import OAI.Analysis.HotSpots.Basic namespace OAI section DouglasLipschitzBase noncomputable section section FullBoundaryCombinedLayer section SingularLayer namespace ReciprocalKernel open Matrix Filter Topology open scoped BigOperators variable {ι : Type*} [Fintype ι] def pairing (A : Matrix ι ι ℝ) (x y : ι → ℝ) : ℝ := x ⬝ᵥ A *ᵥ y lemma pairing_symm {A : Matrix ι ι ℝ} (hA : A.IsSymm) (x y : ι → ℝ) : pairing A x y = pairing A y x := by simpa only [pairing, hA.eq] using Matrix.dotProduct_transpose_mulVec A x y lemma pairing_sub_smul (A : Matrix ι ι ℝ) (x y : ι → ℝ) (t : ℝ) : pairing A (y - t • x) (y - t • x) = pairing A y y - t * pairing A y x - t * pairing A x y + t ^ 2 * pairing A x x := by simp only [pairing, Matrix.mulVec_sub, Matrix.mulVec_smul, sub_dotProduct, dotProduct_sub, dotProduct_smul, smul_dotProduct, smul_eq_mul] ring lemma pairing_sub_smul_right (A : Matrix ι ι ℝ) (x y z : ι → ℝ) (t : ℝ) : pairing A x (y - t • z) = pairing A x y - t * pairing A x z := by simp only [pairing, Matrix.mulVec_sub, Matrix.mulVec_smul, dotProduct_sub, dotProduct_smul, smul_eq_mul] lemma pairing_add_smul_right (A : Matrix ι ι ℝ) (x y z : ι → ℝ) (t : ℝ) : pairing A x (y + t • z) = pairing A x y + t * pairing A x z := by simp only [pairing, Matrix.mulVec_add, Matrix.mulVec_smul, dotProduct_add, dotProduct_smul, smul_eq_mul] lemma pairing_add_smul (A : Matrix ι ι ℝ) (x y : ι → ℝ) (t : ℝ) : pairing A (y + t • x) (y + t • x) = pairing A y y + t * pairing A y x + t * pairing A x y + t ^ 2 * pairing A x x := by simp only [pairing, Matrix.mulVec_add, Matrix.mulVec_smul, add_dotProduct, dotProduct_add, dotProduct_smul, smul_dotProduct, smul_eq_mul] ring lemma pairing_tendsto {A : ℕ → Matrix ι ι ℝ} {B : Matrix ι ι ℝ} (hlim : ∀ i j, Tendsto (fun n => A n i j) atTop (𝓝 (B i j))) (x y : ι → ℝ) : Tendsto (fun n => pairing (A n) x y) atTop (𝓝 (pairing B x y)) := by unfold pairing dotProduct Matrix.mulVec exact tendsto_finsetSum _ (fun i _ => tendsto_const_nhds.mul (tendsto_finsetSum _ fun j _ => (hlim i j).mul tendsto_const_nhds)) def OnePositiveSquare (A : Matrix ι ι ℝ) : Prop := A.IsSymm ∧ ∀ x y : ι → ℝ, 0 < pairing A x x → pairing A x y = 0 → pairing A y y ≤ 0 lemma onePositiveSquare_reverse_cauchy {A : Matrix ι ι ℝ} (hA : OnePositiveSquare A) (x y : ι → ℝ) (hx : 0 < pairing A x x) : pairing A x x * pairing A y y ≤ (pairing A x y) ^ 2 := by have hz : pairing A x (y - (pairing A x y / pairing A x x) • x) = 0 := by rw [pairing_sub_smul_right] field_simp ring have hn := hA.2 x _ hx hz rw [pairing_sub_smul, pairing_symm hA.1 y x] at hn have he : pairing A y y - (pairing A x y / pairing A x x) * pairing A x y - (pairing A x y / pairing A x x) * pairing A x y + (pairing A x y / pairing A x x) ^ 2 * pairing A x x = pairing A y y - (pairing A x y) ^ 2 / pairing A x x := by field_simp ring rw [he] at hn have := (le_div_iff₀ hx).mp (sub_nonpos.mp hn) nlinarith lemma onePositiveSquare_of_entrywise_tendsto {A : ℕ → Matrix ι ι ℝ} {B : Matrix ι ι ℝ} (hA : ∀ n, OnePositiveSquare (A n)) (hlim : ∀ i j, Tendsto (fun n => A n i j) atTop (𝓝 (B i j))) : OnePositiveSquare B := by refine ⟨?_, ?_⟩ · ext i j exact tendsto_nhds_unique (hlim j i) ((hlim i j).congr fun n => (hA n).1.apply i j |>.symm) · intro x y hx hxy have hxx := pairing_tendsto hlim x x have hyy := pairing_tendsto hlim y y have hxy' := pairing_tendsto hlim x y have hp : ∀ᶠ n in atTop, 0 < pairing (A n) x x := hxx.eventually_const_lt hx have hn : pairing B x x * pairing B y y - (pairing B x y) ^ 2 ≤ 0 := by apply le_of_tendsto ((hxx.mul hyy).sub (hxy'.pow 2)) filter_upwards [hp] with n hn exact sub_nonpos.mpr (onePositiveSquare_reverse_cauchy (hA n) x y hn) rw [hxy] at hn nlinarith lemma onePositiveSquare_isotropic_hyperplane {A : Matrix ι ι ℝ} (hA : OnePositiveSquare A) (z : ι → ℝ) (hz : pairing A z z = 0) (hnonradical : ∃ y, pairing A z y ≠ 0) : ∀ x, pairing A z x = 0 → pairing A x x ≤ 0 := by intro x hzx by_contra hx have hx : 0 < pairing A x x := lt_of_not_ge hx obtain ⟨y, hy⟩ := hnonradical have hxz : pairing A x z = 0 := (pairing_symm hA.1 x z).trans hzx let t := ((pairing A x y) ^ 2 / pairing A x x - pairing A y y + 1) / (2 * pairing A z y) have hn := onePositiveSquare_reverse_cauchy hA x (y + t • z) hx rw [pairing_add_smul, pairing_add_smul_right, hz, hxz, pairing_symm hA.1 y z] at hn dsimp [t] at hn have he : pairing A y y + (((pairing A x y) ^ 2 / pairing A x x - pairing A y y + 1) / (2 * pairing A z y)) * pairing A z y + (((pairing A x y) ^ 2 / pairing A x x - pairing A y y + 1) / (2 * pairing A z y)) * pairing A z y = (pairing A x y) ^ 2 / pairing A x x + 1 := by field_simp ring simp only [mul_zero, add_zero] at hn rw [he] at hn have he' : pairing A x x * ((pairing A x y) ^ 2 / pairing A x x + 1) = (pairing A x y) ^ 2 + pairing A x x := by field_simp rw [he'] at hn linarith lemma onePositiveSquare_of_nonpos_on_kernel {A : Matrix ι ι ℝ} (hA : A.IsSymm) (l : ι → ℝ) (h : ∀ x : ι → ℝ, l ⬝ᵥ x = 0 → pairing A x x ≤ 0) : OnePositiveSquare A := by refine ⟨hA, ?_⟩ intro x y hx hxy have hlx : l ⬝ᵥ x ≠ 0 := by intro he exact (not_le_of_gt hx) (h x he) have hyx : pairing A y x = 0 := (pairing_symm hA y x).trans hxy have hz : l ⬝ᵥ (y - ((l ⬝ᵥ y) / (l ⬝ᵥ x)) • x) = 0 := by simp only [dotProduct_sub, dotProduct_smul, smul_eq_mul] field_simp ring have hh := h _ hz rw [pairing_sub_smul, hxy, hyx] at hh nlinarith [sq_nonneg ((l ⬝ᵥ y) / (l ⬝ᵥ x))] lemma onePositiveSquare_rankOne_sub_psd (l : ι → ℝ) {U : Matrix ι ι ℝ} (hU : U.PosSemidef) : OnePositiveSquare (Matrix.vecMulVec l l - U) := by apply onePositiveSquare_of_nonpos_on_kernel (l := l) · apply Matrix.isHermitian_iff_isSymm.mp exact (Matrix.posSemidef_vecMulVec_self_star l |>.isHermitian).sub hU.isHermitian · intro x hx have hnon := hU.dotProduct_mulVec_nonneg x simp only [star_trivial] at hnon simp [pairing, Matrix.sub_mulVec, Matrix.vecMulVec_mulVec, hx, hnon] lemma pairing_const (a : ℝ) (x y : ι → ℝ) : pairing (Matrix.of (fun _ _ => a)) x y = a * (∑ i, x i) * (∑ j, y j) := by simp only [pairing, Matrix.mulVec, dotProduct, Matrix.of_apply] simp_rw [← Finset.mul_sum, ← Finset.sum_mul] ring lemma pairing_scale (A : Matrix ι ι ℝ) (l x y : ι → ℝ) : pairing (Matrix.of (fun i j => l i * A i j * l j)) x y = pairing A (l * x) (l * y) := by simp only [pairing, Matrix.mulVec, dotProduct, Matrix.of_apply, Pi.mul_apply, Finset.mul_sum] apply Finset.sum_congr rfl intro i _ apply Finset.sum_congr rfl intro j _ ring lemma onePositiveSquare_scale {A : Matrix ι ι ℝ} (hA : OnePositiveSquare A) (l : ι → ℝ) : OnePositiveSquare (Matrix.of (fun i j => l i * A i j * l j)) := by refine ⟨?_, ?_⟩ · ext i j simp only [Matrix.transpose_apply, Matrix.of_apply] rw [hA.1.apply] ring · intro x y hx hxy rw [pairing_scale] at hx hxy ⊢ exact hA.2 _ _ hx hxy lemma onePositiveSquare_const_sub_psd (a : ℝ) {U : Matrix ι ι ℝ} (hU : U.PosSemidef) : OnePositiveSquare (Matrix.of (fun _ _ => a) - U) := by apply onePositiveSquare_of_nonpos_on_kernel (l := fun _ => 1) · ext i j simp only [Matrix.transpose_apply, Matrix.sub_apply, Matrix.of_apply] have hij : U j i = U i j := by simpa using hU.isHermitian.apply i j rw [hij] · intro x hx have hsum : ∑ i, x i = 0 := by simpa [dotProduct] using hx have hn := hU.dotProduct_mulVec_nonneg x simp only [star_trivial] at hn change pairing (Matrix.of (fun _ _ => a) - U) x x ≤ 0 change x ⬝ᵥ (Matrix.of (fun _ _ => a) - U) *ᵥ x ≤ 0 rw [Matrix.sub_mulVec, dotProduct_sub] change pairing (Matrix.of (fun _ _ => a)) x x - pairing U x x ≤ 0 rw [pairing_const, hsum] simpa only [pairing, mul_zero, zero_sub, neg_nonpos] using hn lemma normalized_complement_posSemidef {H : Matrix ι ι ℝ} (hH : OnePositiveSquare H) (z : ι) {a : ℝ} (ha : 0 < a) (hrow : ∀ j, H z j = a) : (Matrix.of (fun _ _ => a) - H).PosSemidef := by classical have hpair (v : ι → ℝ) : pairing H (Pi.single z 1) v = a * ∑ i, v i := by unfold pairing rw [single_dotProduct, one_mul] change (∑ j, H z j * v j) = a * ∑ i, v i simp only [hrow, Finset.mul_sum] have he : pairing H (Pi.single z 1) (Pi.single z 1) = a := by simp [hpair] apply Matrix.PosSemidef.of_dotProduct_mulVec_nonneg · apply Matrix.isHermitian_iff_isSymm.mpr apply Matrix.ext intro i j simp only [Matrix.transpose_apply, Matrix.sub_apply, Matrix.of_apply] rw [hH.1.apply] · intro v have hz : pairing H (Pi.single z 1) (v - (∑ i, v i) • Pi.single z 1) = 0 := by rw [hpair] simp only [Pi.sub_apply, Pi.smul_apply, smul_eq_mul, Finset.sum_sub_distrib, ← Finset.mul_sum] simp have hyn := hH.2 (Pi.single z 1) _ (he ▸ ha) hz rw [pairing_sub_smul, he, hpair, pairing_symm hH.1 v (Pi.single z 1), hpair] at hyn have heval : star v ⬝ᵥ (Matrix.of (fun _ _ => a) - H) *ᵥ v = a * (∑ i, v i) ^ 2 - pairing H v v := by simp only [star_trivial, Matrix.sub_mulVec, dotProduct_sub] change pairing (Matrix.of (fun _ _ => a)) v v - pairing H v v = _ rw [pairing_const] ring rw [heval] nlinarith omit [Fintype ι] in lemma posSemidef_entry_sq_le {U : Matrix ι ι ℝ} (hU : U.PosSemidef) (i j : ι) : (U i j) ^ 2 ≤ U i i * U j j := by have h := (hU.submatrix ![i, j]).det_nonneg simp only [Matrix.det_fin_two, Matrix.submatrix_apply, Matrix.cons_val_zero, Matrix.cons_val_one, Matrix.cons_val_fin_one] at h have hij : U j i = U i j := by simpa using hU.isHermitian.apply i j rw [hij] at h nlinarith omit [Fintype ι] in lemma posSemidef_abs_entry_lt {U : Matrix ι ι ℝ} (hU : U.PosSemidef) {a : ℝ} (ha : 0 < a) (hdiag : ∀ i, U i i < a) (i j : ι) : |U i j| < a := by have hp : U i i * U j j < a * a := calc U i i * U j j ≤ U i i * a := mul_le_mul_of_nonneg_left (hdiag j).le hU.diag_nonneg _ < a * a := mul_lt_mul_of_pos_right (hdiag i) ha have hs := posSemidef_entry_sq_le hU i j nlinarith [sq_abs (U i j), abs_nonneg (U i j)] lemma posSemidef_entry_pow {A : Matrix ι ι ℝ} (hA : A.PosSemidef) (n : ℕ) : (Matrix.of (fun i j => A i j ^ n)).PosSemidef := by induction n with | zero => simpa [Matrix.vecMulVec] using (Matrix.posSemidef_vecMulVec_self_star (fun _ : ι => (1 : ℝ))) | succ n ih => have he : Matrix.of (fun i j => A i j ^ (n + 1)) = Matrix.of (fun i j => A i j ^ n) ⊙ A := by ext i j simp [pow_succ] rw [he] exact ih.hadamard hA lemma posSemidef_of_entrywise_tendsto {A : ℕ → Matrix ι ι ℝ} {B : Matrix ι ι ℝ} (hA : ∀ n, (A n).PosSemidef) (hlim : ∀ i j, Tendsto (fun n => A n i j) atTop (𝓝 (B i j))) : B.PosSemidef := by apply Matrix.PosSemidef.of_dotProduct_mulVec_nonneg · apply Matrix.isHermitian_iff_isSymm.mpr ext i j exact tendsto_nhds_unique (hlim j i) ((hlim i j).congr fun n => by simpa using (hA n).isHermitian.apply j i) · intro x have hlimq : Tendsto (fun n => star x ⬝ᵥ A n *ᵥ x) atTop (𝓝 (star x ⬝ᵥ B *ᵥ x)) := by unfold dotProduct Matrix.mulVec exact tendsto_finsetSum _ (fun i _ => tendsto_const_nhds.mul (tendsto_finsetSum _ fun j _ => (hlim i j).mul tendsto_const_nhds)) exact ge_of_tendsto hlimq (Filter.Eventually.of_forall fun n => (hA n).dotProduct_mulVec_nonneg x) lemma posSemidef_entry_resolvent {A : Matrix ι ι ℝ} (hA : A.PosSemidef) (hsmall : ∀ i j, |A i j| < 1) : (Matrix.of (fun i j => A i j / (1 - A i j))).PosSemidef := by apply posSemidef_of_entrywise_tendsto (A := fun n => ∑ k ∈ Finset.range n, Matrix.of (fun i j => A i j ^ (k + 1))) · intro n exact Matrix.posSemidef_sum _ (fun k _ => posSemidef_entry_pow hA (k + 1)) · intro i j have hs := (hasSum_geometric_of_norm_lt_one (show ‖A i j‖ < 1 by simpa only [Real.norm_eq_abs] using hsmall i j)).mul_left (A i j) simpa only [Matrix.sum_apply, Matrix.of_apply, div_eq_mul_inv, pow_succ, mul_comm] using hs.tendsto_sum_nat lemma onePositiveSquare_normalized_update {H : Matrix ι ι ℝ} (hH : OnePositiveSquare H) (hpos : ∀ i j, 0 < H i j) (z : ι) {a : ℝ} (hrow : ∀ j, H z j = a) {c : ℝ} (hc : 0 ≤ c) : OnePositiveSquare (Matrix.of (fun i j => H i j / (1 + c * H i j))) := by rcases hc.eq_or_lt with hc | hc · subst c have he : Matrix.of (fun i j => H i j / (1 + 0 * H i j)) = H := by ext i j simp rwa [he] have ha : 0 < a := hrow z ▸ hpos z z let U := Matrix.of (fun _ _ => a) - H have hU : U.PosSemidef := normalized_complement_posSemidef hH z ha hrow have hdiag : ∀ i, U i i < a := by intro i change a - H i i < a exact sub_lt_self _ (hpos i i) have hUa : ∀ i j, |U i j| < a := posSemidef_abs_entry_lt hU ha hdiag have hd : 0 < 1 + c * a := by positivity let r := c / (1 + c * a) have hr : 0 < r := div_pos hc hd let V := r • U have hV : V.PosSemidef := hU.smul hr.le have hra : r * a < 1 := by dsimp [r] rw [div_mul_eq_mul_div, div_lt_iff₀ hd] linarith have hsmall : ∀ i j, |V i j| < 1 := by intro i j change |r * U i j| < 1 rw [abs_mul, abs_of_pos hr] exact (mul_lt_mul_of_pos_left (hUa i j) hr).trans hra have hW := (posSemidef_entry_resolvent hV hsmall).smul (show 0 ≤ (1 / (c * (1 + c * a)) : ℝ) by positivity) have hP := onePositiveSquare_const_sub_psd (a / (1 + c * a)) hW have he : Matrix.of (fun i j => H i j / (1 + c * H i j)) = Matrix.of (fun _ _ => a / (1 + c * a)) - (1 / (c * (1 + c * a))) • Matrix.of (fun i j => V i j / (1 - V i j)) := by ext i j have hij := hpos i j have hden : 0 < 1 + c * H i j := by positivity have hone : 1 - V i j ≠ 0 := ne_of_gt (sub_pos.mpr (abs_lt.mp (hsmall i j)).2) simp only [Matrix.of_apply, Matrix.sub_apply, Matrix.smul_apply, smul_eq_mul] dsimp [V, U, r] at hone ⊢ field_simp [ne_of_gt hden, ne_of_gt hd, ne_of_gt hc] ring_nf field_simp [show 1 + H i j * c ≠ 0 by nlinarith [hden]] ring rw [he] exact hP theorem reciprocal_rankOne_update {M : Matrix ι ι ℝ} (hM : M.IsSymm) (hpos : ∀ i j, 0 < M i j) (hrec : OnePositiveSquare (Matrix.of (fun i j => (M i j)⁻¹))) (z : ι) {c : ℝ} (hc : 0 ≤ c) : OnePositiveSquare (Matrix.of (fun i j => (M i j + c * M i z * M z j)⁻¹)) := by let H := Matrix.of (fun i j => M i z * (M i j)⁻¹ * M j z) have hH : OnePositiveSquare H := onePositiveSquare_scale hrec (fun i => M i z) have hHp : ∀ i j, 0 < H i j := by intro i j change 0 < M i z * (M i j)⁻¹ * M j z exact mul_pos (mul_pos (hpos i z) (inv_pos.mpr (hpos i j))) (hpos j z) have hrow : ∀ j, H z j = M z z := by intro j change M z z * (M z j)⁻¹ * M j z = M z z rw [hM.apply j z] field_simp [ne_of_gt (hpos j z)] have hnew := onePositiveSquare_normalized_update hH hHp z hrow hc have hscaled := onePositiveSquare_scale hnew (fun i => (M i z)⁻¹) have he : Matrix.of (fun i j => (M i j + c * M i z * M z j)⁻¹) = Matrix.of (fun i j => (M i z)⁻¹ * (H i j / (1 + c * H i j)) * (M j z)⁻¹) := by ext i j have hij := hpos i j have hiz := hpos i z have hjz := hpos j z have hn : 0 < M i j + c * M i z * M j z := by positivity simp only [Matrix.of_apply] dsimp [H] rw [hM.apply z j] have hzj := hpos z j have hn' : 0 < M i j + c * M i z * M z j := by positivity field_simp [ne_of_gt hzj, ne_of_gt hn'] rw [he] exact hscaled def bordered (B : Matrix ι ι ℝ) (l : ι → ℝ) : Matrix (ι ⊕ Unit) (ι ⊕ Unit) ℝ := Matrix.fromBlocks B (Matrix.of (fun i _ => l i)) (Matrix.of (fun _ j => l j)) 0 lemma pairing_bordered (B : Matrix ι ι ℝ) (l x y : ι → ℝ) (a b : ℝ) : pairing (bordered B l) (Sum.elim x (fun _ => a)) (Sum.elim y (fun _ => b)) = pairing B x y + b * (l ⬝ᵥ x) + a * (l ⬝ᵥ y) := by simp only [pairing, bordered, Matrix.mulVec, dotProduct, Fintype.sum_sum_type, Matrix.fromBlocks_apply₁₁, Matrix.fromBlocks_apply₁₂, Matrix.fromBlocks_apply₂₁, Matrix.fromBlocks_apply₂₂, Matrix.of_apply, Matrix.zero_apply, Sum.elim_inl, Sum.elim_inr, Fintype.sum_unique, zero_mul, mul_add, Finset.sum_add_distrib] ring_nf congr 1 rw [Finset.mul_sum] apply Finset.sum_congr rfl intro i _ ring lemma bordered_nonpos_on_kernel {B : Matrix ι ι ℝ} {l : ι → ℝ} (hB : OnePositiveSquare (bordered B l)) (hl : l ≠ 0) : ∀ x, l ⬝ᵥ x = 0 → pairing B x x ≤ 0 := by classical have hz : pairing (bordered B l) (Sum.elim 0 (fun _ => 1)) (Sum.elim 0 (fun _ => 1)) = 0 := by rw [pairing_bordered] simp [pairing] have hrad : ∃ y, pairing (bordered B l) (Sum.elim 0 (fun _ => 1)) y ≠ 0 := by obtain ⟨i, hi⟩ := Function.ne_iff.mp hl refine ⟨Sum.elim (Pi.single i 1) (fun _ => 0), ?_⟩ rw [pairing_bordered] simpa [pairing] using hi have hn := onePositiveSquare_isotropic_hyperplane hB _ hz hrad intro x hx have horth : pairing (bordered B l) (Sum.elim 0 (fun _ => 1)) (Sum.elim x (fun _ => 0)) = 0 := by rw [pairing_bordered] simp [pairing, hx] have hh := hn _ horth simpa [pairing_bordered] using hh def ConditionallyNegative (D : Matrix ι ι ℝ) : Prop := D.IsSymm ∧ ∀ x : ι → ℝ, (∑ i, x i) = 0 → pairing D x x ≤ 0 theorem conditionalNegative_scaled_of_bordered {B : Matrix ι ι ℝ} (k : ι → ℝ) (hk : ∀ i, k i ≠ 0) (hB : OnePositiveSquare (bordered B (fun i => (k i)⁻¹))) : ConditionallyNegative (Matrix.of (fun i j => k i * B i j * k j)) := by have hsym : B.IsSymm := by ext i j exact hB.1.apply (Sum.inl i) (Sum.inl j) refine ⟨?_, ?_⟩ · ext i j simp only [Matrix.transpose_apply, Matrix.of_apply] rw [hsym.apply] ring · intro x hx rw [pairing_scale] by_cases hi : IsEmpty ι · simp [pairing, dotProduct] · have : Nonempty ι := not_isEmpty_iff.mp hi have hl : (fun i => (k i)⁻¹) ≠ 0 := by intro he obtain ⟨i⟩ := ‹Nonempty ι› have := congrFun he i exact (inv_ne_zero (hk i)) this apply bordered_nonpos_on_kernel hB hl simpa [dotProduct, hk] using hx end ReciprocalKernel namespace ReciprocalKernel open Matrix open scoped BigOperators variable {ι : Type*} [Fintype ι] [DecidableEq ι] def eliminate (M : Matrix ι ι ℝ) (w : ι → ℝ) (z : ι) : Matrix ι ι ℝ := Matrix.of fun i j => M i j + (w z / (1 - w z * M z z)) * M i z * M z j omit [Fintype ι] [DecidableEq ι] in lemma eliminate_denominator_pos {M : Matrix ι ι ℝ} {w f : ι → ℝ} (hM : ∀ i j, 0 < M i j) (hw : ∀ i, 0 ≤ w i) (hf : ∀ i, 0 < f i) {S : Finset ι} {z : ι} (hz : z ∈ S) (hb : ∀ i, ∑ j ∈ S, M i j * w j * f j < f i) : 0 < 1 - w z * M z z := by have hs : M z z * w z * f z ≤ ∑ j ∈ S, M z j * w j * f j := Finset.single_le_sum (f := fun j => M z j * w j * f j) (fun i hi => mul_nonneg (mul_nonneg (hM z i).le (hw i)) (hf i).le) hz have hh := lt_of_le_of_lt hs (hb z) nlinarith [hf z] omit [Fintype ι] [DecidableEq ι] in lemma eliminate_symm {M : Matrix ι ι ℝ} (hM : M.IsSymm) (w : ι → ℝ) (z : ι) : (eliminate M w z).IsSymm := by ext i j simp only [eliminate, Matrix.transpose_apply, Matrix.of_apply] rw [hM.apply j i, hM.apply j z, hM.apply z i] ring omit [Fintype ι] [DecidableEq ι] in lemma eliminate_pos {M : Matrix ι ι ℝ} {w : ι → ℝ} {z : ι} (hM : ∀ i j, 0 < M i j) (hw : 0 ≤ w z) (hz : 0 < 1 - w z * M z z) : ∀ i j, 0 < eliminate M w z i j := by intro i j dsimp [eliminate] exact add_pos_of_pos_of_nonneg (hM i j) (mul_nonneg (mul_nonneg (div_nonneg hw hz.le) (hM i z).le) (hM z j).le) omit [Fintype ι] in lemma eliminate_bound {M : Matrix ι ι ℝ} {w f : ι → ℝ} (hM : ∀ i j, 0 < M i j) (hw : ∀ i, 0 ≤ w i) (hf : ∀ i, 0 < f i) {S : Finset ι} {z : ι} (hz : z ∉ S) (hb : ∀ i, ∑ j ∈ insert z S, M i j * w j * f j < f i) : ∀ i, ∑ j ∈ S, eliminate M w z i j * w j * f j < f i := by have hd := eliminate_denominator_pos hM hw hf (Finset.mem_insert_self z S) hb have hc : 0 ≤ w z / (1 - w z * M z z) := div_nonneg (hw z) hd.le have hb' : ∀ i, M i z * w z * f z + ∑ j ∈ S, M i j * w j * f j < f i := by simpa only [Finset.sum_insert hz] using hb intro i have he : (∑ j ∈ S, eliminate M w z i j * w j * f j) = (∑ j ∈ S, M i j * w j * f j) + (w z / (1 - w z * M z z)) * M i z * (∑ j ∈ S, M z j * w j * f j) := by simp only [eliminate, Matrix.of_apply, add_mul, Finset.sum_add_distrib, Finset.mul_sum] congr 1 apply Finset.sum_congr rfl intro j hj ring rw [he] have hz' : ∑ j ∈ S, M z j * w j * f j ≤ (1 - w z * M z z) * f z := by linarith [hb' z] have hmul := mul_le_mul_of_nonneg_left hz' (mul_nonneg hc (hM i z).le) have he' : w z / (1 - w z * M z z) * M i z * ((1 - w z * M z z) * f z) = M i z * w z * f z := by field_simp rw [he'] at hmul linarith [hb' i] omit [DecidableEq ι] in lemma eliminate_reciprocal {M : Matrix ι ι ℝ} {w : ι → ℝ} {z : ι} (hM : M.IsSymm) (hp : ∀ i j, 0 < M i j) (hP : OnePositiveSquare (Matrix.of fun i j => (M i j)⁻¹)) (hw : 0 ≤ w z) (hd : 0 < 1 - w z * M z z) : OnePositiveSquare (Matrix.of fun i j => (eliminate M w z i j)⁻¹) := reciprocal_rankOne_update hM hp hP z (div_nonneg hw hd.le) theorem finite_resolvent_of_superharmonic {M : Matrix ι ι ℝ} {w f : ι → ℝ} (hM : M.IsSymm) (hp : ∀ i j, 0 < M i j) (hP : OnePositiveSquare (Matrix.of fun i j => (M i j)⁻¹)) (hw : ∀ i, 0 ≤ w i) (hf : ∀ i, 0 < f i) (S : Finset ι) (hb : ∀ i, ∑ j ∈ S, M i j * w j * f j < f i) : ∃ R : Matrix ι ι ℝ, R.IsSymm ∧ (∀ i j, 0 < R i j) ∧ OnePositiveSquare (Matrix.of fun i j => (R i j)⁻¹) ∧ ∀ i j, R i j = M i j + ∑ k ∈ S, M i k * w k * R k j := by induction S using Finset.induction_on generalizing M with | empty => exact ⟨M, hM, hp, hP, by simp⟩ | @insert z S hz ih => have hd := eliminate_denominator_pos hp hw hf (Finset.mem_insert_self z S) hb obtain ⟨R, hRs, hRp, hRP, hReq⟩ := ih (eliminate_symm hM w z) (eliminate_pos hp (hw z) hd) (eliminate_reciprocal hM hp hP (hw z) hd) (eliminate_bound hp hw hf hz hb) refine ⟨R, hRs, hRp, hRP, ?_⟩ have heq (i j) : R i j = M i j + (∑ k ∈ S, M i k * w k * R k j) + (w z / (1 - w z * M z z)) * M i z * (M z j + ∑ k ∈ S, M z k * w k * R k j) := by rw [hReq i j] simp only [eliminate, Matrix.of_apply, add_mul, Finset.sum_add_distrib, Finset.mul_sum, mul_add] ring_nf have hez (j) : (1 - w z * M z z) * R z j = M z j + ∑ k ∈ S, M z k * w k * R k j := by have hh := heq z j field_simp at hh nlinarith intro i j rw [Finset.sum_insert hz, heq i j, ← hez j] field_simp ring open scoped Matrix.Norms.L2Operator lemma matrix_geometric_nonneg {T : Matrix ι ι ℝ} (hT : ∀ i j, 0 ≤ T i j) (hn : ‖T‖ < 1) : ∀ i j, 0 ≤ (∑' n : ℕ, T ^ n) i j := by intro i j have hs := summable_geometric_of_norm_lt_one hn have ht := ((continuous_id.matrix_elem i j).tendsto (∑' n : ℕ, T ^ n)).comp hs.hasSum apply ge_of_tendsto ht filter_upwards [] with S change 0 ≤ (∑ n ∈ S, T ^ n) i j rw [Matrix.sum_apply] exact Finset.sum_nonneg fun n _ => Matrix.pow_apply_nonneg hT n i j lemma exists_superharmonic_of_weighted_norm_lt_one {M : Matrix ι ι ℝ} {w : ι → ℝ} (hM : ∀ i j, 0 ≤ M i j) (hw : ∀ i, 0 ≤ w i) (hn : ‖Matrix.diagonal (fun i => Real.sqrt (w i)) * M * Matrix.diagonal (fun i => Real.sqrt (w i))‖ < 1) : ∃ f : ι → ℝ, (∀ i, 0 < f i) ∧ ∀ i, ∑ j, M i j * w j * f j < f i := by let D : Matrix ι ι ℝ := Matrix.diagonal (fun i => Real.sqrt (w i)) let N : Matrix ι ι ℝ := M * D let T : Matrix ι ι ℝ := D * N let B : Matrix ι ι ℝ := ∑' n : ℕ, T ^ n let Q : Matrix ι ι ℝ := N * B * D have hnorm : ‖T‖ < 1 := by simpa only [T, N, Matrix.mul_assoc] using hn have hT : ∀ i j, 0 ≤ T i j := by intro i j simp only [T, N, D, Matrix.diagonal_mul, Matrix.mul_diagonal] exact mul_nonneg (Real.sqrt_nonneg _) (mul_nonneg (hM i j) (Real.sqrt_nonneg _)) have hB := matrix_geometric_nonneg hT hnorm have hQ : ∀ i j, 0 ≤ Q i j := by intro i j simp only [Q, D, Matrix.mul_diagonal] rw [Matrix.mul_apply] simp only [N, D, Matrix.mul_diagonal] apply mul_nonneg _ (Real.sqrt_nonneg _) apply Finset.sum_nonneg intro k hk exact mul_nonneg (mul_nonneg (hM i k) (Real.sqrt_nonneg _)) (hB k j) have hB' : (1 : Matrix ι ι ℝ) + T * B = B := by have hh := mul_neg_geom_series T hnorm change (1 - T) * B = 1 at hh rw [sub_mul, one_mul] at hh exact (sub_eq_iff_eq_add.mp hh).symm have hD : D * D = Matrix.diagonal w := by dsimp [D] rw [Matrix.diagonal_mul_diagonal] congr 1 funext i exact Real.mul_self_sqrt (hw i) have hA : (M * Matrix.diagonal w) * (1 + Q) = Q := by rw [← hD] change (M * (D * D)) * (1 + N * B * D) = N * B * D calc _ = N * (1 + T * B) * D := by dsimp [N, T]; noncomm_ring _ = _ := by rw [hB'] let f : ι → ℝ := (1 + Q) *ᵥ (fun _ => 1) have hf (i) : f i = 1 + ∑ j, Q i j := by simp [f, Matrix.add_mulVec, Matrix.mulVec, dotProduct] refine ⟨f, ?_, ?_⟩ · intro i rw [hf] have : 0 ≤ ∑ j, Q i j := Finset.sum_nonneg fun j _ => hQ i j linarith · intro i have he := congrArg (fun A : Matrix ι ι ℝ => (A *ᵥ (fun _ => 1)) i) hA change (((M * Matrix.diagonal w) * (1 + Q)) *ᵥ (fun _ : ι => (1 : ℝ))) i = _ at he rw [← Matrix.mulVec_mulVec] at he change ((M * Matrix.diagonal w) *ᵥ f) i = _ at he simp only [Matrix.mulVec, dotProduct, Matrix.mul_diagonal, mul_one] at he rw [he, hf] linarith theorem finite_resolvent {M : Matrix ι ι ℝ} {w : ι → ℝ} (hM : M.IsSymm) (hp : ∀ i j, 0 < M i j) (hP : OnePositiveSquare (Matrix.of fun i j => (M i j)⁻¹)) (hw : ∀ i, 0 ≤ w i) (hn : ‖Matrix.diagonal (fun i => Real.sqrt (w i)) * M * Matrix.diagonal (fun i => Real.sqrt (w i))‖ < 1) : ∃ R : Matrix ι ι ℝ, R.IsSymm ∧ (∀ i j, 0 < R i j) ∧ OnePositiveSquare (Matrix.of fun i j => (R i j)⁻¹) ∧ R = M + M * Matrix.diagonal w * R := by obtain ⟨f, hf, hb⟩ := exists_superharmonic_of_weighted_norm_lt_one (fun i j => (hp i j).le) hw hn obtain ⟨R, hRs, hRp, hRP, hReq⟩ := finite_resolvent_of_superharmonic hM hp hP hw hf Finset.univ (by simpa using hb) refine ⟨R, hRs, hRp, hRP, ?_⟩ ext i j change R i j = M i j + ∑ k, (M * Matrix.diagonal w) i k * R k j simp only [Matrix.mul_diagonal] exact hReq i j def finiteResolventKernel (M : Matrix ι ι ℝ) (w : ι → ℝ) : Matrix ι ι ℝ := let D := Matrix.diagonal (fun i => Real.sqrt (w i)) M + M * D * (1 - D * M * D)⁻¹ * D * M lemma resolvent_eq_finiteResolventKernel {M R : Matrix ι ι ℝ} {w : ι → ℝ} (hw : ∀ i, 0 ≤ w i) (hn : ‖Matrix.diagonal (fun i => Real.sqrt (w i)) * M * Matrix.diagonal (fun i => Real.sqrt (w i))‖ < 1) (hR : R = M + M * Matrix.diagonal w * R) : R = finiteResolventKernel M w := by let D : Matrix ι ι ℝ := Matrix.diagonal (fun i => Real.sqrt (w i)) let N : Matrix ι ι ℝ := M * D let T : Matrix ι ι ℝ := D * N let B : Matrix ι ι ℝ := (1 - T)⁻¹ let U : Matrix ι ι ℝ := 1 + N * B * D have hnorm : ‖T‖ < 1 := by simpa only [T, N, Matrix.mul_assoc] using hn have hu : IsUnit (1 - T) := isUnit_one_sub_of_norm_lt_one hnorm have hBinv : (1 - T) * B = 1 := Matrix.mul_nonsing_inv _ ((Matrix.isUnit_iff_isUnit_det _).mp hu) have hD : D * D = Matrix.diagonal w := by dsimp [D] rw [Matrix.diagonal_mul_diagonal] congr 1 funext i exact Real.mul_self_sqrt (hw i) have hA : M * Matrix.diagonal w = N * D := by rw [← hD]; exact (Matrix.mul_assoc ..).symm have hKU : (1 - N * D) * U = 1 := by calc _ = 1 + N * ((1 - T) * B - 1) * D := by dsimp [U, T]; noncomm_ring _ = 1 := by rw [hBinv, sub_self, mul_zero, zero_mul, add_zero] have hUK : U * (1 - N * D) = 1 := mul_eq_one_comm.mp hKU have hKR : (1 - N * D) * R = M := by rw [sub_mul, one_mul] rw [hA] at hR exact sub_eq_iff_eq_add.mpr hR calc R = (U * (1 - N * D)) * R := by rw [hUK, one_mul] _ = U * M := by rw [Matrix.mul_assoc, hKR] _ = finiteResolventKernel M w := by simp only [U, N, B, T, finiteResolventKernel, D] noncomm_ring theorem finite_quadrature_reciprocal {M : Matrix ι ι ℝ} {w : ι → ℝ} (hM : M.IsSymm) (hp : ∀ i j, 0 < M i j) (hP : OnePositiveSquare (Matrix.of fun i j => (M i j)⁻¹)) (hw : ∀ i, 0 ≤ w i) (hn : ‖Matrix.diagonal (fun i => Real.sqrt (w i)) * M * Matrix.diagonal (fun i => Real.sqrt (w i))‖ < 1) : (∀ i j, 0 < finiteResolventKernel M w i j) ∧ OnePositiveSquare (Matrix.of fun i j => (finiteResolventKernel M w i j)⁻¹) := by obtain ⟨R, _, hpos, hrec, hReq⟩ := finite_resolvent hM hp hP hw hn rw [resolvent_eq_finiteResolventKernel hw hn hReq] at hpos hrec exact ⟨hpos, hrec⟩ end ReciprocalKernel open Filter open scoped Topology namespace StrictHotSpots.BanachResolvent variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] def resolvent (A : E →L[ℝ] E) : E →L[ℝ] E := ∑' n : ℕ, A ^ n lemma resolvent_left (A : E →L[ℝ] E) (hA : ‖A‖ < 1) : (1 - A) * resolvent A = 1 := mul_neg_geom_series A hA lemma resolvent_right (A : E →L[ℝ] E) (hA : ‖A‖ < 1) : resolvent A * (1 - A) = 1 := geom_series_mul_neg A hA omit [CompleteSpace E] in lemma resolvent_norm_le (A : E →L[ℝ] E) {d : ℝ} (hA : ‖A‖ ≤ d) (hd : d < 1) : ‖resolvent A‖ ≤ (1 - d)⁻¹ := by have h := tsum_geometric_le_of_norm_lt_one A (hA.trans_lt hd) have h1 : ‖(1 : E →L[ℝ] E)‖ ≤ 1 := ContinuousLinearMap.norm_id_le have hi : (1 - ‖A‖)⁻¹ ≤ (1 - d)⁻¹ := by apply inv_anti₀ (sub_pos.mpr hd) linarith dsimp only [resolvent] linarith lemma resolvent_identity (A B : E →L[ℝ] E) (hA : ‖A‖ < 1) (hB : ‖B‖ < 1) : resolvent B - resolvent A = resolvent B * (B - A) * resolvent A := by have ht : B - A = (1 - A) - (1 - B) := by abel rw [ht, mul_sub, sub_mul, mul_assoc, resolvent_left A hA, resolvent_right B hB, mul_one, one_mul] lemma resolvent_strong_tendsto (A : E →L[ℝ] E) (Q : ℕ → E →L[ℝ] E) {d : ℝ} (hd : d < 1) (hA : ‖A‖ < 1) (hQ : ∀ᶠ n in atTop, ‖Q n‖ ≤ d) (hs : ∀ v, Tendsto (fun n => Q n v) atTop (𝓝 (A v))) (v : E) : Tendsto (fun n => resolvent (Q n) v) atTop (𝓝 (resolvent A v)) := by rw [← tendsto_sub_nhds_zero_iff] have hz : Tendsto (fun n => (1 - d)⁻¹ * ‖(Q n - A) (resolvent A v)‖) atTop (𝓝 0) := by have ht := ((hs (resolvent A v)).sub (tendsto_const_nhds (x := A (resolvent A v)))).norm.const_mul (1 - d)⁻¹ simpa using ht apply squeeze_zero_norm' _ hz filter_upwards [hQ] with n hn have he := congrArg (fun T : E →L[ℝ] E => T v) (resolvent_identity A (Q n) hA (hn.trans_lt hd)) simp only [sub_apply, mul_apply_eq_comp] at he rw [he] exact (ContinuousLinearMap.le_opNorm _ _).trans (mul_le_mul_of_nonneg_right (resolvent_norm_le (Q n) hn hd) (norm_nonneg _)) lemma resolvent_solution (A : E →L[ℝ] E) (hA : ‖A‖ < 1) {u f : E} (hu : u - A u = f) : u = resolvent A f := by have h := congrArg (fun T : E →L[ℝ] E => T u) (resolvent_right A hA) simpa only [mul_apply_eq_comp, sub_apply, one_apply_eq_self, hu] using h.symm lemma resolvent_nonneg [PartialOrder E] [IsOrderedAddMonoid E] [OrderClosedTopology E] (A : E →L[ℝ] E) (hA : ‖A‖ < 1) (hpos : ∀ v, 0 ≤ v → 0 ≤ A v) {f : E} (hf : 0 ≤ f) : 0 ≤ resolvent A f := by have ht : ∀ n : ℕ, 0 ≤ (A^n) f := by intro n induction n with | zero => simpa using hf | succ n ih => simpa only [pow_succ', mul_apply_eq_comp] using hpos _ ih have hs0 : HasSum (fun n : ℕ => A^n) (resolvent A) := (summable_geometric_of_norm_lt_one hA).hasSum have hs := (ContinuousLinearMap.apply ℝ E f).hasSum hs0 change HasSum (fun n : ℕ => (A^n) f) (resolvent A f) at hs rw [← hs.tsum_eq] exact tsum_nonneg ht lemma resolvent_ge_self [PartialOrder E] [IsOrderedAddMonoid E] [OrderClosedTopology E] (A : E →L[ℝ] E) (hA : ‖A‖ < 1) (hpos : ∀ v, 0 ≤ v → 0 ≤ A v) {f : E} (hf : 0 ≤ f) : f ≤ resolvent A f := by have h := congrArg (fun T : E →L[ℝ] E => T f) (resolvent_left A hA) simp only [mul_apply_eq_comp, sub_apply, one_apply_eq_self] at h have hn := hpos _ (resolvent_nonneg A hA hpos hf) calc f = resolvent A f - A (resolvent A f) := h.symm _ ≤ resolvent A f := sub_le_self _ hn end StrictHotSpots.BanachResolvent open MeasureTheory Filter Set TopologicalSpace open scoped Topology ENNReal namespace StrictHotSpots.CompactKernel variable {X : Type*} [MetricSpace X] [CompactSpace X] [MeasurableSpace X] [BorelSpace X] (ν : Measure X) [IsFiniteMeasure ν] lemma kernel_integrable (C : C(X × X, ℝ)) (f : C(X, ℝ)) : Integrable (fun z => f z • C.curry z) ν := (f.continuous.smul C.curry.continuous).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _) def kernelOp (C : C(X × X, ℝ)) : C(X, ℝ) →L[ℝ] C(X, ℝ) := LinearMap.mkContinuous { toFun := fun f => ∫ z, f z • C.curry z ∂ν map_add' := by intro f g simp only [ContinuousMap.add_apply, add_smul] exact integral_add (kernel_integrable ν C f) (kernel_integrable ν C g) map_smul' := by intro r f simp only [ContinuousMap.smul_apply, smul_eq_mul, mul_smul] exact integral_smul r _ } (ν.real univ * ‖C‖) (by intro f calc ‖∫ z, f z • C.curry z ∂ν‖ ≤ ∫ z, ‖f z • C.curry z‖ ∂ν := norm_integral_le_integral_norm _ _ ≤ ∫ _ : X, ‖f‖ * ‖C‖ ∂ν := by apply integral_mono (kernel_integrable ν C f).norm (integrable_const _) intro z dsimp only rw [norm_smul] apply mul_le_mul (f.norm_coe_le_norm z) _ (norm_nonneg _) (norm_nonneg _) exact (ContinuousMap.norm_le _ (norm_nonneg C)).mpr fun x => C.norm_coe_le_norm (z,x) _ = ν.real univ * ‖C‖ * ‖f‖ := by simp [integral_const]; ring) lemma kernelOp_apply (C : C(X × X, ℝ)) (f : C(X, ℝ)) (x : X) : kernelOp ν C f x = ∫ z, C (z,x) * f z ∂ν := by change (∫ z, f z • C.curry z ∂ν) x = _ rw [ContinuousMap.integral_apply (kernel_integrable ν C f)] simp only [ContinuousMap.smul_apply, smul_eq_mul, ContinuousMap.curry_apply, mul_comm] lemma kernelOp_nonneg (C : C(X × X, ℝ)) (hC : ∀ z x, 0 ≤ C (z,x)) {f : C(X, ℝ)} (hf : 0 ≤ f) : 0 ≤ kernelOp ν C f := by intro x change 0 ≤ kernelOp ν C f x rw [kernelOp_apply] exact integral_nonneg fun z => mul_nonneg (hC z x) (hf z) lemma kernelOp_norm_le (C : C(X × X, ℝ)) (hC : ∀ z x, 0 ≤ C (z,x)) {d : ℝ} (hd : 0 ≤ d) (hrow : ∀ x, ∫ z, C (z,x) ∂ν ≤ d) : ‖kernelOp ν C‖ ≤ d := by apply ContinuousLinearMap.opNorm_le_bound _ hd intro f apply (ContinuousMap.norm_le _ (mul_nonneg hd (norm_nonneg f))).mpr intro x rw [kernelOp_apply] calc ‖∫ z, C (z,x) * f z ∂ν‖ ≤ ∫ z, ‖C (z,x) * f z‖ ∂ν := norm_integral_le_integral_norm _ _ ≤ ∫ z, C (z,x) * ‖f‖ ∂ν := by apply integral_mono · exact ((C.continuous.comp (continuous_id.prodMk continuous_const)).mul f.continuous).norm.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _) · exact ((C.continuous.comp (continuous_id.prodMk continuous_const)).mul continuous_const).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _) · intro z dsimp only rw [norm_mul, Real.norm_of_nonneg (hC z x)] exact mul_le_mul_of_nonneg_left (f.norm_coe_le_norm z) (hC z x) _ = (∫ z, C (z,x) ∂ν) * ‖f‖ := integral_mul_const _ _ _ ≤ d * ‖f‖ := mul_le_mul_of_nonneg_right (hrow x) (norm_nonneg f) omit [MetricSpace X] [CompactSpace X] [BorelSpace X] in lemma integral_comp_simpleFunc {Y : Type*} {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] (π : SimpleFunc X Y) (F : Y → E) : ∫ x, F (π x) ∂ν = ∑ y ∈ π.range, ν.real (π ⁻¹' {y}) • F y := by classical have he : (fun x => F (π x)) = fun x => ∑ y ∈ π.range, (π ⁻¹' {y}).indicator (fun _ => F y) x := by funext x simp [Set.indicator_apply, eq_comm] rw [he, integral_finsetSum] · apply Finset.sum_congr rfl intro y hy exact integral_indicator_const _ (π.measurableSet_fiber y) · intro y hy exact (integrable_const (F y)).indicator (π.measurableSet_fiber y) def sampleOp (π : SimpleFunc X X) (C : C(X × X, ℝ)) : C(X, ℝ) →L[ℝ] C(X, ℝ) := ∑ z ∈ π.range, ν.real (π ⁻¹' {z}) • (ContinuousMap.evalCLM ℝ z).smulRight (C.curry z) omit [BorelSpace X] in lemma sampleOp_eq_integral (π : SimpleFunc X X) (C : C(X × X, ℝ)) (f : C(X, ℝ)) : sampleOp ν π C f = ∫ z, f (π z) • C.curry (π z) ∂ν := by rw [integral_comp_simpleFunc ν π (fun z => f z • C.curry z)] simp [sampleOp, sum_apply] omit [BorelSpace X] [IsFiniteMeasure ν] in lemma sampleOp_apply (π : SimpleFunc X X) (C : C(X × X, ℝ)) (f : C(X, ℝ)) (x : X) : sampleOp ν π C f x = ∑ z ∈ π.range, ν.real (π ⁻¹' {z}) * C (z,x) * f z := by simp only [sampleOp, sum_apply, smul_apply, ContinuousMap.sum_apply, ContinuousMap.smul_apply, ContinuousLinearMap.smulRight_apply, ContinuousMap.evalCLM_apply, ContinuousMap.curry_apply, smul_eq_mul] apply Finset.sum_congr rfl intro z hz ring omit [BorelSpace X] [IsFiniteMeasure ν] in lemma sampleOp_nonneg (π : SimpleFunc X X) (C : C(X × X, ℝ)) (hC : ∀ z x, 0 ≤ C (z,x)) {f : C(X, ℝ)} (hf : 0 ≤ f) : 0 ≤ sampleOp ν π C f := by intro x change 0 ≤ sampleOp ν π C f x rw [sampleOp_apply] exact Finset.sum_nonneg fun z _ => mul_nonneg (mul_nonneg (measureReal_nonneg) (hC z x)) (hf z) omit [BorelSpace X] [IsFiniteMeasure ν] in lemma sampleOp_norm_le (π : SimpleFunc X X) (C : C(X × X, ℝ)) (hC : ∀ z x, 0 ≤ C (z,x)) {d : ℝ} (hd : 0 ≤ d) (hrow : ∀ x, sampleOp ν π C 1 x ≤ d) : ‖sampleOp ν π C‖ ≤ d := by apply ContinuousLinearMap.opNorm_le_bound _ hd intro f apply (ContinuousMap.norm_le _ (mul_nonneg hd (norm_nonneg f))).mpr intro x rw [sampleOp_apply] calc ‖∑ z ∈ π.range, ν.real (π ⁻¹' {z}) * C (z,x) * f z‖ ≤ ∑ z ∈ π.range, ‖ν.real (π ⁻¹' {z}) * C (z,x) * f z‖ := norm_sum_le _ _ _ ≤ ∑ z ∈ π.range, ν.real (π ⁻¹' {z}) * C (z,x) * ‖f‖ := by apply Finset.sum_le_sum intro z hz rw [norm_mul, Real.norm_of_nonneg (mul_nonneg measureReal_nonneg (hC z x))] exact mul_le_mul_of_nonneg_left (f.norm_coe_le_norm z) (mul_nonneg measureReal_nonneg (hC z x)) _ = sampleOp ν π C 1 x * ‖f‖ := by simp [sampleOp_apply, Finset.sum_mul] _ ≤ d * ‖f‖ := mul_le_mul_of_nonneg_right (hrow x) (norm_nonneg f) variable [Nonempty X] def quantize (n : ℕ) : SimpleFunc X X := SimpleFunc.nearestPt (denseSeq X) n lemma quantize_tendsto (x : X) : Tendsto (fun n => quantize (X := X) n x) atTop (𝓝 x) := SimpleFunc.tendsto_nearestPt (by rw [(denseRange_denseSeq X).closure_range]; trivial) lemma integral_quantize_tendsto {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] (F : C(X, E)) : Tendsto (fun n => ∫ z, F (quantize n z) ∂ν) atTop (𝓝 (∫ z, F z ∂ν)) := by apply tendsto_integral_of_dominated_convergence (fun _ => ‖F‖) · intro n exact ((quantize n).map F).stronglyMeasurable.aestronglyMeasurable · exact integrable_const _ · intro n exact Filter.Eventually.of_forall fun z => F.norm_coe_le_norm (quantize n z) · exact Filter.Eventually.of_forall fun z => F.continuous.continuousAt.tendsto.comp (quantize_tendsto z) lemma sampleOp_strong_tendsto (C : C(X × X, ℝ)) (f : C(X, ℝ)) : Tendsto (fun n => sampleOp ν (quantize n) C f) atTop (𝓝 (kernelOp ν C f)) := by simp only [sampleOp_eq_integral] exact integral_quantize_tendsto ν ⟨_, f.continuous.smul C.curry.continuous⟩ lemma sampleOp_eventually_norm_le (C : C(X × X, ℝ)) (hC : ∀ z x, 0 ≤ C (z,x)) {d d' : ℝ} (hd : 0 ≤ d) (hdd' : d < d') (hrow : ∀ x, ∫ z, C (z,x) ∂ν ≤ d) : ∀ᶠ n in atTop, ‖sampleOp ν (quantize n) C‖ ≤ d' := by obtain ⟨N,hN⟩ := Metric.tendsto_atTop.mp (sampleOp_strong_tendsto ν C 1) (d' - d) (sub_pos.mpr hdd') filter_upwards [eventually_ge_atTop N] with n hn apply sampleOp_norm_le ν _ C hC (hd.trans hdd'.le) intro x have hh := (sampleOp ν (quantize n) C 1 - kernelOp ν C 1).norm_coe_le_norm x have hk : kernelOp ν C 1 x ≤ d := by simpa [kernelOp_apply] using hrow x have hc : sampleOp ν (quantize n) C 1 x - kernelOp ν C 1 x ≤ ‖sampleOp ν (quantize n) C 1 - kernelOp ν C 1‖ := (le_abs_self _).trans hh have he := hN n hn rw [dist_eq_norm] at he linarith end StrictHotSpots.CompactKernel namespace ReciprocalKernel open Matrix lemma onePositiveSquare_submatrix_sum_inl {ι κ : Type*} [Fintype ι] [Fintype κ] {A : Matrix (ι ⊕ κ) (ι ⊕ κ) ℝ} (hA : OnePositiveSquare A) : OnePositiveSquare (A.submatrix Sum.inl Sum.inl) := by have he (u v : ι → ℝ) : pairing (A.submatrix Sum.inl Sum.inl) u v = pairing A (Sum.elim u (fun _ => 0)) (Sum.elim v (fun _ => 0)) := by simp [pairing, dotProduct, Matrix.mulVec, Fintype.sum_sum_type] refine ⟨?_, ?_⟩ · ext i j exact hA.1.apply _ _ · intro u v hu huv rw [he] at hu huv ⊢ exact hA.2 _ _ hu huv end ReciprocalKernel namespace StrictHotSpots.CompactKernel open MeasureTheory Filter Set TopologicalSpace BanachResolvent ReciprocalKernel open scoped Topology ENNReal variable {X : Type} [MetricSpace X] [CompactSpace X] [MeasurableSpace X] [BorelSpace X] (ν : Measure X) [IsFiniteMeasure ν] def resolvedKernel (C : C(X × X, ℝ)) (A : C(X, ℝ) →L[ℝ] C(X, ℝ)) (x y : X) : ℝ := resolvent A (C.curry y) x omit [BorelSpace X] [IsFiniteMeasure ν] in lemma sampled_reciprocal (π : SimpleFunc X X) (C : C(X × X, ℝ)) (hC : ∀ z x, 0 < C (z,x)) (hsym : ∀ x y, C (x,y) = C (y,x)) (hP : ∀ (ι : Type) [Fintype ι] (s : ι → X), OnePositiveSquare (Matrix.of fun i j => (C (s i,s j))⁻¹)) {d : ℝ} (hd : 0 ≤ d) (hd1 : d < 1) (hrow : ∀ x, sampleOp ν π C 1 x ≤ d) {ι : Type} [Fintype ι] (s : ι → X) : (∀ i j, 0 < resolvedKernel C (sampleOp ν π C) (s i) (s j)) ∧ OnePositiveSquare (Matrix.of fun i j => (resolvedKernel C (sampleOp ν π C) (s i) (s j))⁻¹) := by classical let t : ι ⊕ π.range → X := Sum.elim s Subtype.val let w : ι ⊕ π.range → ℝ := Sum.elim (fun _ => 0) (fun z => ν.real (π ⁻¹' {z.val})) let M : Matrix (ι ⊕ π.range) (ι ⊕ π.range) ℝ := Matrix.of fun i j => C (t j,t i) have hM : M.IsSymm := by ext i j; exact hsym _ _ have hp : ∀ i j, 0 < M i j := fun i j => hC _ _ have hrec : OnePositiveSquare (Matrix.of fun i j => (M i j)⁻¹) := by have he : (Matrix.of fun i j => (M i j)⁻¹) = (Matrix.of fun i j => (C (t i,t j))⁻¹) := by ext i j; simp [M, hsym] rw [he] exact hP _ t have hw : ∀ i, 0 ≤ w i := by intro i; cases i <;> simp [w, measureReal_nonneg] have hsum (i : ι ⊕ π.range) (f : C(X, ℝ)) : ∑ j, M i j * w j * f (t j) = sampleOp ν π C f (t i) := by simp only [Fintype.sum_sum_type, M, w, t, Sum.elim_inl, Sum.elim_inr, Matrix.of_apply, mul_zero, zero_mul, Finset.sum_const_zero, zero_add, sampleOp_apply] rw [Finset.sum_subtype π.range (fun _ => Iff.rfl)] apply Finset.sum_congr rfl intro z hz ring have hb : ∀ i, ∑ j ∈ Finset.univ, M i j * w j * (1 : ℝ) < 1 := by intro i have he := hsum i 1 simp only [ContinuousMap.one_apply] at he exact (he ▸ hrow (t i)).trans_lt hd1 obtain ⟨R,hRs,hRp,hRrec,hReq⟩ := finite_resolvent_of_superharmonic hM hp hrec hw (fun _ => one_pos) Finset.univ hb have hn : ‖sampleOp ν π C‖ < 1 := (sampleOp_norm_le ν π C (fun z x => (hC z x).le) hd hrow).trans_lt hd1 have hmatch (j : ι ⊕ π.range) : ∀ i, R i j = resolvedKernel C (sampleOp ν π C) (t i) (t j) := by let u : C(X, ℝ) := C.curry (t j) + ∑ z : π.range, (ν.real (π ⁻¹' {z.val}) * R (Sum.inr z) j) • C.curry z.val have hu (i : ι ⊕ π.range) : u (t i) = R i j := by rw [hReq i j] simp only [u, ContinuousMap.add_apply, ContinuousMap.sum_apply, ContinuousMap.smul_apply, ContinuousMap.curry_apply, smul_eq_mul, Fintype.sum_sum_type, M, w, t, Sum.elim_inl, Sum.elim_inr, Matrix.of_apply, mul_zero, zero_mul, Finset.sum_const_zero, zero_add] congr 1 apply Finset.sum_congr rfl intro z hz ring have he : u - sampleOp ν π C u = C.curry (t j) := by ext x simp only [ContinuousMap.sub_apply, sampleOp_apply] rw [Finset.sum_subtype π.range (fun _ => Iff.rfl)] have hz (z : π.range) : u z.val = R (Sum.inr z) j := hu (Sum.inr z) simp only [hz, u, ContinuousMap.add_apply, ContinuousMap.sum_apply, ContinuousMap.smul_apply, ContinuousMap.curry_apply, smul_eq_mul] have hsum' : (∑ z : π.range, (ν.real (π ⁻¹' {↑z}) * R (Sum.inr z) j) * C (↑z,x)) = ∑ z : π.range, ν.real (π ⁻¹' {↑z}) * C (↑z,x) * R (Sum.inr z) j := by apply Finset.sum_congr rfl intro z hz ring rw [hsum'] abel intro i rw [← hu i, resolvent_solution _ hn he] rfl refine ⟨fun i j => ?_, ?_⟩ · simpa only [t, Sum.elim_inl, hmatch] using hRp (Sum.inl i) (Sum.inl j) · have hsub := onePositiveSquare_submatrix_sum_inl hRrec convert hsub using 1 ext i j simp only [Matrix.submatrix_apply, Matrix.of_apply, hmatch, t, Sum.elim_inl] omit [CompactSpace X] [MeasurableSpace X] [BorelSpace X] in lemma continuousMap_orderClosed : OrderClosedTopology C(X, ℝ) := by constructor have he : {p : C(X, ℝ) × C(X, ℝ) | p.1 ≤ p.2} = ⋂ x : X, {p : C(X, ℝ) × C(X, ℝ) | p.1 x ≤ p.2 x} := by ext p simp only [Set.mem_ofPred_eq, Set.mem_iInter] rfl rw [he] exact isClosed_iInter fun x => isClosed_le ((ContinuousMap.evalCLM ℝ x).continuous.comp continuous_fst) ((ContinuousMap.evalCLM ℝ x).continuous.comp continuous_snd) omit [MeasurableSpace X] [BorelSpace X] in lemma continuous_resolvent_nonneg (A : C(X, ℝ) →L[ℝ] C(X, ℝ)) (hA : ‖A‖ < 1) (hpos : ∀ f : C(X, ℝ), 0 ≤ f → 0 ≤ A f) {f : C(X, ℝ)} (hf : 0 ≤ f) : 0 ≤ resolvent A f := by let : OrderClosedTopology C(X, ℝ) := continuousMap_orderClosed exact resolvent_nonneg A hA hpos hf omit [MeasurableSpace X] [BorelSpace X] in lemma resolvedKernel_ge (C : C(X × X, ℝ)) {A : C(X, ℝ) →L[ℝ] C(X, ℝ)} (hA : ‖A‖ < 1) (hpos : ∀ f : C(X, ℝ), 0 ≤ f → 0 ≤ A f) (hC : ∀ z x, 0 ≤ C (z,x)) (x y : X) : C (y,x) ≤ resolvedKernel C A x y := by have hp : 0 ≤ C.curry y := fun x => hC y x have hn := hpos _ (continuous_resolvent_nonneg A hA hpos hp) have he := congrArg (fun T : C(X, ℝ) →L[ℝ] C(X, ℝ) => T (C.curry y)) (resolvent_left A hA) simp only [mul_apply_eq_comp, sub_apply, one_apply_eq_self] at he have hex := congrArg (fun f : C(X, ℝ) => f x) he change resolvedKernel C A x y - A (resolvent A (C.curry y)) x = C (y,x) at hex have hnx := hn x change 0 ≤ A (resolvent A (C.curry y)) x at hnx linarith lemma resolvedKernel_tendsto [Nonempty X] (C : C(X × X, ℝ)) (hC : ∀ z x, 0 ≤ C (z,x)) {d : ℝ} (hd : 0 ≤ d) (hd1 : d < 1) (hrow : ∀ x, ∫ z, C (z,x) ∂ν ≤ d) (x y : X) : Tendsto (fun n => resolvedKernel C (sampleOp ν (quantize n) C) x y) atTop (𝓝 (resolvedKernel C (kernelOp ν C) x y)) := by have hn := kernelOp_norm_le ν C hC hd hrow have hh := BanachResolvent.resolvent_strong_tendsto (kernelOp ν C) (fun n => sampleOp ν (quantize n) C) (d := (1+d)/2) (by linarith) (hn.trans_lt hd1) (sampleOp_eventually_norm_le ν C hC hd (by linarith) hrow) (sampleOp_strong_tendsto ν C) (C.curry y) exact ((ContinuousMap.evalCLM ℝ x).continuous.tendsto _).comp hh theorem compact_reciprocal_of_row_bound [Nonempty X] (C : C(X × X, ℝ)) (hC : ∀ z x, 0 < C (z,x)) (hsym : ∀ x y, C (x,y) = C (y,x)) (hP : ∀ (ι : Type) [Fintype ι] (s : ι → X), OnePositiveSquare (Matrix.of fun i j => (C (s i,s j))⁻¹)) {d : ℝ} (hd : 0 ≤ d) (hd1 : d < 1) (hrow : ∀ x, ∫ z, C (z,x) ∂ν ≤ d) {ι : Type} [Fintype ι] (s : ι → X) : (∀ i j, 0 < resolvedKernel C (kernelOp ν C) (s i) (s j)) ∧ OnePositiveSquare (Matrix.of fun i j => (resolvedKernel C (kernelOp ν C) (s i) (s j))⁻¹) := by have hn := kernelOp_norm_le ν C (fun z x => (hC z x).le) hd hrow have hp (x y : X) : 0 < resolvedKernel C (kernelOp ν C) x y := (hC y x).trans_le (resolvedKernel_ge C (hn.trans_lt hd1) (fun f hf => kernelOp_nonneg ν C (fun z x => (hC z x).le) hf) (fun z x => (hC z x).le) x y) refine ⟨fun i j => hp _ _, ?_⟩ have he := sampleOp_eventually_norm_le ν C (fun z x => (hC z x).le) hd (show d < (1+d)/2 by linarith) hrow have hr : ∀ᶠ n in atTop, ∀ x, sampleOp ν (quantize n) C 1 x ≤ (1+d)/2 := by filter_upwards [he] with n hn x calc sampleOp ν (quantize n) C 1 x ≤ ‖sampleOp ν (quantize n) C 1 x‖ := le_abs_self _ _ ≤ ‖sampleOp ν (quantize n) C 1‖ := ContinuousMap.norm_coe_le_norm _ _ _ ≤ ‖sampleOp ν (quantize n) C‖ * ‖(1 : C(X, ℝ))‖ := ContinuousLinearMap.le_opNorm _ _ _ ≤ (1+d)/2 := by simpa using hn obtain ⟨N,hN⟩ := eventually_atTop.mp hr apply onePositiveSquare_of_entrywise_tendsto (A := fun n => Matrix.of fun i j => (resolvedKernel C (sampleOp ν (quantize (n+N)) C) (s i) (s j))⁻¹) · intro n exact (sampled_reciprocal ν (quantize (n+N)) C hC hsym hP (show 0 ≤ (1+d)/2 by linarith) (by linarith) (hN _ (Nat.le_add_left _ _)) s).2 · intro i j exact ((resolvedKernel_tendsto ν C (fun z x => (hC z x).le) hd hd1 hrow (s i) (s j)).inv₀ (hp _ _).ne').comp (tendsto_add_atTop_nat N) end StrictHotSpots.CompactKernel namespace StrictHotSpots.CompactKernel open MeasureTheory Filter Set TopologicalSpace BanachResolvent open scoped Topology ENNReal variable {X : Type*} [MetricSpace X] [CompactSpace X] [MeasurableSpace X] [BorelSpace X] (ν : Measure X) [IsFiniteMeasure ν] lemma integral_norm_L2_le (f : Lp ℝ 2 ν) : ∫ z, ‖f z‖ ∂ν ≤ ‖f‖ * ‖ContinuousMap.toLp 2 ν ℝ (1 : C(X, ℝ))‖ := by have he : ∫ z, ‖f z‖ ∂ν = inner ℝ |f| (ContinuousMap.toLp 2 ν ℝ (1 : C(X, ℝ))) := by rw [L2.inner_def] apply integral_congr_ae filter_upwards [Lp.coeFn_abs f, ContinuousMap.coeFn_toLp (p := 2) (𝕜 := ℝ) ν (1 : C(X, ℝ))] with z hz h1 change ‖f z‖ = (ContinuousMap.toLp 2 ν ℝ (1 : C(X, ℝ))) z * (|f|) z rw [hz, h1] simp [Real.norm_eq_abs] rw [he] simpa only [norm_abs_eq_norm] using real_inner_le_norm |f| (ContinuousMap.toLp 2 ν ℝ (1 : C(X, ℝ))) lemma l2kernel_integrable (C : C(X × X, ℝ)) (f : Lp ℝ 2 ν) : Integrable (fun z => f z • C.curry z) ν := by apply ((Lp.memLp f).integrable (by norm_num)).norm.mul_const ‖C‖ |>.mono' ((Lp.aestronglyMeasurable f).smul C.curry.continuous.aestronglyMeasurable) exact Eventually.of_forall fun z => by change ‖f z • C.curry z‖ ≤ _ rw [norm_smul] apply mul_le_mul_of_nonneg_left _ (norm_nonneg _) exact (ContinuousMap.norm_le _ (norm_nonneg C)).mpr fun x => C.norm_coe_le_norm (z,x) def l2SectionOp (C : C(X × X, ℝ)) : Lp ℝ 2 ν →L[ℝ] C(X, ℝ) := LinearMap.mkContinuous { toFun := fun f => ∫ z, f z • C.curry z ∂ν map_add' := by intro f g change (∫ z, (f+g) z • C.curry z ∂ν) = _ calc _ = ∫ z, f z • C.curry z + g z • C.curry z ∂ν := by apply integral_congr_ae filter_upwards [Lp.coeFn_add f g] with z hz rw [hz, Pi.add_apply, add_smul] _ = _ := integral_add (l2kernel_integrable ν C f) (l2kernel_integrable ν C g) map_smul' := by intro r f calc _ = ∫ z, r • (f z • C.curry z) ∂ν := by apply integral_congr_ae filter_upwards [Lp.coeFn_smul r f] with z hz simp [hz, mul_smul] _ = _ := integral_smul r _ } (‖C‖ * ‖ContinuousMap.toLp 2 ν ℝ (1 : C(X, ℝ))‖) (by intro f calc ‖∫ z, f z • C.curry z ∂ν‖ ≤ ∫ z, ‖f z • C.curry z‖ ∂ν := norm_integral_le_integral_norm _ _ ≤ ∫ z, ‖f z‖ * ‖C‖ ∂ν := by apply integral_mono (l2kernel_integrable ν C f).norm (((Lp.memLp f).integrable (by norm_num)).norm.mul_const _) intro z dsimp only rw [norm_smul] apply mul_le_mul_of_nonneg_left _ (norm_nonneg _) exact (ContinuousMap.norm_le _ (norm_nonneg C)).mpr fun x => C.norm_coe_le_norm (z,x) _ = (∫ z, ‖f z‖ ∂ν) * ‖C‖ := integral_mul_const _ _ _ ≤ (‖f‖ * ‖ContinuousMap.toLp 2 ν ℝ (1 : C(X, ℝ))‖) * ‖C‖ := mul_le_mul_of_nonneg_right (integral_norm_L2_le ν f) (norm_nonneg _) _ = _ := by ring) lemma l2SectionOp_apply (C : C(X × X, ℝ)) (f : Lp ℝ 2 ν) (x : X) : l2SectionOp ν C f x = ∫ z, C (z,x) * f z ∂ν := by change (∫ z, f z • C.curry z ∂ν) x = _ rw [ContinuousMap.integral_apply (l2kernel_integrable ν C f)] simp only [ContinuousMap.smul_apply, smul_eq_mul, ContinuousMap.curry_apply, mul_comm] lemma l2SectionOp_toLp (C : C(X × X, ℝ)) (f : C(X, ℝ)) : l2SectionOp ν C (ContinuousMap.toLp 2 ν ℝ f) = kernelOp ν C f := by ext x rw [kernelOp_apply, l2SectionOp_apply] apply integral_congr_ae filter_upwards [ContinuousMap.coeFn_toLp (p := 2) (𝕜 := ℝ) ν f] with z hz rw [hz] lemma l2SectionOp_nonneg (C : C(X × X, ℝ)) (hC : ∀ z x, 0 ≤ C (z,x)) {f : Lp ℝ 2 ν} (hf : 0 ≤ f) : 0 ≤ l2SectionOp ν C f := by intro x change 0 ≤ l2SectionOp ν C f x rw [l2SectionOp_apply] apply integral_nonneg_of_ae filter_upwards [(Lp.coeFn_nonneg f).mpr hf] with z hz exact mul_nonneg (hC z x) hz def l2Op (C : C(X × X, ℝ)) : Lp ℝ 2 ν →L[ℝ] Lp ℝ 2 ν := (ContinuousMap.toLp 2 ν ℝ).comp (l2SectionOp ν C) lemma l2Op_nonneg (C : C(X × X, ℝ)) (hC : ∀ z x, 0 ≤ C (z,x)) {f : Lp ℝ 2 ν} (hf : 0 ≤ f) : 0 ≤ l2Op ν C f := by apply (Lp.coeFn_nonneg _).mp have he := ContinuousMap.coeFn_toLp (p := 2) (𝕜 := ℝ) ν (l2SectionOp ν C f) filter_upwards [he] with x hx exact (l2SectionOp_nonneg ν C hC hf x).trans_eq hx.symm theorem exists_superharmonic (C : C(X × X, ℝ)) (hC : ∀ z x, 0 ≤ C (z,x)) (hn : ‖l2Op ν C‖ < 1) : ∃ f : C(X, ℝ), (∀ x, 1 ≤ f x) ∧ kernelOp ν C f = f - 1 := by let a := ContinuousMap.toLp 2 ν ℝ (1 : C(X, ℝ)) let g := resolvent (l2Op ν C) a let f : C(X, ℝ) := 1 + l2SectionOp ν C g have ha : 0 ≤ a := by apply (Lp.coeFn_nonneg _).mp filter_upwards [ContinuousMap.coeFn_toLp (p := 2) (𝕜 := ℝ) ν (1 : C(X, ℝ))] with x hx change 0 ≤ (ContinuousMap.toLp 2 ν ℝ (1 : C(X, ℝ))) x rw [hx] exact zero_le_one have hg : 0 ≤ g := resolvent_nonneg _ hn (fun v hv => l2Op_nonneg ν C hC hv) ha have hf : ContinuousMap.toLp 2 ν ℝ f = g := by have he := congrArg (fun T : Lp ℝ 2 ν →L[ℝ] Lp ℝ 2 ν => T a) (resolvent_left (l2Op ν C) hn) simp only [mul_apply_eq_comp, sub_apply, one_apply_eq_self] at he change g - l2Op ν C g = a at he dsimp only [f] rw [map_add] change a + l2Op ν C g = g exact (sub_eq_iff_eq_add.mp he).symm refine ⟨f, ?_, ?_⟩ · intro x have hp := l2SectionOp_nonneg ν C hC hg x change 0 ≤ l2SectionOp ν C g x at hp change 1 ≤ 1 + l2SectionOp ν C g x linarith · rw [← l2SectionOp_toLp, hf] dsimp only [f] abel end StrictHotSpots.CompactKernel namespace StrictHotSpots.CompactKernel open MeasureTheory Filter Set TopologicalSpace BanachResolvent open scoped Topology ENNReal variable {X : Type} [MetricSpace X] [CompactSpace X] [MeasurableSpace X] [BorelSpace X] (ν : Measure X) [IsFiniteMeasure ν] def invMap (f : C(X, ℝ)) (hf : ∀ x, 0 < f x) : C(X, ℝ) := ⟨fun x => (f x)⁻¹, f.continuous.inv₀ (fun x => (hf x).ne')⟩ def scaledKernel (C : C(X × X, ℝ)) (f : C(X, ℝ)) (hf : ∀ x, 0 < f x) : C(X × X, ℝ) := ⟨fun p => C p * f p.1 / f p.2, (C.continuous.mul (f.continuous.comp continuous_fst)).div (f.continuous.comp continuous_snd) (fun p => (hf p.2).ne')⟩ omit [CompactSpace X] [MeasurableSpace X] [BorelSpace X] in lemma scaledKernel_pos (C : C(X × X, ℝ)) (hC : ∀ z x, 0 < C (z,x)) (f : C(X, ℝ)) (hf : ∀ x, 0 < f x) (z x : X) : 0 < scaledKernel C f hf (z,x) := div_pos (mul_pos (hC z x) (hf z)) (hf x) lemma kernelOp_scaled (C : C(X × X, ℝ)) (f : C(X, ℝ)) (hf : ∀ x, 0 < f x) (u : C(X, ℝ)) : kernelOp ν (scaledKernel C f hf) u = invMap f hf * kernelOp ν C (f*u) := by ext x simp only [kernelOp_apply, ContinuousMap.mul_apply] change (∫ z, (C (z,x) * f z / f x) * u z ∂ν) = (f x)⁻¹ * ∫ z, C (z,x) * (f z * u z) ∂ν rw [← integral_const_mul] apply integral_congr_ae exact Eventually.of_forall fun z => by ring omit [BorelSpace X] [IsFiniteMeasure ν] in lemma sampleOp_scaled (π : SimpleFunc X X) (C : C(X × X, ℝ)) (f : C(X, ℝ)) (hf : ∀ x, 0 < f x) (u : C(X, ℝ)) : sampleOp ν π (scaledKernel C f hf) u = invMap f hf * sampleOp ν π C (f*u) := by ext x simp only [sampleOp_apply, ContinuousMap.mul_apply] change (∑ z ∈ π.range, ν.real (π ⁻¹' {z}) * (C (z,x) * f z / f x) * u z) = (f x)⁻¹ * ∑ z ∈ π.range, ν.real (π ⁻¹' {z}) * C (z,x) * (f z * u z) rw [Finset.mul_sum] apply Finset.sum_congr rfl intro z hz ring omit [CompactSpace X] [MeasurableSpace X] [BorelSpace X] in lemma mul_invMap (f : C(X, ℝ)) (hf : ∀ x, 0 < f x) : f * invMap f hf = 1 := by ext x exact mul_inv_cancel₀ (hf x).ne' omit [CompactSpace X] [MeasurableSpace X] [BorelSpace X] in lemma invMap_mul_cancel (f : C(X, ℝ)) (hf : ∀ x, 0 < f x) (u : C(X, ℝ)) : f * (invMap f hf * u) = u := by rw [← mul_assoc, mul_invMap, one_mul] def rescaledKernel (C : C(X × X, ℝ)) (f : C(X, ℝ)) (hf : ∀ x, 0 < f x) (A : C(X, ℝ) →L[ℝ] C(X, ℝ)) (x y : X) : ℝ := f x * resolvent A (invMap f hf * C.curry y) x lemma scaled_row_bound [Nonempty X] (C : C(X × X, ℝ)) (f : C(X, ℝ)) (hf : ∀ x, 1 ≤ f x) (he : kernelOp ν C f = f - 1) : 0 ≤ 1 - ‖f‖⁻¹ ∧ 1 - ‖f‖⁻¹ < 1 ∧ ∀ x, ∫ z, scaledKernel C f (fun x => (hf x).trans_lt' zero_lt_one) (z,x) ∂ν ≤ 1 - ‖f‖⁻¹ := by have hfn : 1 ≤ ‖f‖ := by obtain ⟨x⟩ := ‹Nonempty X› exact (hf x).trans ((le_abs_self _).trans (f.norm_coe_le_norm x)) have hfnpos : 0 < ‖f‖ := lt_of_lt_of_le zero_lt_one hfn refine ⟨sub_nonneg.mpr (inv_le_one_of_one_le₀ hfn), by linarith [inv_pos.mpr hfnpos], ?_⟩ intro x have hx : 0 < f x := lt_of_lt_of_le zero_lt_one (hf x) have hfx : f x ≤ ‖f‖ := (le_abs_self _).trans (f.norm_coe_le_norm x) have hxf := inv_anti₀ hx hfx have hr := congrArg (fun u : C(X, ℝ) => u x) (kernelOp_scaled ν C f (fun x => lt_of_lt_of_le zero_lt_one (hf x)) 1) simp only [mul_one, he, ContinuousMap.mul_apply, ContinuousMap.sub_apply, ContinuousMap.one_apply, kernelOp_apply] at hr change (∫ z, scaledKernel C f _ (z,x) ∂ν) = (f x)⁻¹ * (f x - 1) at hr rw [hr, mul_sub, inv_mul_cancel₀ hx.ne', mul_one] linarith end StrictHotSpots.CompactKernel namespace StrictHotSpots.CompactKernel open MeasureTheory Filter Set TopologicalSpace BanachResolvent ReciprocalKernel open scoped Topology ENNReal variable {X : Type} [MetricSpace X] [CompactSpace X] [MeasurableSpace X] [BorelSpace X] (ν : Measure X) [IsFiniteMeasure ν] def l2ResolvedKernel (C : C(X × X, ℝ)) (x y : X) : ℝ := C (y,x) + l2SectionOp ν C (resolvent (l2Op ν C) (ContinuousMap.toLp 2 ν ℝ (C.curry y))) x omit [MeasurableSpace X] [BorelSpace X] in lemma scaled_solution (f : C(X, ℝ)) (hf : ∀ x, 0 < f x) (A B : C(X, ℝ) →L[ℝ] C(X, ℝ)) (hB : ‖B‖ < 1) (hscale : ∀ u, B u = invMap f hf * A (f * u)) {u v : C(X, ℝ)} (heq : u - A u = v) : u = f * resolvent B (invMap f hf * v) := by have h : invMap f hf * u - B (invMap f hf * u) = invMap f hf * v := by rw [hscale, invMap_mul_cancel, ← mul_sub, heq] have hr := congrArg (fun z : C(X, ℝ) => f * z) (resolvent_solution B hB h) rwa [invMap_mul_cancel] at hr lemma rescaledKernel_eq_l2ResolvedKernel (C : C(X × X, ℝ)) (f : C(X, ℝ)) (hf : ∀ x, 0 < f x) (hL : ‖l2Op ν C‖ < 1) (hC : ‖kernelOp ν (scaledKernel C f hf)‖ < 1) (x y : X) : rescaledKernel C f hf (kernelOp ν (scaledKernel C f hf)) x y = l2ResolvedKernel ν C x y := by let g := resolvent (l2Op ν C) (ContinuousMap.toLp 2 ν ℝ (C.curry y)) let u : C(X, ℝ) := C.curry y + l2SectionOp ν C g have hg : ContinuousMap.toLp 2 ν ℝ u = g := by have he := congrArg (fun T : Lp ℝ 2 ν →L[ℝ] Lp ℝ 2 ν => T (ContinuousMap.toLp 2 ν ℝ (C.curry y))) (resolvent_left (l2Op ν C) hL) simp only [mul_apply_eq_comp, sub_apply, one_apply_eq_self] at he change g - l2Op ν C g = ContinuousMap.toLp 2 ν ℝ (C.curry y) at he change ContinuousMap.toLp 2 ν ℝ (C.curry y + l2SectionOp ν C g) = g rw [map_add] exact (sub_eq_iff_eq_add.mp he).symm have hu : u - kernelOp ν C u = C.curry y := by rw [← l2SectionOp_toLp, hg] dsimp only [u] abel have hu' := scaled_solution f hf (kernelOp ν C) (kernelOp ν (scaledKernel C f hf)) hC (kernelOp_scaled ν C f hf) hu have h := congrArg (fun v : C(X, ℝ) => v x) hu' exact h.symm lemma l2ResolvedKernel_ge (C : C(X × X, ℝ)) (hC : ∀ z x, 0 ≤ C (z,x)) (hn : ‖l2Op ν C‖ < 1) (x y : X) : C (y,x) ≤ l2ResolvedKernel ν C x y := by have hy : 0 ≤ ContinuousMap.toLp 2 ν ℝ (C.curry y) := by apply (Lp.coeFn_nonneg _).mp filter_upwards [ContinuousMap.coeFn_toLp (p := 2) (𝕜 := ℝ) ν (C.curry y)] with z hz rw [hz] exact hC y z have hp := l2SectionOp_nonneg ν C hC (resolvent_nonneg (l2Op ν C) hn (fun v hv => l2Op_nonneg ν C hC hv) hy) x exact le_add_of_nonneg_right hp omit [BorelSpace X] [IsFiniteMeasure ν] in lemma sampled_rescaled_reciprocal (π : SimpleFunc X X) (C : C(X × X, ℝ)) (hC : ∀ z x, 0 < C (z,x)) (hsym : ∀ x y, C (x,y) = C (y,x)) (hP : ∀ (ι : Type) [Fintype ι] (s : ι → X), OnePositiveSquare (Matrix.of fun i j => (C (s i,s j))⁻¹)) (f : C(X, ℝ)) (hf : ∀ x, 0 < f x) {d : ℝ} (hd : 0 ≤ d) (hd1 : d < 1) (hrow : ∀ x, sampleOp ν π (scaledKernel C f hf) 1 x ≤ d) {ι : Type} [Fintype ι] (s : ι → X) : (∀ i j, 0 < rescaledKernel C f hf (sampleOp ν π (scaledKernel C f hf)) (s i) (s j)) ∧ OnePositiveSquare (Matrix.of fun i j => (rescaledKernel C f hf (sampleOp ν π (scaledKernel C f hf)) (s i) (s j))⁻¹) := by classical let t : ι ⊕ π.range → X := Sum.elim s Subtype.val let w : ι ⊕ π.range → ℝ := Sum.elim (fun _ => 0) (fun z => ν.real (π ⁻¹' {z.val})) let M : Matrix (ι ⊕ π.range) (ι ⊕ π.range) ℝ := Matrix.of fun i j => C (t j,t i) have hM : M.IsSymm := by ext i j; exact hsym _ _ have hp : ∀ i j, 0 < M i j := fun i j => hC _ _ have hrec : OnePositiveSquare (Matrix.of fun i j => (M i j)⁻¹) := by have he : (Matrix.of fun i j => (M i j)⁻¹) = (Matrix.of fun i j => (C (t i,t j))⁻¹) := by ext i j; simp [M, hsym] rw [he] exact hP _ t have hw : ∀ i, 0 ≤ w i := by intro i; cases i <;> simp [w, measureReal_nonneg] have hsum (i : ι ⊕ π.range) (v : C(X, ℝ)) : ∑ j, M i j * w j * v (t j) = sampleOp ν π C v (t i) := by simp only [Fintype.sum_sum_type, M, w, t, Sum.elim_inl, Sum.elim_inr, Matrix.of_apply, mul_zero, zero_mul, Finset.sum_const_zero, zero_add, sampleOp_apply] rw [Finset.sum_subtype π.range (fun _ => Iff.rfl)] apply Finset.sum_congr rfl intro z hz ring have hb : ∀ i, ∑ j ∈ Finset.univ, M i j * w j * f (t j) < f (t i) := by intro i rw [hsum] have he := congrArg (fun v : C(X, ℝ) => f * v) (sampleOp_scaled ν π C f hf 1) rw [mul_one, invMap_mul_cancel] at he have hx := congrArg (fun v : C(X, ℝ) => v (t i)) he change f (t i) * sampleOp ν π (scaledKernel C f hf) 1 (t i) = sampleOp ν π C f (t i) at hx rw [← hx] exact (mul_le_mul_of_nonneg_left (hrow (t i)) (hf _).le).trans_lt (by simpa using (mul_lt_mul_of_pos_left hd1 (hf (t i)))) obtain ⟨R,hRs,hRp,hRrec,hReq⟩ := finite_resolvent_of_superharmonic hM hp hrec hw (fun i => hf (t i)) Finset.univ hb have hn : ‖sampleOp ν π (scaledKernel C f hf)‖ < 1 := (sampleOp_norm_le ν π _ (fun z x => (scaledKernel_pos C hC f hf z x).le) hd hrow).trans_lt hd1 have hmatch (j : ι ⊕ π.range) : ∀ i, R i j = rescaledKernel C f hf (sampleOp ν π (scaledKernel C f hf)) (t i) (t j) := by let u : C(X, ℝ) := C.curry (t j) + ∑ z : π.range, (ν.real (π ⁻¹' {z.val}) * R (Sum.inr z) j) • C.curry z.val have hu (i : ι ⊕ π.range) : u (t i) = R i j := by rw [hReq i j] simp only [u, ContinuousMap.add_apply, ContinuousMap.sum_apply, ContinuousMap.smul_apply, ContinuousMap.curry_apply, smul_eq_mul, Fintype.sum_sum_type, M, w, t, Sum.elim_inl, Sum.elim_inr, Matrix.of_apply, mul_zero, zero_mul, Finset.sum_const_zero, zero_add] congr 1 apply Finset.sum_congr rfl intro z hz ring have he : u - sampleOp ν π C u = C.curry (t j) := by ext x simp only [ContinuousMap.sub_apply, sampleOp_apply] rw [Finset.sum_subtype π.range (fun _ => Iff.rfl)] have hz (z : π.range) : u z.val = R (Sum.inr z) j := hu (Sum.inr z) simp only [hz, u, ContinuousMap.add_apply, ContinuousMap.sum_apply, ContinuousMap.smul_apply, ContinuousMap.curry_apply, smul_eq_mul] have hsum' : (∑ z : π.range, (ν.real (π ⁻¹' {↑z}) * R (Sum.inr z) j) * C (↑z,x)) = ∑ z : π.range, ν.real (π ⁻¹' {↑z}) * C (↑z,x) * R (Sum.inr z) j := by apply Finset.sum_congr rfl intro z hz ring rw [hsum'] abel have he' := scaled_solution f hf (sampleOp ν π C) (sampleOp ν π (scaledKernel C f hf)) hn (sampleOp_scaled ν π C f hf) he intro i rw [← hu i, he'] rfl refine ⟨fun i j => ?_, ?_⟩ · simpa only [t, Sum.elim_inl, hmatch] using hRp (Sum.inl i) (Sum.inl j) · have hsub := onePositiveSquare_submatrix_sum_inl hRrec convert hsub using 1 ext i j simp only [Matrix.submatrix_apply, Matrix.of_apply, hmatch, t, Sum.elim_inl] lemma rescaledKernel_tendsto [Nonempty X] (C : C(X × X, ℝ)) (hC : ∀ z x, 0 ≤ C (z,x)) (f : C(X, ℝ)) (hf : ∀ x, 0 < f x) {d : ℝ} (hd : 0 ≤ d) (hd1 : d < 1) (hrow : ∀ x, ∫ z, scaledKernel C f hf (z,x) ∂ν ≤ d) (x y : X) : Tendsto (fun n => rescaledKernel C f hf (sampleOp ν (quantize n) (scaledKernel C f hf)) x y) atTop (𝓝 (rescaledKernel C f hf (kernelOp ν (scaledKernel C f hf)) x y)) := by have hp (z x : X) : 0 ≤ scaledKernel C f hf (z,x) := div_nonneg (mul_nonneg (hC z x) (hf z).le) (hf x).le have hn := kernelOp_norm_le ν (scaledKernel C f hf) hp hd hrow have hh := BanachResolvent.resolvent_strong_tendsto (kernelOp ν (scaledKernel C f hf)) (fun n => sampleOp ν (quantize n) (scaledKernel C f hf)) (d := (1+d)/2) (by linarith) (hn.trans_lt hd1) (sampleOp_eventually_norm_le ν (scaledKernel C f hf) hp hd (by linarith) hrow) (sampleOp_strong_tendsto ν (scaledKernel C f hf)) (invMap f hf * C.curry y) exact (((ContinuousMap.evalCLM ℝ x).continuous.tendsto _).comp hh).const_mul (f x) end StrictHotSpots.CompactKernel namespace StrictHotSpots.CompactKernel open MeasureTheory Filter Set TopologicalSpace BanachResolvent ReciprocalKernel open scoped Topology ENNReal variable {X : Type} [MetricSpace X] [CompactSpace X] [MeasurableSpace X] [BorelSpace X] (ν : Measure X) [IsFiniteMeasure ν] theorem compact_reciprocal_of_l2_norm_lt_one [Nonempty X] (C : C(X × X, ℝ)) (hC : ∀ z x, 0 < C (z,x)) (hsym : ∀ x y, C (x,y) = C (y,x)) (hP : ∀ (ι : Type) [Fintype ι] (s : ι → X), OnePositiveSquare (Matrix.of fun i j => (C (s i,s j))⁻¹)) (hL : ‖l2Op ν C‖ < 1) {ι : Type} [Fintype ι] (s : ι → X) : (∀ i j, 0 < l2ResolvedKernel ν C (s i) (s j)) ∧ OnePositiveSquare (Matrix.of fun i j => (l2ResolvedKernel ν C (s i) (s j))⁻¹) := by obtain ⟨f,hf,heq⟩ := exists_superharmonic ν C (fun z x => (hC z x).le) hL let hf' : ∀ x, 0 < f x := fun x => lt_of_lt_of_le zero_lt_one (hf x) let C' := scaledKernel C f hf' let d : ℝ := 1 - ‖f‖⁻¹ obtain ⟨hd,hd1,hrow⟩ := scaled_row_bound ν C f hf heq have hp' (z x : X) : 0 ≤ C' (z,x) := (scaledKernel_pos C hC f hf' z x).le have hn : ‖kernelOp ν C'‖ < 1 := (kernelOp_norm_le ν C' hp' hd hrow).trans_lt hd1 have hkernel (x y : X) : rescaledKernel C f hf' (kernelOp ν C') x y = l2ResolvedKernel ν C x y := rescaledKernel_eq_l2ResolvedKernel ν C f hf' hL hn x y have hp (x y : X) : 0 < l2ResolvedKernel ν C x y := (hC y x).trans_le (l2ResolvedKernel_ge ν C (fun z x => (hC z x).le) hL x y) refine ⟨fun i j => hp _ _, ?_⟩ have hev := sampleOp_eventually_norm_le ν C' hp' hd (show d < (1+d)/2 by dsimp [d]; linarith) hrow have hr : ∀ᶠ n in atTop, ∀ x, sampleOp ν (quantize n) C' 1 x ≤ (1+d)/2 := by filter_upwards [hev] with n hnn x calc sampleOp ν (quantize n) C' 1 x ≤ ‖sampleOp ν (quantize n) C' 1 x‖ := le_abs_self _ _ ≤ ‖sampleOp ν (quantize n) C' 1‖ := ContinuousMap.norm_coe_le_norm _ _ _ ≤ ‖sampleOp ν (quantize n) C'‖ * ‖(1 : C(X, ℝ))‖ := ContinuousLinearMap.le_opNorm _ _ _ ≤ (1+d)/2 := by simpa using hnn obtain ⟨N,hN⟩ := eventually_atTop.mp hr apply onePositiveSquare_of_entrywise_tendsto (A := fun n => Matrix.of fun i j => (rescaledKernel C f hf' (sampleOp ν (quantize (n+N)) C') (s i) (s j))⁻¹) · intro n exact (sampled_rescaled_reciprocal ν (quantize (n+N)) C hC hsym hP f hf' (show 0 ≤ (1+d)/2 by dsimp [d]; linarith) (show (1+d)/2 < 1 by dsimp [d]; linarith) (hN _ (Nat.le_add_left _ _)) s).2 · intro i j have ht := rescaledKernel_tendsto ν C (fun z x => (hC z x).le) f hf' hd hd1 hrow (s i) (s j) rw [hkernel] at ht exact (ht.inv₀ (hp _ _).ne').comp (tendsto_add_atTop_nat N) lemma l2ResolvedKernel_eq_inner (C : C(X × X, ℝ)) (hsym : ∀ x y, C (x,y) = C (y,x)) (x y : X) : l2ResolvedKernel ν C x y = C (x,y) + inner ℝ (ContinuousMap.toLp 2 ν ℝ (C.curry x)) (resolvent (l2Op ν C) (ContinuousMap.toLp 2 ν ℝ (C.curry y))) := by unfold l2ResolvedKernel rw [hsym y x, l2SectionOp_apply, L2.inner_def] congr 1 apply integral_congr_ae filter_upwards [ContinuousMap.coeFn_toLp (p := 2) (𝕜 := ℝ) ν (C.curry x)] with z hz rw [hz] change C (z,x) * _ = _ * C (x,z) rw [hsym z x, mul_comm] end StrictHotSpots.CompactKernel open MeasureTheory Filter open scoped Topology ENNReal namespace StrictHotSpots.L2Limits variable {X : Type*} [MeasurableSpace X] {ν : Measure X} lemma norm_sq_eq_integral_sq (f : Lp ℝ 2 ν) : ‖f‖ ^ 2 = ∫ x, (f x)^2 ∂ν := by rw [← real_inner_self_eq_norm_sq, L2.inner_def] change (∫ x, f x * f x ∂ν) = _ simp only [pow_two] lemma norm_toLp_sub_sq {f g : X → ℝ} (hf : MemLp f 2 ν) (hg : MemLp g 2 ν) : ‖hf.toLp f - hg.toLp g‖ ^ 2 = ∫ x, (f x - g x)^2 ∂ν := by rw [norm_sq_eq_integral_sq] apply integral_congr_ae filter_upwards [Lp.coeFn_sub (hf.toLp f) (hg.toLp g), hf.coeFn_toLp, hg.coeFn_toLp] with x hx hfx hgx simp only [hx, hfx, hgx, Pi.sub_apply] lemma tendsto_toLp_of_dominated {ι : Type*} {l : Filter ι} [l.IsCountablyGenerated] {f : ι → X → ℝ} {g b : X → ℝ} (hf : ∀ n, MemLp (f n) 2 ν) (hg : MemLp g 2 ν) (hb : MemLp b 2 ν) (hdom : ∀ᶠ n in l, ∀ᵐ x ∂ν, ‖f n x‖ ≤ b x) (hlim : ∀ᵐ x ∂ν, Tendsto (fun n => f n x) l (𝓝 (g x))) : Tendsto (fun n => (hf n).toLp (f n)) l (𝓝 (hg.toLp g)) := by have ht : Tendsto (fun n => ∫ x, (f n x - g x)^2 ∂ν) l (𝓝 0) := by have hm : ∀ᶠ n in l, AEStronglyMeasurable (fun x => (f n x-g x)^2) ν := Eventually.of_forall fun n => ((hf n).sub hg).aestronglyMeasurable.pow 2 have hd : ∀ᶠ n in l, ∀ᵐ x ∂ν, ‖(f n x-g x)^2‖ ≤ (b x+‖g x‖)^2 := by filter_upwards [hdom] with n hn filter_upwards [hn] with x hx have h1 : ‖f n x-g x‖ ≤ b x+‖g x‖ := by linarith [norm_sub_le (f n x) (g x)] rw [Real.norm_eq_abs, abs_pow, ← Real.norm_eq_abs] exact pow_le_pow_left₀ (norm_nonneg _) h1 2 have hl : ∀ᵐ x ∂ν, Tendsto (fun n => (f n x-g x)^2) l (𝓝 (0 : ℝ)) := by filter_upwards [hlim] with x hx simpa using (hx.sub_const (g x)).pow 2 simpa using tendsto_integral_filter_of_dominated_convergence (fun x => (b x+‖g x‖)^2) hm hd (hb.add hg.norm).integrable_sq hl have hs : Tendsto (fun n => ‖(hf n).toLp (f n)-hg.toLp g‖^2) l (𝓝 0) := by simpa only [norm_toLp_sub_sq] using ht rw [tendsto_iff_norm_sub_tendsto_zero] have hr := Real.continuous_sqrt.continuousAt.tendsto.comp hs simpa only [Function.comp_def, Real.sqrt_sq (norm_nonneg _), Real.sqrt_zero] using hr lemma tendsto_L2_of_dominated {ι : Type*} {l : Filter ι} [l.IsCountablyGenerated] (f : ι → Lp ℝ 2 ν) (g b : Lp ℝ 2 ν) (hdom : ∀ᶠ n in l, ∀ᵐ x ∂ν, ‖f n x‖ ≤ b x) (hlim : ∀ᵐ x ∂ν, Tendsto (fun n => f n x) l (𝓝 (g x))) : Tendsto f l (𝓝 g) := by have ht := tendsto_toLp_of_dominated (fun n => Lp.memLp (f n)) (Lp.memLp g) (Lp.memLp b) hdom hlim simpa only [Lp.toLp_coeFn] using ht end StrictHotSpots.L2Limits namespace StrictHotSpots.CompactKernel open MeasureTheory Filter Set BanachResolvent ReciprocalKernel open scoped Topology ENNReal variable {X : Type} [MetricSpace X] [CompactSpace X] [MeasurableSpace X] [BorelSpace X] (ν : Measure X) [IsFiniteMeasure ν] lemma l2Op_ae_integral (C : C(X × X, ℝ)) (f : Lp ℝ 2 ν) : ∀ᵐ x ∂ν, l2Op ν C f x = ∫ z, C (z,x) * f z ∂ν := by filter_upwards [ContinuousMap.coeFn_toLp (p := 2) (𝕜 := ℝ) ν (l2SectionOp ν C f)] with x hx exact hx.trans (l2SectionOp_apply ν C f x) lemma l2Op_abs_domination (C : C(X × X, ℝ)) (hC : ∀ z x, 0 ≤ C (z,x)) (G : X → X → ℝ) (hG : ∀ x, MemLp (fun z => G z x) 2 ν) (hdom : ∀ x, ∀ᵐ z ∂ν, C (z,x) ≤ G z x) (T : Lp ℝ 2 ν →L[ℝ] Lp ℝ 2 ν) (hT : ∀ f : Lp ℝ 2 ν, ∀ᵐ x ∂ν, T f x = ∫ z, G z x * f z ∂ν) (f : Lp ℝ 2 ν) : |l2Op ν C f| ≤ T |f| := by apply (Lp.coeFn_le _ _).mp filter_upwards [l2Op_ae_integral ν C f, hT |f|, Lp.coeFn_abs (l2Op ν C f)] with x hx htx hax rw [hax, hx, htx] calc |∫ z, C (z,x) * f z ∂ν| ≤ ∫ z, ‖C (z,x) * f z‖ ∂ν := by simpa only [Real.norm_eq_abs] using norm_integral_le_integral_norm (fun z => C (z,x) * f z) _ ≤ ∫ z, G z x * (|f|) z ∂ν := by apply integral_mono_ae · exact (ContinuousMap.memLp (p := (2 : ℝ≥0∞)) ν ℝ (⟨fun z => C (z,x), C.continuous.comp (continuous_id.prodMk continuous_const)⟩ : C(X,ℝ))).integrable_mul (Lp.memLp f) |>.norm · exact (hG x).integrable_mul (Lp.memLp |f|) · filter_upwards [hdom x, Lp.coeFn_abs f] with z hz hfz rw [hfz, norm_mul, Real.norm_eq_abs, abs_of_nonneg (hC z x), Real.norm_eq_abs] exact mul_le_mul_of_nonneg_right hz (abs_nonneg _) lemma norm_l2Op_le_of_domination (C : C(X × X, ℝ)) (hC : ∀ z x, 0 ≤ C (z,x)) (G : X → X → ℝ) (hG : ∀ x, MemLp (fun z => G z x) 2 ν) (hdom : ∀ x, ∀ᵐ z ∂ν, C (z,x) ≤ G z x) (T : Lp ℝ 2 ν →L[ℝ] Lp ℝ 2 ν) (hT : ∀ f : Lp ℝ 2 ν, ∀ᵐ x ∂ν, T f x = ∫ z, G z x * f z ∂ν) : ‖l2Op ν C‖ ≤ ‖T‖ := by apply ContinuousLinearMap.opNorm_le_bound _ (norm_nonneg T) intro f calc ‖l2Op ν C f‖ ≤ ‖T |f|‖ := norm_le_norm_of_abs_le_abs ((l2Op_abs_domination ν C hC G hG hdom T hT f).trans (le_abs_self _)) _ ≤ ‖T‖ * ‖f‖ := by simpa only [norm_abs_eq_norm] using T.le_opNorm |f| lemma l2Op_strong_tendsto_of_dominated (C : ℕ → C(X × X, ℝ)) (hC : ∀ n z x, 0 ≤ C n (z,x)) (G : X → X → ℝ) (hG : ∀ x, MemLp (fun z => G z x) 2 ν) (hdom : ∀ n x, ∀ᵐ z ∂ν, C n (z,x) ≤ G z x) (hlim : ∀ x, ∀ᵐ z ∂ν, Tendsto (fun n => C n (z,x)) atTop (𝓝 (G z x))) (T : Lp ℝ 2 ν →L[ℝ] Lp ℝ 2 ν) (hT : ∀ f : Lp ℝ 2 ν, ∀ᵐ x ∂ν, T f x = ∫ z, G z x * f z ∂ν) (f : Lp ℝ 2 ν) : Tendsto (fun n => l2Op ν (C n) f) atTop (𝓝 (T f)) := by apply L2Limits.tendsto_L2_of_dominated _ _ (T |f|) · apply Eventually.of_forall intro n have hd := (Lp.coeFn_le _ _).mpr (l2Op_abs_domination ν (C n) (hC n) G hG (hdom n) T hT f) filter_upwards [hd, Lp.coeFn_abs (l2Op ν (C n) f)] with x hx hax simpa only [hax, Real.norm_eq_abs] using hx · have he : ∀ᵐ x ∂ν, ∀ n, l2Op ν (C n) f x = ∫ z, C n (z,x) * f z ∂ν := ae_all_iff.mpr fun n => l2Op_ae_integral ν (C n) f filter_upwards [he, hT f] with x hx htx simp only [hx, htx] apply tendsto_integral_of_dominated_convergence (fun z => G z x * ‖f z‖) · intro n exact ((C n).continuous.comp (continuous_id.prodMk continuous_const)).aestronglyMeasurable.mul (Lp.aestronglyMeasurable f) · exact (hG x).integrable_mul (Lp.memLp f).norm · intro n filter_upwards [hdom n x] with z hz rw [norm_mul, Real.norm_eq_abs, abs_of_nonneg (hC n z x)] exact mul_le_mul_of_nonneg_right hz (norm_nonneg _) · filter_upwards [hlim x] with z hz exact hz.mul_const (f z) end StrictHotSpots.CompactKernel namespace StrictHotSpots.BanachResolvent open Filter open scoped Topology variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] lemma resolvent_strong_tendsto_moving (A : E →L[ℝ] E) (Q : ℕ → E →L[ℝ] E) {d : ℝ} (hd : d < 1) (hA : ‖A‖ < 1) (hQ : ∀ᶠ n in atTop, ‖Q n‖ ≤ d) (hs : ∀ v, Tendsto (fun n => Q n v) atTop (𝓝 (A v))) {v : ℕ → E} {w : E} (hv : Tendsto v atTop (𝓝 w)) : Tendsto (fun n => resolvent (Q n) (v n)) atTop (𝓝 (resolvent A w)) := by have hz : Tendsto (fun n => resolvent (Q n) (v n-w)) atTop (𝓝 0) := by have hb : Tendsto (fun n => (1-d)⁻¹ * ‖v n-w‖) atTop (𝓝 0) := by simpa using (hv.sub_const w).norm.const_mul (1-d)⁻¹ apply squeeze_zero_norm' _ hb filter_upwards [hQ] with n hn exact (ContinuousLinearMap.le_opNorm _ _).trans (mul_le_mul_of_nonneg_right (resolvent_norm_le (Q n) hn hd) (norm_nonneg _)) have ht := hz.add (resolvent_strong_tendsto A Q hd hA hQ hs w) simpa only [map_sub, sub_add_cancel, zero_add] using ht end StrictHotSpots.BanachResolvent namespace StrictHotSpots.CompactKernel open MeasureTheory Filter Set BanachResolvent ReciprocalKernel open scoped Topology ENNReal variable {X : Type} [MetricSpace X] [CompactSpace X] [MeasurableSpace X] [BorelSpace X] (ν : Measure X) [IsFiniteMeasure ν] lemma l2Sections_tendsto_of_dominated (C : ℕ → C(X × X, ℝ)) (hC : ∀ n z x, 0 ≤ C n (z,x)) (hsym : ∀ n x y, C n (x,y) = C n (y,x)) (G : X → X → ℝ) (hG : ∀ x, MemLp (fun z => G z x) 2 ν) (hdom : ∀ n x, ∀ᵐ z ∂ν, C n (z,x) ≤ G z x) (hlim : ∀ x, ∀ᵐ z ∂ν, Tendsto (fun n => C n (z,x)) atTop (𝓝 (G z x))) (x : X) : Tendsto (fun n => ContinuousMap.toLp 2 ν ℝ ((C n).curry x)) atTop (𝓝 ((hG x).toLp (fun z => G z x))) := by apply L2Limits.tendsto_L2_of_dominated _ _ ((hG x).toLp (fun z => G z x)) · apply Eventually.of_forall intro n filter_upwards [ContinuousMap.coeFn_toLp (p := 2) (𝕜 := ℝ) ν ((C n).curry x), (hG x).coeFn_toLp, hdom n x] with z hz hgz hdz rw [hz, hgz] change ‖C n (x,z)‖ ≤ G z x rw [hsym n x z, Real.norm_eq_abs, abs_of_nonneg (hC n z x)] exact hdz · have he : ∀ᵐ z ∂ν, ∀ n, ContinuousMap.toLp 2 ν ℝ ((C n).curry x) z = C n (x,z) := ae_all_iff.mpr fun n => ContinuousMap.coeFn_toLp (p := 2) (𝕜 := ℝ) ν ((C n).curry x) filter_upwards [he, (hG x).coeFn_toLp, hlim x] with z hz hgz hlz simpa only [hz, hgz, hsym] using hlz def singularResolvedKernel (G : X → X → ℝ) (hG : ∀ x, MemLp (fun z => G z x) 2 ν) (T : Lp ℝ 2 ν →L[ℝ] Lp ℝ 2 ν) (x y : X) : ℝ := G x y + inner ℝ ((hG x).toLp (fun z => G z x)) (resolvent T ((hG y).toLp (fun z => G z y))) lemma l2ResolvedKernel_tendsto_of_dominated (C : ℕ → C(X × X, ℝ)) (hC : ∀ n z x, 0 ≤ C n (z,x)) (hsym : ∀ n x y, C n (x,y) = C n (y,x)) (G : X → X → ℝ) (hG : ∀ x, MemLp (fun z => G z x) 2 ν) (hdom : ∀ n x, ∀ᵐ z ∂ν, C n (z,x) ≤ G z x) (hlim : ∀ x, ∀ᵐ z ∂ν, Tendsto (fun n => C n (z,x)) atTop (𝓝 (G z x))) (T : Lp ℝ 2 ν →L[ℝ] Lp ℝ 2 ν) (hT : ∀ f : Lp ℝ 2 ν, ∀ᵐ x ∂ν, T f x = ∫ z, G z x * f z ∂ν) (hsub : ‖T‖ < 1) (x y : X) (hxy : Tendsto (fun n => C n (x,y)) atTop (𝓝 (G x y))) : Tendsto (fun n => l2ResolvedKernel ν (C n) x y) atTop (𝓝 (singularResolvedKernel ν G hG T x y)) := by have hs := l2Sections_tendsto_of_dominated ν C hC hsym G hG hdom hlim have hop := l2Op_strong_tendsto_of_dominated ν C hC G hG hdom hlim T hT have hb : ∀ᶠ n in atTop, ‖l2Op ν (C n)‖ ≤ ‖T‖ := Eventually.of_forall fun n => norm_l2Op_le_of_domination ν (C n) (hC n) G hG (hdom n) T hT have hr := resolvent_strong_tendsto_moving T (fun n => l2Op ν (C n)) hsub hsub hb hop (hs y) have ht := hxy.add ((hs x).inner hr) simpa only [singularResolvedKernel, l2ResolvedKernel_eq_inner ν _ (hsym _)] using ht lemma singularResolvedKernel_ge (C : ℕ → C(X × X, ℝ)) (hC : ∀ n z x, 0 ≤ C n (z,x)) (hsym : ∀ n x y, C n (x,y) = C n (y,x)) (G : X → X → ℝ) (hG : ∀ x, MemLp (fun z => G z x) 2 ν) (hdom : ∀ n x, ∀ᵐ z ∂ν, C n (z,x) ≤ G z x) (hlim : ∀ x, ∀ᵐ z ∂ν, Tendsto (fun n => C n (z,x)) atTop (𝓝 (G z x))) (T : Lp ℝ 2 ν →L[ℝ] Lp ℝ 2 ν) (hT : ∀ f : Lp ℝ 2 ν, ∀ᵐ x ∂ν, T f x = ∫ z, G z x * f z ∂ν) (hsub : ‖T‖ < 1) (x y : X) (hxy : Tendsto (fun n => C n (x,y)) atTop (𝓝 (G x y))) : G x y ≤ singularResolvedKernel ν G hG T x y := by apply le_of_tendsto_of_tendsto hxy (l2ResolvedKernel_tendsto_of_dominated ν C hC hsym G hG hdom hlim T hT hsub x y hxy) exact Eventually.of_forall fun n => by dsimp only rw [hsym n x y] exact l2ResolvedKernel_ge ν (C n) (hC n) ((norm_l2Op_le_of_domination ν (C n) (hC n) G hG (hdom n) T hT).trans_lt hsub) x y theorem singular_reciprocal_of_regularization [Nonempty X] [DecidableEq X] (C : ℕ → C(X × X, ℝ)) (hC : ∀ n z x, 0 < C n (z,x)) (hsym : ∀ n x y, C n (x,y) = C n (y,x)) (hP : ∀ n (ι : Type) [Fintype ι] (s : ι → X), OnePositiveSquare (Matrix.of fun i j => (C n (s i,s j))⁻¹)) (G : X → X → ℝ) (hG : ∀ x, MemLp (fun z => G z x) 2 ν) (hpos : ∀ x y, x ≠ y → 0 < G x y) (hdom : ∀ n x, ∀ᵐ z ∂ν, C n (z,x) ≤ G z x) (hlim : ∀ x, ∀ᵐ z ∂ν, Tendsto (fun n => C n (z,x)) atTop (𝓝 (G z x))) (hoff : ∀ x y, x ≠ y → Tendsto (fun n => C n (x,y)) atTop (𝓝 (G x y))) (hdiag : ∀ x, Tendsto (fun n => C n (x,x)) atTop atTop) (T : Lp ℝ 2 ν →L[ℝ] Lp ℝ 2 ν) (hT : ∀ f : Lp ℝ 2 ν, ∀ᵐ x ∂ν, T f x = ∫ z, G z x * f z ∂ν) (hsub : ‖T‖ < 1) {ι : Type} [Fintype ι] (s : ι → X) : OnePositiveSquare (Matrix.of fun i j => if s i = s j then 0 else (singularResolvedKernel ν G hG T (s i) (s j))⁻¹) := by classical have hCn : ∀ n z x, 0 ≤ C n (z,x) := fun n z x => (hC n z x).le have hb (n : ℕ) : ‖l2Op ν (C n)‖ < 1 := (norm_l2Op_le_of_domination ν (C n) (hCn n) G hG (hdom n) T hT).trans_lt hsub apply onePositiveSquare_of_entrywise_tendsto (A := fun n => Matrix.of fun i j => (l2ResolvedKernel ν (C n) (s i) (s j))⁻¹) · intro n exact (compact_reciprocal_of_l2_norm_lt_one ν (C n) (hC n) (hsym n) (hP n) (hb n) s).2 · intro i j dsimp only [Matrix.of_apply] split_ifs with hij · have ht : Tendsto (fun n => l2ResolvedKernel ν (C n) (s i) (s j)) atTop atTop := by apply tendsto_atTop_mono _ (hdiag (s i)) intro n simpa only [← hij] using l2ResolvedKernel_ge ν (C n) (hCn n) (hb n) (s i) (s i) exact tendsto_inv_atTop_zero.comp ht · have hp := (hpos (s i) (s j) hij).trans_le (singularResolvedKernel_ge ν C hCn hsym G hG hdom hlim T hT hsub (s i) (s j) (hoff (s i) (s j) hij)) exact (l2ResolvedKernel_tendsto_of_dominated ν C hCn hsym G hG hdom hlim T hT hsub (s i) (s j) (hoff (s i) (s j) hij)).inv₀ hp.ne' end StrictHotSpots.CompactKernel end SingularLayer end FullBoundaryCombinedLayer end end DouglasLipschitzBase section DouglasLipschitzBase noncomputable section section FullBoundaryCombinedLayer section SquaredDistanceKernelLayer noncomputable section open Set Filter Topology open scoped BigOperators InnerProductSpace open scoped NNReal namespace SquaredDistanceKernel universe u variable {X : Type u} def ConditionallyNegative (D : Matrix X X ℝ) : Prop := D.IsHermitian ∧ (∀ x, D x x = 0) ∧ ∀ c : X →₀ ℝ, c.sum (fun _ a => a) = 0 → c.sum (fun x a => c.sum (fun y b => a * D x y * b)) ≤ 0 def anchored (D : Matrix X X ℝ) (o : X) : Matrix X X ℝ := Matrix.of fun x y => (D x o + D y o - D x y) / 2 lemma anchor_sum_identity (D : X → X → ℝ) (o : X) (c : X →₀ ℝ) : (c.sum (fun x a => c.sum (fun y b => a * ((D x o + D y o - D x y) / 2) * b))) = ((c.sum (fun x a => a * D x o * (c.sum fun _ b => b))) + (c.sum (fun y b => (c.sum fun _ a => a) * D y o * b)) - (c.sum (fun x a => c.sum (fun y b => a * D x y * b)))) / 2 := by calc _ = c.sum (fun x a => c.sum (fun y b => (a * D x o * b + a * D y o * b - a * D x y * b) / 2)) := by apply Finsupp.sum_congr intro x hx apply Finsupp.sum_congr intro y hy ring _ = _ := by simp only [Finsupp.sum] simp only [← Finset.sum_div, Finset.sum_add_distrib, Finset.sum_sub_distrib] congr 2 congr 1 · simp only [← Finset.mul_sum] · rw [Finset.sum_comm] simp only [← Finset.sum_mul] lemma anchored_posSemidef {D : Matrix X X ℝ} (hD : ConditionallyNegative D) (o : X) : (anchored D o).PosSemidef := by classical have hs (x y : X) : D x y = D y x := by simpa using (hD.1.apply x y).symm refine ⟨?_, ?_⟩ · ext x y simp [Matrix.conjTranspose_apply, anchored, hs y x, add_comm] · intro c change 0 ≤ c.sum (fun x a => c.sum (fun y b => a * ((D x o + D y o - D x y) / 2) * b)) let m := c.sum (fun _ a => a) have hm : (c - Finsupp.single o m).sum (fun _ a => a) = 0 := by simp [Finsupp.sum_sub_index, m] have hh := hD.2.2 (c - Finsupp.single o m) hm simp [Finsupp.sum_sub_index, sub_mul, mul_sub, Finsupp.sum_sub, zero_mul, mul_zero, Finsupp.sum_single_index, hD.2.1, hs o, sub_zero] at hh dsimp [m] at hh rw [anchor_sum_identity (fun x y => D x y) o c] linarith def lifted (G : Matrix X X ℝ) : Matrix X X (ℝ →L[ℝ] ℝ) := Matrix.of fun x y => G x y • ContinuousLinearMap.id ℝ ℝ lemma lifted_posSemidef {G : Matrix X X ℝ} (hG : G.PosSemidef) : (lifted G).PosSemidef := by have hs (x y : X) : G y x = G x y := by simpa using hG.1.apply x y apply (RKHS.posSemidef_tfae.out 1 3).mpr refine ⟨?_, ?_⟩ · ext x y simp [Matrix.conjTranspose_apply, lifted, star_smul, ContinuousLinearMap.star_eq_adjoint, ContinuousLinearMap.adjoint_id, hs] · intro c simpa [lifted, RCLike.inner_apply, hs, mul_comm, mul_left_comm, mul_assoc] using hG.2 c lemma positive_kernel_realization {G : Matrix X X ℝ} (hG : G.PosSemidef) : ∃ (H : Type u) (_ : NormedAddCommGroup H) (_ : InnerProductSpace ℝ H) (_ : CompleteSpace H) (b : X → H), ∀ x y, inner ℝ (b x) (b y) = G x y := by let K := lifted G let : Fact K.PosSemidef := ⟨lifted_posSemidef hG⟩ let H := RKHS.OfKernel K refine ⟨H, inferInstance, inferInstance, inferInstance, fun x => RKHS.kerFun H x 1, ?_⟩ intro x y rw [← RKHS.kernel_inner] change inner ℝ (RKHS.kernel (RKHS.OfKernel K) y x 1) 1 = G x y rw [RKHS.OfKernel.kernel_ofKernel (K := K)] simp [K, lifted, RCLike.inner_apply, show G y x = G x y by simpa using hG.1.apply x y] lemma squared_distance_realization {D : Matrix X X ℝ} (hD : ConditionallyNegative D) (o : X) : ∃ (H : Type u) (_ : NormedAddCommGroup H) (_ : InnerProductSpace ℝ H) (_ : CompleteSpace H) (b : X → H), ∀ x y, ‖b x - b y‖ ^ 2 = D x y := by obtain ⟨H, hn, hi, hc, b, hb⟩ := positive_kernel_realization (anchored_posSemidef hD o) let := hn let := hi let := hc refine ⟨H, hn, hi, hc, b, ?_⟩ intro x y rw [norm_sub_sq_real, ← real_inner_self_eq_norm_sq, ← real_inner_self_eq_norm_sq, hb, hb, hb] simp only [anchored, Matrix.of_apply, hD.2.1] ring lemma injective_of_distance_positive {H : Type*} [NormedAddCommGroup H] {D : Matrix X X ℝ} {b : X → H} (hb : ∀ x y, ‖b x - b y‖ ^ 2 = D x y) (hpos : ∀ x y, x ≠ y → 0 < D x y) : Function.Injective b := by intro x y heq by_contra hne have hh := hpos x y hne rw [← hb x y, heq, sub_self, norm_zero, zero_pow (by decide)] at hh exact lt_irrefl _ hh lemma lipschitz_of_distance_bound [PseudoMetricSpace X] {H : Type*} [NormedAddCommGroup H] {D : Matrix X X ℝ} {b : X → H} {C : ℝ≥0} (hb : ∀ x y, ‖b x - b y‖ ^ 2 = D x y) (hbound : ∀ x y, D x y ≤ C ^ 2 * dist x y ^ 2) : LipschitzWith C b := by apply LipschitzWith.of_dist_le_mul intro x y rw [dist_eq_norm] apply (sq_le_sq₀ (norm_nonneg _) (mul_nonneg C.coe_nonneg dist_nonneg)).mp rw [mul_pow, hb] exact hbound x y lemma separable_lipschitz_embedding [PseudoMetricSpace X] [TopologicalSpace.SeparableSpace X] {D : Matrix X X ℝ} (hD : ConditionallyNegative D) (o : X) {C : ℝ≥0} (hpos : ∀ x y, x ≠ y → 0 < D x y) (hbound : ∀ x y, D x y ≤ (C : ℝ) ^ 2 * dist x y ^ 2) : ∃ (H : Type u) (_ : NormedAddCommGroup H) (_ : InnerProductSpace ℝ H) (_ : CompleteSpace H) (_ : TopologicalSpace.SeparableSpace H) (b : X → H), Function.Injective b ∧ LipschitzWith C b ∧ ∀ x y, ‖b x - b y‖ ^ 2 = D x y := by obtain ⟨H, hn, hi, hc, b, hb⟩ := squared_distance_realization hD o let := hn let := hi let := hc have hl := lipschitz_of_distance_bound hb hbound let S := (Submodule.span ℝ (Set.range b)).topologicalClosure have hsep : TopologicalSpace.IsSeparable (S : Set H) := by rw [Submodule.topologicalClosure_coe] exact ((TopologicalSpace.isSeparable_range hl.continuous).span (R := ℝ)).closure let := hsep.separableSpace let b' : X → S := fun x => ⟨b x, (Submodule.span ℝ (Set.range b)).le_topologicalClosure (Submodule.subset_span ⟨x, rfl⟩)⟩ have hb' (x y : X) : ‖b' x - b' y‖ ^ 2 = D x y := hb x y refine ⟨S, inferInstance, inferInstance, inferInstance, inferInstance, b', injective_of_distance_positive hb' hpos, lipschitz_of_distance_bound hb' hbound, hb'⟩ end SquaredDistanceKernel end end SquaredDistanceKernelLayer end FullBoundaryCombinedLayer end end DouglasLipschitzBase end OAI