem Mathlib.MeasureTheory.Covering.Differentiation Mathlib.MeasureTheory.Covering.LiminfLimsup Mathlib.MeasureTheory.Covering.OneDim Mathlib.MeasureTheory.Decomposition.IntegralRNDeriv Mathlib.MeasureTheory.Decomposition.Lebesgue Mathlib.MeasureTheory.Decomposition.RadonNikodym 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Mathlib.MeasureTheory.Integral.DivergenceTheorem Mathlib.MeasureTheory.Integral.DominatedConvergence Mathlib.MeasureTheory.Integral.ExpDecay Mathlib.MeasureTheory.Integral.FundThmCalculus Mathlib.MeasureTheory.Integral.Gamma Mathlib.MeasureTheory.Integral.IntegrableOn Mathlib.MeasureTheory.Integral.IntegralEqImproper Mathlib.MeasureTheory.Integral.IntervalAverage Mathlib.MeasureTheory.Integral.IntervalIntegral Mathlib.MeasureTheory.Integral.Layercake Mathlib.MeasureTheory.Integral.PeakFunction Mathlib.MeasureTheory.Integral.Periodic Mathlib.MeasureTheory.Integral.Pi Mathlib.MeasureTheory.Integral.Prod Mathlib.MeasureTheory.Integral.SetIntegral Mathlib.MeasureTheory.Integral.SetToL1 Mathlib.MeasureTheory.Integral.TorusIntegral Mathlib.MeasureTheory.Integral.VitaliCaratheodory Mathlib.MeasureTheory.Measure.DiracProba Mathlib.MeasureTheory.Measure.FiniteMeasureProd Mathlib.MeasureTheory.Measure.FiniteMeasure Mathlib.MeasureTheory.Measure.Haar.Disintegration Mathlib.MeasureTheory.Measure.Haar.NormedSpace Mathlib.MeasureTheory.Measure.Haar.Quotient Mathlib.MeasureTheory.Measure.Haar.Unique Mathlib.MeasureTheory.Measure.Lebesgue.Integral Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls Mathlib.MeasureTheory.Measure.LevyProkhorovMetric Mathlib.MeasureTheory.Measure.LogLikelihoodRatio Mathlib.MeasureTheory.Measure.Portmanteau Mathlib.MeasureTheory.Measure.ProbabilityMeasure Mathlib.MeasureTheory.Measure.SeparableMeasure Mathlib.MeasureTheory.Measure.Tilted Mathlib.MeasureTheory.Measure.WithDensityFinite Mathlib.MeasureTheory.Order.UpperLower Mathlib.MeasureTheory.VectorMeasure.WithDensity Mathlib.NumberTheory.AbelSummation Mathlib.NumberTheory.Cyclotomic.Embeddings Mathlib.NumberTheory.Cyclotomic.PID Mathlib.NumberTheory.Cyclotomic.Three Mathlib.NumberTheory.DirichletCharacter.GaussSum Mathlib.NumberTheory.EulerProduct.DirichletLSeries Mathlib.NumberTheory.FLT.Three Mathlib.NumberTheory.Fermat Mathlib.NumberTheory.GaussSum Mathlib.NumberTheory.Harmonic.Bounds Mathlib.NumberTheory.Harmonic.GammaDeriv Mathlib.NumberTheory.Harmonic.ZetaAsymp Mathlib.NumberTheory.JacobiSum.Basic Mathlib.NumberTheory.LSeries.AbstractFuncEq Mathlib.NumberTheory.LSeries.Deriv Mathlib.NumberTheory.LSeries.DirichletContinuation Mathlib.NumberTheory.LSeries.Dirichlet Mathlib.NumberTheory.LSeries.HurwitzZetaEven Mathlib.NumberTheory.LSeries.HurwitzZetaOdd Mathlib.NumberTheory.LSeries.HurwitzZetaValues Mathlib.NumberTheory.LSeries.HurwitzZeta Mathlib.NumberTheory.LSeries.MellinEqDirichlet Mathlib.NumberTheory.LSeries.Nonvanishing Mathlib.NumberTheory.LSeries.Positivity Mathlib.NumberTheory.LSeries.PrimesInAP Mathlib.NumberTheory.LSeries.RiemannZeta Mathlib.NumberTheory.LSeries.SumCoeff Mathlib.NumberTheory.LSeries.ZMod Mathlib.NumberTheory.LegendreSymbol.AddCharacter Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity Mathlib.NumberTheory.ModularForms.EisensteinSeries.Basic Mathlib.NumberTheory.ModularForms.EisensteinSeries.MDifferentiable Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds Mathlib.NumberTheory.ModularForms.JacobiTheta.Manifold Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable Mathlib.NumberTheory.ModularForms.LevelOne Mathlib.NumberTheory.ModularForms.QExpansion Mathlib.NumberTheory.NumberField.AdeleRing Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody Mathlib.NumberTheory.NumberField.CanonicalEmbedding.FundamentalCone Mathlib.NumberTheory.NumberField.ClassNumber Mathlib.NumberTheory.NumberField.Completion Mathlib.NumberTheory.NumberField.Discriminant.Basic Mathlib.NumberTheory.NumberField.Embeddings Mathlib.NumberTheory.NumberField.EquivReindex Mathlib.NumberTheory.NumberField.FinitePlaces Mathlib.NumberTheory.NumberField.House Mathlib.NumberTheory.NumberField.ProductFormula Mathlib.NumberTheory.NumberField.Units.Basic Mathlib.NumberTheory.NumberField.Units.DirichletTheorem Mathlib.NumberTheory.NumberField.Units.Regulator Mathlib.NumberTheory.Transcendental.Lindemann.Init.AnalyticalPart Mathlib.NumberTheory.WellApproximable Mathlib.NumberTheory.ZetaValues Mathlib.Probability.BorelCantelli Mathlib.Probability.CDF Mathlib.Probability.CondVar Mathlib.Probability.ConditionalExpectation Mathlib.Probability.Density Mathlib.Probability.Distributions.Exponential Mathlib.Probability.Distributions.Gamma Mathlib.Probability.Distributions.Gaussian Mathlib.Probability.Distributions.Pareto Mathlib.Probability.Distributions.Uniform Mathlib.Probability.IdentDistrib Mathlib.Probability.Independence.Conditional Mathlib.Probability.Independence.Integrable Mathlib.Probability.Independence.ZeroOne Mathlib.Probability.Integration Mathlib.Probability.Kernel.Composition.IntegralCompProd Mathlib.Probability.Kernel.Composition.Lemmas Mathlib.Probability.Kernel.Composition.MeasureCompProd Mathlib.Probability.Kernel.Composition.MeasureComp Mathlib.Probability.Kernel.CondDistrib Mathlib.Probability.Kernel.Condexp Mathlib.Probability.Kernel.Disintegration.Basic Mathlib.Probability.Kernel.Disintegration.CDFToKernel Mathlib.Probability.Kernel.Disintegration.CondCDF Mathlib.Probability.Kernel.Disintegration.Density Mathlib.Probability.Kernel.Disintegration.Integral Mathlib.Probability.Kernel.Disintegration.StandardBorel Mathlib.Probability.Kernel.Disintegration.Unique Mathlib.Probability.Kernel.Integral Mathlib.Probability.Kernel.MeasurableIntegral Mathlib.Probability.Kernel.RadonNikodym Mathlib.Probability.Kernel.WithDensity Mathlib.Probability.Martingale.Basic Mathlib.Probability.Martingale.BorelCantelli Mathlib.Probability.Martingale.Centering Mathlib.Probability.Martingale.Convergence Mathlib.Probability.Martingale.OptionalSampling Mathlib.Probability.Martingale.OptionalStopping Mathlib.Probability.Martingale.Upcrossing Mathlib.Probability.Moments.Basic Mathlib.Probability.Moments.ComplexMGF Mathlib.Probability.Moments.IntegrableExpMul Mathlib.Probability.Moments.MGFAnalytic Mathlib.Probability.Notation Mathlib.Probability.ProbabilityMassFunction.Integrals Mathlib.Probability.Process.Adapted Mathlib.Probability.Process.Filtration Mathlib.Probability.Process.HittingTime Mathlib.Probability.Process.PartitionFiltration Mathlib.Probability.Process.Stopping Mathlib.Probability.StrongLaw Mathlib.Probability.Variance Mathlib.RingTheory.Polynomial.Selmer Mathlib.Tactic.NormNum.LegendreSymbol	2
Mathlib.Data.ENNReal.Holder (new file)	694
Mathlib.MeasureTheory.Function.Holder (new file)	1941

---

Declarations diff

+ HolderConjugate
+ HolderTriple
+ MeasureTheory.Lp.toDualCLM
+ MeasureTheory.Lp.toDualCLM_eq_integral
+ Memℒp.of_bilin
+ _root_.ContinuousLinearMap.holderL
+ _root_.ContinuousLinearMap.holderₗ
+ _root_.ContinuousLinearMap.norm_holderL_le
+ add_smul
+ coeFn_holder
+ coeFn_lp_smul
+ eq_top_iff_eq_one
+ holder
+ holder_add_left
+ holder_add_right
+ holder_smul_left
+ holder_smul_right
+ instInfty
+ instOneInfty
+ instTwoTwo
+ instZero
+ instance : HSMul (Lp 𝕜 p μ) (Lp E q μ) (Lp E r μ)
+ inv_add_inv_eq_inv
+ inv_add_inv_eq_one
+ inv_eq
+ inv_inv_add_inv
+ inv_le_inv
+ inv_sub_inv_eq_inv
+ inv_sub_inv_eq_inv'
+ le
+ lt_top_iff_one_lt
+ ne_top_iff_ne_one
+ neg_smul
+ neg_smul_neg
+ nnnorm_holder_apply_apply_le
+ norm_holder_apply_apply_le
+ norm_smul_le
+ of
+ one_div_add_one_div
+ one_div_eq
+ one_le
+ one_sub_inv
+ pos
+ smul_add
+ smul_comm
+ smul_def
+ smul_neg
+ smul_smul_assoc
+ smul_zero
+ sub_one_mul_inv
+ symm
+ zero_smul

You can run this locally as follows

## summary with just the declaration names:
./scripts/declarations_diff.sh <optional_commit>

## more verbose report:
./scripts/declarations_diff.sh long <optional_commit>

The doc-module for script/declarations_diff.sh contains some details about this script.

---

No changes to technical debt.

You can run this locally as

./scripts/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

[YaelDillies]

Can you show the relationship to conjugate exponents?

[j-loreaux]

p q are Hölder conjugate exponents if HolderTriple 1 p q. Is that your question?

[mathlib4-dependent-issues-bot]

This PR/issue depends on:

• refactor: allow for bilinear maps of norm greater than one in Hölder's inequality and use HolderTriple #21792
• [Merged by Bors] - feat: Hölder triples / conjugates as a type class #21854
  By Dependent Issues (🤖). Happy coding!