import Mathlib namespace OAI noncomputable section open MeasureTheory Set open scoped ENNReal namespace CubicTorus end CubicTorus end section noncomputable section open MeasureTheory Set Complex open scoped ENNReal Real namespace CubicTorus theorem periodic_wirtinger {f g : ℝ → ℂ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivAt f (g x) x) (hg : IntervalIntegrable g volume 0 1) (hf2 : MemLp f 2 (volume.restrict (Ioc (0 : ℝ) 1))) (hg2 : MemLp g 2 (volume.restrict (Ioc (0 : ℝ) 1))) (hp : f 1=f 0) (hm : ∫ x in (0 : ℝ)..1,f x=0) : 4*Real.pi^2*(∫ x in (0 : ℝ)..1,‖f x‖^2) ≤ ∫ x in (0 : ℝ)..1,‖g x‖^2 := by have h01 : (0 : ℝ)<1 := by norm_num have hcf0 : fourierCoeffOn h01 f 0=0 := by rw [fourierCoeffOn_eq_integral] simpa using hm have hcoef (n : ℤ) : 4*Real.pi^2*‖fourierCoeffOn h01 f n‖^2 ≤ ‖fourierCoeffOn h01 g n‖^2 := by by_cases hn : n=0 · subst n rw [hcf0] simp have he := fourierCoeffOn_of_hasDerivAt h01 hn (by simpa using hf) hg have he' : fourierCoeffOn h01 g n= (2*(Real.pi : ℂ)*Complex.I*(n : ℂ))*fourierCoeffOn h01 f n := by simp only [hp,sub_self,mul_zero,sub_zero,zero_sub,Complex.ofReal_one,Complex.ofReal_zero,sub_zero,one_mul] at he have hp0 : (Real.pi : ℂ)≠0 := by exact_mod_cast Real.pi_ne_zero have hn0 : (n : ℂ)≠0 := by exact_mod_cast hn field_simp at he linear_combination -he have hen := congrArg (fun z : ℂ => ‖z‖^2) he' simp only [norm_mul,Complex.norm_ofNat,Complex.norm_real,Real.norm_eq_abs, abs_of_pos Real.pi_pos,Complex.norm_I,mul_one,Complex.norm_intCast] at hen have hn1 : (1 : ℝ)≤|(n : ℝ)| := by exact_mod_cast (Int.one_le_abs hn) have hsq : 1≤|(n : ℝ)|^2 := by nlinarith [sq_nonneg (|(n : ℝ)|-1)] nlinarith [mul_nonneg (sq_nonneg (2*Real.pi*‖fourierCoeffOn h01 f n‖)) (sub_nonneg.mpr hsq)] have hfs := hasSum_sq_fourierCoeffOn h01 hf2 have hgs := hasSum_sq_fourierCoeffOn h01 hg2 have hle := hasSum_le hcoef (hfs.mul_left (4*Real.pi^2)) hgs simpa using hle end CubicTorus end end section noncomputable section open MeasureTheory Set Complex open scoped ENNReal Real namespace CubicTorus theorem mean_zero_curve_area_energy {f g : ℝ → ℂ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivAt f (g x) x) (hg : IntervalIntegrable g volume 0 1) (hf2 : MemLp f 2 (volume.restrict (Ioc (0 : ℝ) 1))) (hg2 : MemLp g 2 (volume.restrict (Ioc (0 : ℝ) 1))) (hp : f 1=f 0) (hm : ∫ x in (0 : ℝ)..1,f x=0) : 2*Real.pi*|∫ x in (0 : ℝ)..1,(star (f x)*g x).im| ≤ ∫ x in (0 : ℝ)..1,‖g x‖^2 := by have hw := periodic_wirtinger hf hg hf2 hg2 hp hm have hfs := hf2.integrable_norm_pow (by norm_num : (2 : ℕ)≠0) have hgs := hg2.integrable_norm_pow (by norm_num : (2 : ℕ)≠0) have hmul : Integrable (fun x => (star (f x)*g x).im) (volume.restrict (Ioc (0 : ℝ) 1)) := (hf2.star.integrable_mul hg2).im have hyp (x : ℝ) : 4*Real.pi*|(star (f x)*g x).im| ≤ 4*Real.pi^2*‖f x‖^2+‖g x‖^2 := by have hi := Complex.abs_im_le_norm (star (f x)*g x) simp only [norm_mul,norm_star] at hi have hi' := mul_le_mul_of_nonneg_left hi (by positivity : 0≤4*Real.pi) nlinarith [sq_nonneg (2*Real.pi*‖f x‖-‖g x‖)] have hle := integral_mono (hmul.abs.const_mul (4*Real.pi)) ((hfs.const_mul (4*Real.pi^2)).add hgs) hyp have habs := norm_integral_le_integral_norm (μ := volume.restrict (Ioc (0 : ℝ) 1)) (fun x => (star (f x)*g x).im) simp only [Real.norm_eq_abs] at habs simp only [Pi.add_apply] at hle rw [integral_const_mul,integral_add (hfs.const_mul _) hgs,integral_const_mul] at hle simp only [intervalIntegral.integral_of_le (by norm_num : (0 : ℝ)≤1)] at hw ⊢ have hh := mul_le_mul_of_nonneg_left habs (by positivity : 0≤4*Real.pi) linarith theorem closed_curve_area_energy {f g : ℝ → ℂ} (hf : ∀ x ∈ Icc (0 : ℝ) 1, HasDerivAt f (g x) x) (hg : IntervalIntegrable g volume 0 1) (hf2 : MemLp f 2 (volume.restrict (Ioc (0 : ℝ) 1))) (hg2 : MemLp g 2 (volume.restrict (Ioc (0 : ℝ) 1))) (hp : f 1=f 0) : 4*Real.pi*|(1/2 : ℝ)*(∫ x in (0 : ℝ)..1,(star (f x)*g x).im)| ≤ ∫ x in (0 : ℝ)..1,‖g x‖^2 := by let c : ℂ := ∫ x in (0 : ℝ)..1,f x let F : ℝ → ℂ := fun x => f x-c have hF : ∀ x ∈ Icc (0 : ℝ) 1,HasDerivAt F (g x) x := fun x hx => (hf x hx).sub_const c have hfi : IntervalIntegrable f volume 0 1 := by apply ContinuousOn.intervalIntegrable simpa only [uIcc_of_le (by norm_num : (0 : ℝ)≤1)] using (show ContinuousOn f (Icc (0 : ℝ) 1) from fun x hx => (hf x hx).continuousAt.continuousWithinAt) have hmean : ∫ x in (0 : ℝ)..1,F x=0 := by rw [show F=(fun x => f x-c) from rfl,intervalIntegral.integral_sub hfi intervalIntegrable_const] simp [c] have hFp : F 1=F 0 := by simp only [F,hp] have hF2 : MemLp F 2 (volume.restrict (Ioc (0 : ℝ) 1)) := hf2.sub (memLp_const c) have hg0 : ∫ x in (0 : ℝ)..1,g x=0 := by rw [intervalIntegral.integral_eq_sub_of_hasDerivAt (by simpa using hf) hg,hp,sub_self] have htrans : (∫ x in (0 : ℝ)..1,(star (F x)*g x).im)= ∫ x in (0 : ℝ)..1,(star (f x)*g x).im := by have hmi := hf2.star.integrable_mul hg2 have hgi : Integrable g (volume.restrict (Ioc (0 : ℝ) 1)) := hg.1 simp only [intervalIntegral.integral_of_le (by norm_num : (0 : ℝ)≤1)] at hg0 ⊢ have hA := integral_im (hF2.star.integrable_mul hg2) have hB := integral_im hmi change (∫ x in Ioc (0 : ℝ) 1,(star (F x)*g x).im)= (∫ x in Ioc (0 : ℝ) 1,star (F x)*g x).im at hA change (∫ x in Ioc (0 : ℝ) 1,(star (f x)*g x).im)= (∫ x in Ioc (0 : ℝ) 1,star (f x)*g x).im at hB rw [hA,hB] congr 1 simp only [F,star_sub,sub_mul] change Integrable (fun x => star (f x)*g x) (volume.restrict (Ioc (0 : ℝ) 1)) at hmi rw [integral_sub hmi (hgi.const_mul _),integral_const_mul,hg0,mul_zero,sub_zero] have hh := mean_zero_curve_area_energy hF hg hF2 hg2 hFp hmean rw [htrans] at hh rw [abs_mul,abs_of_pos (by norm_num : (0 : ℝ)<1/2)] nlinarith only [hh] end CubicTorus end end section noncomputable section open MeasureTheory Set Filter open scoped Topology Real namespace CubicTorus theorem positive_periodic_primitive_inverse {h : ℝ → ℝ} (hc : Continuous h) (hp : Function.Periodic h 1) (hpos : ∀ x,0 ∫ t in (0 : ℝ)..x,h t have hA (x : ℝ) : HasDerivAt A (h x) x := (hc.integral_hasStrictDerivAt 0 x).hasDerivAt have hAc : Continuous A := continuous_iff_continuousAt.mpr fun x => (hA x).continuousAt have hmon : StrictMono A := strictMono_of_deriv_pos fun x => by rw [(hA x).deriv]; exact hpos x have hsurj : Function.Surjective A := hAc.surjective (hp.tendsto_atTop_intervalIntegral_of_pos' (hc.intervalIntegrable 0 1) hpos (by norm_num)) (hp.tendsto_atBot_intervalIntegral_of_pos' (hc.intervalIntegrable 0 1) hpos (by norm_num)) let e : ℝ ≃o ℝ := hmon.orderIsoOfSurjective A hsurj refine ⟨e,fun _ => rfl,hA,?_⟩ intro x exact HasDerivAt.of_local_left_inverse e.symm.continuous.continuousAt (hA (e.symm x)) (hpos _).ne' (Eventually.of_forall fun y => e.apply_symm_apply y) theorem positive_periodic_arclength_parameter {h : ℝ → ℝ} (hc : Continuous h) (hp : Function.Periodic h 1) (hpos : ∀ x,0 e.symm (L*x) let d : ℝ → ℝ := fun x => L/h (q x) have hqc : Continuous q := e.symm.continuous.comp (continuous_const.mul continuous_id) have hdc : Continuous d := continuous_const.div (hc.comp hqc) (fun x => (hpos _).ne') have hqD (x : ℝ) : HasDerivAt q (d x) x := by simpa only [Function.comp_def,mul_one,one_mul,d,q,div_eq_mul_inv,mul_comm] using (heI (L*x)).comp x ((hasDerivAt_id x).const_mul L) have hq0 : q 0=0 := by apply e.injective change e (e.symm (L*0))=e 0 rw [e.apply_symm_apply,he] simp have hq1 : q 1=1 := by apply e.injective change e (e.symm (L*1))=e 1 rw [e.apply_symm_apply,he] simp [L] refine ⟨q,d,hqc,hdc,?_,hqD,hq0,hq1,fun x => div_pos hL (hpos _),?_⟩ · intro x y hxy exact e.symm.strictMono (mul_lt_mul_of_pos_left hxy hL) · intro x dsimp [d] exact div_mul_cancel₀ L (hpos _).ne' end CubicTorus end end section noncomputable section open MeasureTheory Set Complex open scoped ENNReal Real namespace CubicTorus lemma continuous_memLp_two_Ioc {f : ℝ → ℂ} (hf : Continuous f) : MemLp f 2 (volume.restrict (Ioc (0 : ℝ) 1)) := by apply (memLp_two_iff_integrable_sq_norm hf.aestronglyMeasurable).2 exact ((hf.norm.pow 2).continuousOn.integrableOn_compact isCompact_Icc).mono_set Ioc_subset_Icc_self theorem planar_curve_isoperimetric {f g : ℝ → ℂ} (hf : ∀ x,HasDerivAt f (g x) x) (hgc : Continuous g) (hp : Function.Periodic f 1) (hreg : ∀ x,g x≠0) : 4*Real.pi*|(1/2 : ℝ)*(∫ x in (0 : ℝ)..1,(star (f x)*g x).im)| ≤ (∫ x in (0 : ℝ)..1,‖g x‖)^2 := by have hfc : Continuous f := continuous_iff_continuousAt.mpr fun x => (hf x).continuousAt have hgp : Function.Periodic g 1 := by intro x have hd : HasDerivAt f (g (x+1)) x := by have hh := (hf (x+1)).scomp x ((hasDerivAt_id x).add_const 1) have hpeq : (fun x => f (x+1))=f := funext hp simpa only [Function.comp_def,id_eq,hpeq,one_smul] using hh exact hd.unique (hf x) have hspeedp : Function.Periodic (fun x => ‖g x‖) 1 := fun x => congrArg norm (hgp x) obtain ⟨q,d,hqc,hdc,hmon,hqD,hq0,hq1,hdpos,hsp⟩ := positive_periodic_arclength_parameter hgc.norm hspeedp (fun x => norm_pos_iff.mpr (hreg x)) let F : ℝ → ℂ := f ∘ q let G : ℝ → ℂ := fun t => d t • g (q t) have hF : ∀ x,HasDerivAt F (G x) x := fun x => (hf (q x)).scomp x (hqD x) have hFc : Continuous F := hfc.comp hqc have hGc : Continuous G := hdc.smul (hgc.comp hqc) have hFp : F 1=F 0 := by simpa only [F,Function.comp_def,hq0,hq1,zero_add] using hp 0 have hW := closed_curve_area_energy (fun x _ => hF x) (hGc.intervalIntegrable 0 1) (continuous_memLp_two_Ioc hFc) (continuous_memLp_two_Ioc hGc) hFp have hnG (x : ℝ) : ‖G x‖=(∫ t in (0 : ℝ)..1,‖g t‖) := by dsimp only [G] rw [norm_smul,Real.norm_eq_abs,abs_of_pos (hdpos x)] exact hsp x have henergy : (∫ x in (0 : ℝ)..1,‖G x‖^2)=(∫ t in (0 : ℝ)..1,‖g t‖)^2 := by simp only [hnG,intervalIntegral.integral_const,sub_zero,one_smul] have harea : (∫ x in (0 : ℝ)..1,(star (F x)*G x).im)= ∫ x in (0 : ℝ)..1,(star (f x)*g x).im := by have hareaC : Continuous (fun x => (star (f x)*g x).im) := Complex.continuous_im.comp (hfc.star.mul hgc) have hsub := intervalIntegral.integral_deriv_smul_comp (fun x (_ : x∈uIcc (0 : ℝ) 1) => hqD x) hdc.continuousOn hareaC rw [hq0,hq1] at hsub convert hsub using 1 apply intervalIntegral.integral_congr intro x hx simp only [F,G,Function.comp_def,Complex.real_smul,smul_eq_mul] simp ring rwa [henergy,harea] at hW end CubicTorus end end section noncomputable section open MeasureTheory Set Complex open scoped ENNReal Real namespace CubicTorus def unitAngle (t : ℝ) : ℂ := Complex.exp ((2*Real.pi*t : ℝ)*Complex.I) lemma unitAngle_norm (t : ℝ) : ‖unitAngle t‖=1 := by simp [unitAngle,Complex.norm_exp] lemma unitAngle_periodic : Function.Periodic unitAngle 1 := by intro x change Complex.exp ((2*Real.pi*(x+1) : ℝ)*Complex.I)=_ rw [show ((2*Real.pi*(x+1) : ℝ) : ℂ)*Complex.I= ((2*Real.pi*x : ℝ) : ℂ)*Complex.I+2*(Real.pi : ℂ)*Complex.I by push_cast; ring] exact Complex.exp_periodic _ lemma unitAngle_deriv (t : ℝ) : HasDerivAt unitAngle (((2*Real.pi : ℝ)*Complex.I)*unitAngle t) t := by have hd := ((((hasDerivAt_id t).const_mul (2*Real.pi)).ofReal_comp).mul_const Complex.I).cexp convert! hd using 1 simp [unitAngle,id_eq,mul_comm] lemma unitAngle_continuous : Continuous unitAngle := continuous_iff_continuousAt.mpr fun t => (unitAngle_deriv t).continuousAt def radialCurve (r : ℝ → ℝ) (t : ℝ) : ℂ := (r t : ℂ)*unitAngle t def radialVelocity (r dr : ℝ → ℝ) (t : ℝ) : ℂ := ((dr t : ℂ)+(2*Real.pi*r t : ℝ)*Complex.I)*unitAngle t lemma radialCurve_deriv {r dr : ℝ → ℝ} (hr : ∀ t,HasDerivAt r (dr t) t) (t : ℝ) : HasDerivAt (radialCurve r) (radialVelocity r dr t) t := by convert! ((hr t).ofReal_comp).mul (unitAngle_deriv t) using 1 simp only [radialVelocity,Complex.ofReal_mul] ring lemma radialVelocity_norm_sq (r dr : ℝ → ℝ) (t : ℝ) : ‖radialVelocity r dr t‖^2=(dr t)^2+(2*Real.pi*r t)^2 := by rw [radialVelocity,norm_mul,unitAngle_norm,mul_one,Complex.sq_norm] simp [Complex.normSq_apply,pow_two] lemma radialCurve_area_integrand (r dr : ℝ → ℝ) (t : ℝ) : (star (radialCurve r t)*radialVelocity r dr t).im=2*Real.pi*(r t)^2 := by have hu : star (unitAngle t)*unitAngle t=1 := by rw [Complex.star_def,Complex.conj_mul',unitAngle_norm] norm_num have he : star (radialCurve r t)*radialVelocity r dr t= (r t : ℂ)*((dr t : ℂ)+(2*Real.pi*r t : ℝ)*Complex.I) := by simp only [radialCurve,radialVelocity,star_mul,Complex.star_def,Complex.conj_ofReal] calc _ = (r t : ℂ)*((dr t : ℂ)+(2*Real.pi*r t : ℝ)*Complex.I)* (star (unitAngle t)*unitAngle t) := by simp only [Complex.star_def]; ring _ = _ := by rw [hu,mul_one] rw [he] simp only [Complex.mul_im,Complex.ofReal_re,Complex.add_im,Complex.ofReal_im, Complex.I_re,Complex.I_im,mul_zero,zero_mul,zero_add,mul_one,add_zero] ring theorem radial_curve_isoperimetric {r dr : ℝ → ℝ} (hr : ∀ t,HasDerivAt r (dr t) t) (hdc : Continuous dr) (hp : Function.Periodic r 1) (hpos : ∀ t,0 (hr t).continuousAt have hvc : Continuous (radialVelocity r dr) := by exact ((Complex.continuous_ofReal.comp hdc).add ((Complex.continuous_ofReal.comp (continuous_const.mul hrc)).mul continuous_const)).mul unitAngle_continuous have hreg (t : ℝ) : radialVelocity r dr t≠0 := by intro hz have hn := radialVelocity_norm_sq r dr t rw [hz,norm_zero,zero_pow (by norm_num : (2 : ℕ)≠0)] at hn have hrt := hpos t have hh : 0<(2*Real.pi*r t)^2 := sq_pos_of_pos (by positivity) nlinarith [sq_nonneg (dr t)] have hperiod : Function.Periodic (radialCurve r) 1 := by intro x; simp only [radialCurve,hp x,unitAngle_periodic x] have hh := planar_curve_isoperimetric (radialCurve_deriv hr) hvc hperiod hreg have hspeed (t : ℝ) : ‖radialVelocity r dr t‖= Real.sqrt ((dr t)^2+(2*Real.pi*r t)^2) := by rw [← radialVelocity_norm_sq,Real.sqrt_sq (norm_nonneg _)] simp only [radialCurve_area_integrand,hspeed] at hh rw [intervalIntegral.integral_const_mul] at hh have hi : 0≤∫ t in (0 : ℝ)..1,(r t)^2 := intervalIntegral.integral_nonneg (by norm_num) (fun t _ => sq_nonneg (r t)) rw [abs_of_nonneg (by positivity)] at hh nlinarith end CubicTorus end end section noncomputable section open Set Complex open scoped Topology Real namespace CubicTorus def complexSquareGauge (z : ℂ) : ℝ := max |z.re| |z.im| def complexFold (z : ℂ) : ℂ := ⟨|z.re|,|z.im|⟩ def unitSquareScale (t : ℝ) : ℝ := 1/(2*complexSquareGauge (unitAngle t)) def unitSquareEnd (t : ℝ) : ℂ := unitSquareScale t • complexFold (unitAngle t) lemma complexSquareGauge_pos {z : ℂ} (hz : z≠0) : 0 (mul_pos (by norm_num) (complexSquareGauge_pos (unitAngle_ne_zero t))).ne') lemma unitSquareEnd_continuous : Continuous unitSquareEnd := unitSquareScale_continuous.smul (complexFold_continuous.comp unitAngle_continuous) lemma unitSquareScale_periodic : Function.Periodic unitSquareScale 1 := by intro t simp only [unitSquareScale,unitAngle_periodic t] lemma unitSquareEnd_periodic : Function.Periodic unitSquareEnd 1 := by intro t simp only [unitSquareEnd,unitSquareScale_periodic t,unitAngle_periodic t] lemma complexFold_smul {r : ℝ} (hr : 0≤r) (z : ℂ) : complexFold (r • z)=r • complexFold z := by apply Complex.ext <;> simp [complexFold,abs_mul,abs_of_nonneg hr] lemma complexSquareGauge_smul {r : ℝ} (hr : 0≤r) (z : ℂ) : complexSquareGauge (r • z)=r*complexSquareGauge z := by simp only [complexSquareGauge,Complex.smul_re,Complex.smul_im,smul_eq_mul,abs_mul,abs_of_nonneg hr] exact (mul_max_of_nonneg _ _ hr).symm lemma unitSquareEnd_gauge (t : ℝ) : complexSquareGauge (unitSquareEnd t)=1/2 := by rw [unitSquareEnd,complexSquareGauge_smul (unitSquareScale_pos t).le] have he : complexSquareGauge (complexFold (unitAngle t))=complexSquareGauge (unitAngle t) := by simp [complexFold,complexSquareGauge] rw [he,unitSquareScale] field_simp [(complexSquareGauge_pos (unitAngle_ne_zero t)).ne'] lemma unitSquareEnd_coordinates (t : ℝ) : 0≤(unitSquareEnd t).re ∧ 0≤(unitSquareEnd t).im ∧ (unitSquareEnd t).re≤1/2 ∧ (unitSquareEnd t).im≤1/2 ∧ ((unitSquareEnd t).re=1/2 ∨ (unitSquareEnd t).im=1/2) := by have h0 : 0≤(unitSquareEnd t).re := by simp only [unitSquareEnd,Complex.smul_re,smul_eq_mul,complexFold] exact mul_nonneg (unitSquareScale_pos t).le (abs_nonneg _) have h1 : 0≤(unitSquareEnd t).im := by simp only [unitSquareEnd,Complex.smul_im,smul_eq_mul,complexFold] exact mul_nonneg (unitSquareScale_pos t).le (abs_nonneg _) have hg := unitSquareEnd_gauge t rw [complexSquareGauge,abs_of_nonneg h0,abs_of_nonneg h1] at hg refine ⟨h0,h1,?_,?_,?_⟩ · exact (le_max_left _ _).trans_eq hg · exact (le_max_right _ _).trans_eq hg · rcases le_total (unitSquareEnd t).re (unitSquareEnd t).im with h|h · exact Or.inr (by rwa [max_eq_right h] at hg) · exact Or.inl (by rwa [max_eq_left h] at hg) end CubicTorus end end section noncomputable section open MeasureTheory Set open scoped ENNReal namespace CubicTorus end CubicTorus end section noncomputable section open MeasureTheory Set Filter open scoped ENNReal NNReal Interval namespace CubicTorus lemma integral_square_le_mass {X : Type*} [MeasurableSpace X] {μ : Measure X} [IsFiniteMeasure μ] {f : X → ℝ} (hf : Integrable f μ) (hfsq : Integrable (fun x ↦ f x ^ 2) μ) : (∫ x, f x ∂μ) ^ 2 ≤ μ.real univ * ∫ x, f x ^ 2 ∂μ := by let m := μ.real univ let q := ∫ x, f x ∂μ have hm : 0 ≤ m := measureReal_nonneg by_cases hm0 : m = 0 · have hμ0 : μ = 0 := by apply Measure.measure_univ_eq_zero.mp exact (ENNReal.toReal_eq_zero_iff _).mp hm0 |>.resolve_right (measure_ne_top μ univ) simp [hμ0] have hmpos : 0 < m := lt_of_le_of_ne hm (Ne.symm hm0) have hi : Integrable (fun x ↦ f x ^ 2 - (2 * (q / m)) * f x) μ := hfsq.sub (hf.const_mul _) have hp : 0 ≤ ∫ x, f x ^ 2 - (2 * (q / m)) * f x + (q / m) ^ 2 ∂μ := by apply integral_nonneg intro x change 0 ≤ f x ^ 2 - 2 * (q / m) * f x + (q / m) ^ 2 nlinarith [sq_nonneg (f x - q / m)] rw [integral_add hi (integrable_const ((q / m) ^ 2)), integral_sub hfsq (hf.const_mul (2 * (q / m))), integral_const_mul, integral_const, smul_eq_mul] at hp change 0 ≤ (∫ x, f x ^ 2 ∂μ) - 2 * (q / m) * q + m * (q / m) ^ 2 at hp have heq : (∫ x, f x ^ 2 ∂μ) - 2 * (q / m) * q + m * (q / m) ^ 2 = (∫ x, f x ^ 2 ∂μ) - q ^ 2 / m := by field_simp; ring rw [heq] at hp exact (div_le_iff₀ hmpos).mp (by linarith : q ^ 2 / m ≤ ∫ x, f x ^ 2 ∂μ) |>.trans_eq (mul_comm _ _) lemma lipschitz_deriv_sq_integrableOn {f : ℝ → ℝ} {K : ℝ≥0} (hf : LipschitzWith K f) (a b : ℝ) : IntegrableOn (fun x ↦ deriv f x ^ 2) (Icc a b) := by apply (integrable_const ((K : ℝ) ^ 2)).mono' · exact (aestronglyMeasurable_deriv f (volume.restrict (Icc a b))).pow 2 · exact ae_of_all _ fun x ↦ by rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)] have h := norm_deriv_le_of_lipschitz hf (x₀ := x) rw [Real.norm_eq_abs] at h nlinarith [sq_abs (deriv f x), abs_nonneg (deriv f x), NNReal.coe_nonneg K] lemma lipschitz_sq_sub_le_interval_energy {f : ℝ → ℝ} {K : ℝ≥0} (hf : LipschitzWith K f) {a b : ℝ} (hab : a ≤ b) : (f b - f a) ^ 2 ≤ (b - a) * ∫ x in Icc a b, deriv f x ^ 2 := by have hAC := hf.lipschitzOnWith.absolutelyContinuousOnInterval (a := a) (b := b) have hid : IntegrableOn (deriv f) (Icc a b) := by rw [integrableOn_Icc_iff_integrableOn_Ioc] exact hAC.intervalIntegrable_deriv.1 have hcs := integral_square_le_mass hid (lipschitz_deriv_sq_integrableOn hf a b) have hFTC : (∫ x in Icc a b, deriv f x) = f b - f a := by rw [integral_Icc_eq_integral_Ioc] rw [← intervalIntegral.integral_of_le hab] exact hAC.integral_deriv_eq_sub simpa only [hFTC, Measure.restrict_apply_univ, measureReal_def, Real.volume_Icc, ENNReal.toReal_ofReal (sub_nonneg.mpr hab)] using hcs lemma integrable_square_sub_const {X : Type*} [MeasurableSpace X] {μ : Measure X} [IsFiniteMeasure μ] {f : X → ℝ} (hf : Integrable f μ) (hfsq : Integrable (fun x ↦ f x ^ 2) μ) (c : ℝ) : Integrable (fun x ↦ (c - f x) ^ 2) μ := by have h : (fun x ↦ (c - f x) ^ 2) = (fun x ↦ c ^ 2 - (2 * c) * f x + f x ^ 2) := by funext x; ring rw [h] exact ((integrable_const _).sub (hf.const_mul _)).add hfsq lemma square_sub_average_le {X : Type*} [MeasurableSpace X] {μ : Measure X} [IsFiniteMeasure μ] {f : X → ℝ} (hm : 0 < μ.real univ) (hf : Integrable f μ) (hfsq : Integrable (fun x ↦ f x ^ 2) μ) (c : ℝ) : (c - (∫ x, f x ∂μ) / μ.real univ) ^ 2 ≤ (∫ x, (c - f x) ^ 2 ∂μ) / μ.real univ := by have hcs := integral_square_le_mass hf hfsq have hi : Integrable (fun x ↦ c ^ 2 - (2 * c) * f x) μ := (integrable_const _).sub (hf.const_mul _) have he : (∫ x, (c - f x) ^ 2 ∂μ) = μ.real univ * c ^ 2 - (2 * c) * (∫ x, f x ∂μ) + ∫ x, f x ^ 2 ∂μ := by calc _ = ∫ x, c ^ 2 - (2 * c) * f x + f x ^ 2 ∂μ := integral_congr_ae (ae_of_all _ fun x ↦ by ring) _ = _ := by rw [integral_add hi hfsq, integral_sub (integrable_const (c ^ 2)) (hf.const_mul (2 * c)), integral_const, integral_const_mul, smul_eq_mul] rw [he] have halg : (μ.real univ * c ^ 2 - (2 * c) * (∫ x, f x ∂μ) + ∫ x, f x ^ 2 ∂μ) / μ.real univ - (c - (∫ x, f x ∂μ) / μ.real univ) ^ 2 = (μ.real univ * (∫ x, f x ^ 2 ∂μ) - (∫ x, f x ∂μ) ^ 2) / (μ.real univ) ^ 2 := by field_simp ring have hnonneg : 0 ≤ (μ.real univ * (∫ x, f x ^ 2 ∂μ) - (∫ x, f x ∂μ) ^ 2) / (μ.real univ) ^ 2 := div_nonneg (sub_nonneg.mpr hcs) (sq_nonneg _) linarith lemma lipschitz_sq_sub_le_common_interval_energy {f : ℝ → ℝ} {K : ℝ≥0} (hf : LipschitzWith K f) {a b u v : ℝ} (hu : u ∈ Icc a b) (hv : v ∈ Icc a b) : (f u - f v) ^ 2 ≤ (b - a) * ∫ x in Icc a b, deriv f x ^ 2 := by have hab : a ≤ b := hu.1.trans hu.2 have he (u v : ℝ) (hu : u ∈ Icc a b) (hv : v ∈ Icc a b) (huv : u ≤ v) : (f v - f u) ^ 2 ≤ (b - a) * ∫ x in Icc a b, deriv f x ^ 2 := by apply (lipschitz_sq_sub_le_interval_energy hf huv).trans apply mul_le_mul (by linarith [hu.1, hv.2] : v - u ≤ b - a) (setIntegral_mono_set (lipschitz_deriv_sq_integrableOn hf a b) (ae_of_all _ fun x ↦ sq_nonneg (deriv f x)) (ae_of_all _ fun _ hx ↦ Icc_subset_Icc hu.1 hv.2 hx)) (integral_nonneg fun x ↦ sq_nonneg _) (sub_nonneg.mpr hab) rcases le_total u v with huv | hvu · simpa only [show (f v - f u) ^ 2 = (f u - f v) ^ 2 by ring] using he u v hu hv huv · exact he v u hv hu hvu lemma lipschitz_interval_poincare {f : ℝ → ℝ} {K : ℝ≥0} (hf : LipschitzWith K f) {a b : ℝ} (hab : a < b) : (∫ x in Icc a b, (f x - (∫ y in Icc a b, f y) / (b - a)) ^ 2) ≤ (b - a) ^ 2 * ∫ x in Icc a b, deriv f x ^ 2 := by have hmass : (volume.restrict (Icc a b)).real univ = b - a := by simp [measureReal_def, Real.volume_Icc, hab.le] have hi : IntegrableOn f (Icc a b) := hf.continuous.continuousOn.integrableOn_Icc have hi₂ : IntegrableOn (fun x ↦ f x ^ 2) (Icc a b) := (hf.continuous.pow 2).continuousOn.integrableOn_Icc have hpoint (x : ℝ) (hx : x ∈ Icc a b) : (f x - (∫ y in Icc a b, f y) / (b - a)) ^ 2 ≤ (b - a) * ∫ y in Icc a b, deriv f y ^ 2 := by have hj := square_sub_average_le (by rw [hmass]; linarith) hi hi₂ (f x) rw [hmass] at hj apply hj.trans apply (div_le_iff₀ (sub_pos.mpr hab)).mpr calc _ ≤ ∫ _y in Icc a b, (b - a) * ∫ z in Icc a b, deriv f z ^ 2 := by apply setIntegral_mono_on (integrable_square_sub_const hi hi₂ _) (integrable_const _) measurableSet_Icc intro y hy exact lipschitz_sq_sub_le_common_interval_energy hf hx hy _ = _ := by rw [integral_const, smul_eq_mul, hmass]; ring calc _ ≤ ∫ _x in Icc a b, (b - a) * ∫ y in Icc a b, deriv f y ^ 2 := by apply setIntegral_mono_on (((hf.continuous.sub continuous_const).pow 2).continuousOn.integrableOn_Icc) (integrable_const _) measurableSet_Icc hpoint _ = _ := by rw [integral_const, smul_eq_mul, hmass]; ring end CubicTorus end end section noncomputable section open MeasureTheory Set Filter open scoped ENNReal NNReal Topology namespace CubicTorus lemma variance_le_separated_pairwise {X : Type*} [MeasurableSpace X] {μ : Measure X} [IsFiniteMeasure μ] {f A B : X → ℝ} (hm : 0 < μ.real univ) (hf : Integrable f μ) (hfsq : Integrable (fun x ↦ f x ^ 2) μ) (hA : Integrable A μ) (hB : Integrable B μ) (hpair : ∀ᵐ x ∂μ, ∀ᵐ y ∂μ, (f x - f y) ^ 2 ≤ A x + B y) : (∫ x, (f x - (∫ y, f y ∂μ) / μ.real univ) ^ 2 ∂μ) ≤ (∫ x, A x ∂μ) + ∫ y, B y ∂μ := by let m := (∫ y, f y ∂μ) / μ.real univ have hpoint : ∀ᵐ x ∂μ, (f x - m) ^ 2 ≤ A x + (∫ y, B y ∂μ) / μ.real univ := by filter_upwards [hpair] with x hx have hj := square_sub_average_le hm hf hfsq (f x) apply hj.trans apply (div_le_iff₀ hm).mpr calc _ ≤ ∫ y, A x + B y ∂μ := integral_mono_ae (integrable_square_sub_const hf hfsq _) ((integrable_const _).add hB) hx _ = (A x + (∫ y, B y ∂μ) / μ.real univ) * μ.real univ := by rw [integral_add (integrable_const _) hB, integral_const, smul_eq_mul] field_simp have hisq : Integrable (fun x ↦ (f x - m) ^ 2) μ := by convert integrable_square_sub_const hf hfsq m using 1 funext x ring have hh := integral_mono_ae hisq (hA.add (integrable_const ((∫ y, B y ∂μ) / μ.real univ))) hpoint change (∫ x, (f x - m) ^ 2 ∂μ) ≤ ∫ x, A x + (∫ y, B y ∂μ) / μ.real univ ∂μ at hh rw [integral_add hA (integrable_const _), integral_const, smul_eq_mul, mul_div_cancel₀ _ (ne_of_gt hm)] at hh exact hh lemma lipschitz_product_partial_measurable {f : ℝ × ℝ → ℝ} {K : ℝ≥0} (hf : LipschitzWith K f) : Measurable (fun p : ℝ × ℝ ↦ deriv (fun x ↦ f (x, p.2)) p.1) ∧ Measurable (fun p : ℝ × ℝ ↦ deriv (fun y ↦ f (p.1, y)) p.2) := by constructor · exact (measurable_deriv_with_param (f := fun y x ↦ f (x, y)) (hf.continuous.comp continuous_swap)).comp measurable_swap · exact measurable_deriv_with_param (f := fun x y ↦ f (x, y)) hf.continuous lemma lipschitz_product_partial_integrable {f : ℝ × ℝ → ℝ} {K : ℝ≥0} (hf : LipschitzWith K f) (a b : ℝ) : Integrable (fun p : ℝ × ℝ ↦ deriv (fun x ↦ f (x, p.2)) p.1 ^ 2) ((volume.restrict (Icc a b)).prod (volume.restrict (Icc a b))) ∧ Integrable (fun p : ℝ × ℝ ↦ deriv (fun y ↦ f (p.1, y)) p.2 ^ 2) ((volume.restrict (Icc a b)).prod (volume.restrict (Icc a b))) := by have hm := lipschitz_product_partial_measurable hf constructor · apply (integrable_const ((K : ℝ) ^ 2)).mono' (hm.1.pow_const 2).aestronglyMeasurable exact ae_of_all _ fun p ↦ by have hl : LipschitzWith K (fun x ↦ f (x, p.2)) := by simpa only [Function.comp_def, mul_one] using hf.comp (LipschitzWith.prodMk_right p.2) have hb := norm_deriv_le_of_lipschitz hl (x₀ := p.1) rw [Real.norm_eq_abs] at hb ⊢ rw [abs_of_nonneg (sq_nonneg _)] nlinarith [sq_abs (deriv (fun x ↦ f (x, p.2)) p.1), NNReal.coe_nonneg K, abs_nonneg (deriv (fun x ↦ f (x, p.2)) p.1)] · apply (integrable_const ((K : ℝ) ^ 2)).mono' (hm.2.pow_const 2).aestronglyMeasurable exact ae_of_all _ fun p ↦ by have hl : LipschitzWith K (fun y ↦ f (p.1, y)) := by simpa only [Function.comp_def, mul_one] using hf.comp (LipschitzWith.prodMk_left p.1) have hb := norm_deriv_le_of_lipschitz hl (x₀ := p.2) rw [Real.norm_eq_abs] at hb ⊢ rw [abs_of_nonneg (sq_nonneg _)] nlinarith [sq_abs (deriv (fun y ↦ f (p.1, y)) p.2), NNReal.coe_nonneg K, abs_nonneg (deriv (fun y ↦ f (p.1, y)) p.2)] theorem lipschitz_box_poincare {f : ℝ × ℝ → ℝ} {K : ℝ≥0} (hf : LipschitzWith K f) {a b : ℝ} (hab : a < b) : let μ := (volume.restrict (Icc a b)).prod (volume.restrict (Icc a b)) (∫ p, (f p - (∫ q, f q ∂μ) / (b - a) ^ 2) ^ 2 ∂μ) ≤ 2 * (b - a) ^ 2 * ((∫ p, deriv (fun x ↦ f (x, p.2)) p.1 ^ 2 ∂μ) + ∫ p, deriv (fun y ↦ f (p.1, y)) p.2 ^ 2 ∂μ) := by dsimp only let ν : Measure ℝ := volume.restrict (Icc a b) let μ := ν.prod ν let Ex : ℝ → ℝ := fun y ↦ ∫ x, deriv (fun t ↦ f (t, y)) x ^ 2 ∂ν let Ey : ℝ → ℝ := fun x ↦ ∫ y, deriv (fun t ↦ f (x, t)) y ^ 2 ∂ν have hν : ν.real univ = b - a := by simp [ν, measureReal_def, Real.volume_Icc, hab.le] have hμ : μ.real univ = (b - a) ^ 2 := by change ((ν.prod ν) univ).toReal = _ rw [← Set.univ_prod_univ, Measure.prod_prod, ENNReal.toReal_mul] change ν.real univ * ν.real univ = _ rw [hν] ring have hi : Integrable f μ := by rw [show μ = volume.restrict (Icc a b ×ˢ Icc a b) from by simp only [μ, ν, Measure.prod_restrict, ← Measure.volume_eq_prod]] exact ContinuousOn.integrableOn_compact (isCompact_Icc.prod isCompact_Icc) hf.continuous.continuousOn have hisq : Integrable (fun p ↦ f p ^ 2) μ := by rw [show μ = volume.restrict (Icc a b ×ˢ Icc a b) from by simp only [μ, ν, Measure.prod_restrict, ← Measure.volume_eq_prod]] exact ContinuousOn.integrableOn_compact (isCompact_Icc.prod isCompact_Icc) (hf.continuous.pow 2).continuousOn have hip := lipschitz_product_partial_integrable hf a b have hiEx : Integrable Ex ν := hip.1.integral_prod_right have hiEy : Integrable Ey ν := hip.2.integral_prod_left have hmem : ∀ᵐ p ∂μ, p.1 ∈ Icc a b ∧ p.2 ∈ Icc a b := by rw [show μ = volume.restrict (Icc a b ×ˢ Icc a b) from by simp only [μ, ν, Measure.prod_restrict, ← Measure.volume_eq_prod]] exact ae_restrict_mem (measurableSet_Icc.prod measurableSet_Icc) have hpair : ∀ᵐ p ∂μ, ∀ᵐ q ∂μ, (f p - f q) ^ 2 ≤ 2 * (b - a) * Ex p.2 + 2 * (b - a) * Ey q.1 := by filter_upwards [hmem] with p hp filter_upwards [hmem] with q hq have hx : (f p - f (q.1, p.2)) ^ 2 ≤ (b - a) * Ex p.2 := lipschitz_sq_sub_le_common_interval_energy (by simpa using hf.comp (LipschitzWith.prodMk_right p.2)) hp.1 hq.1 have hy : (f (q.1, p.2) - f q) ^ 2 ≤ (b - a) * Ey q.1 := lipschitz_sq_sub_le_common_interval_energy (by simpa using hf.comp (LipschitzWith.prodMk_left q.1)) hp.2 hq.2 nlinarith [sq_nonneg (f p - 2 * f (q.1, p.2) + f q)] have hh := variance_le_separated_pairwise (μ := μ) (by rw [hμ]; positivity) hi hisq ((hiEx.comp_snd ν).const_mul (2 * (b - a))) ((hiEy.comp_fst ν).const_mul (2 * (b - a))) hpair rw [hμ] at hh have hex : (∫ p, Ex p.2 ∂μ) = (b - a) * ∫ y, Ex y ∂ν := by rw [integral_prod _ (hiEx.comp_snd ν)] simp only [integral_const, smul_eq_mul, hν] have hey : (∫ p, Ey p.1 ∂μ) = (b - a) * ∫ x, Ey x ∂ν := by rw [integral_prod _ (hiEy.comp_fst ν)] simp only [integral_const, smul_eq_mul, hν, integral_const_mul] have hEx : (∫ y, Ex y ∂ν) = ∫ p, deriv (fun x ↦ f (x, p.2)) p.1 ^ 2 ∂μ := (integral_prod_symm _ hip.1).symm have hEy : (∫ x, Ey x ∂ν) = ∫ p, deriv (fun y ↦ f (p.1, y)) p.2 ^ 2 ∂μ := (integral_prod _ hip.2).symm rw [integral_const_mul, integral_const_mul, hex, hey, hEx, hEy] at hh convert hh using 1 ring lemma differentiableAt_partial_sq_le {f : ℝ × ℝ → ℝ} {p : ℝ × ℝ} (hf : DifferentiableAt ℝ f p) : deriv (fun x ↦ f (x, p.2)) p.1 ^ 2 + deriv (fun y ↦ f (p.1, y)) p.2 ^ 2 ≤ 2 * ‖fderiv ℝ f p‖ ^ 2 := by have hx : deriv (fun x ↦ f (x, p.2)) p.1 = fderiv ℝ f p (1, 0) := by exact (hf.hasFDerivAt.comp_hasDerivAt p.1 ((hasDerivAt_id p.1).prodMk (hasDerivAt_const p.1 p.2))).deriv have hy : deriv (fun y ↦ f (p.1, y)) p.2 = fderiv ℝ f p (0, 1) := by exact (hf.hasFDerivAt.comp_hasDerivAt p.2 ((hasDerivAt_const p.2 p.1).prodMk (hasDerivAt_id p.2))).deriv rw [hx, hy] have hb₁ : |fderiv ℝ f p (1, 0)| ≤ ‖fderiv ℝ f p‖ := by simpa [Prod.norm_def] using (fderiv ℝ f p).le_opNorm (1, 0) have hb₂ : |fderiv ℝ f p (0, 1)| ≤ ‖fderiv ℝ f p‖ := by simpa [Prod.norm_def] using (fderiv ℝ f p).le_opNorm (0, 1) nlinarith [sq_abs (fderiv ℝ f p (1, 0)), sq_abs (fderiv ℝ f p (0, 1)), abs_nonneg (fderiv ℝ f p (1, 0)), abs_nonneg (fderiv ℝ f p (0, 1))] lemma lipschitz_fderiv_sq_integrable {X : Type*} [NormedAddCommGroup X] [NormedSpace ℝ X] [MeasurableSpace X] [OpensMeasurableSpace X] {μ : Measure X} [IsFiniteMeasure μ] {f : X → ℝ} {K : ℝ≥0} (hf : LipschitzWith K f) : Integrable (fun x ↦ ‖fderiv ℝ f x‖ ^ 2) μ := by apply (integrable_const ((K : ℝ) ^ 2)).mono' ((measurable_fderiv ℝ f).norm.pow_const 2).aestronglyMeasurable exact ae_of_all _ fun x ↦ by rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)] have hh := norm_fderiv_le_of_lipschitz ℝ hf (x₀ := x) nlinarith [norm_nonneg (fderiv ℝ f x), NNReal.coe_nonneg K] theorem lipschitz_box_poincare_fderiv {f : ℝ × ℝ → ℝ} {K : ℝ≥0} (hf : LipschitzWith K f) {a b : ℝ} (hab : a < b) : let μ := (volume.restrict (Icc a b)).prod (volume.restrict (Icc a b)) (∫ p, (f p - (∫ q, f q ∂μ) / (b - a) ^ 2) ^ 2 ∂μ) ≤ 4 * (b - a) ^ 2 * (∫ p, ‖fderiv ℝ f p‖ ^ 2 ∂μ) := by dsimp only let μ := (volume.restrict (Icc a b)).prod (volume.restrict (Icc a b)) have hdiff : ∀ᵐ p ∂μ, DifferentiableAt ℝ f p := by rw [show μ = volume.restrict (Icc a b ×ˢ Icc a b) from by simp only [μ, Measure.prod_restrict, ← Measure.volume_eq_prod]] exact ae_restrict_of_ae hf.ae_differentiableAt have hip := lipschitz_product_partial_integrable hf a b have hb := integral_mono_ae (hip.1.add hip.2) ((lipschitz_fderiv_sq_integrable (μ := μ) hf).const_mul 2) (hdiff.mono fun _ h ↦ differentiableAt_partial_sq_le h) change (∫ p, deriv (fun x ↦ f (x, p.2)) p.1 ^ 2 + deriv (fun y ↦ f (p.1, y)) p.2 ^ 2 ∂μ) ≤ ∫ p, 2 * ‖fderiv ℝ f p‖ ^ 2 ∂μ at hb rw [integral_add hip.1 hip.2, integral_const_mul] at hb apply (lipschitz_box_poincare hf hab).trans have hh := mul_le_mul_of_nonneg_left hb (by positivity : 0 ≤ 2 * (b - a) ^ 2) convert hh using 1 ring end CubicTorus end end end end OAI