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d5b562f | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 | import OAI.MathematicalPhysics.DefocusingNLS.Linear.ExpandingFourier
import OAI.MathematicalPhysics.DefocusingNLS.Linear.SobolevConvolution
namespace OAI
/-! # Multiplication with the exact expanding-torus Fourier weight
The constants below are independent of the expanding radius `L`.
-/
open scoped ENNReal
namespace DefocusingNLS
theorem weight_mul_norm_expandingFourierCoefficient (a k L : ℝ) (hL : 1 ≤ L) (f : FourierL2)
(n : frequencyLattice) :
expandingSobolevWeight a k L n * ‖expandingFourierCoefficient a k L f n‖ = ‖f n‖ := by
simpa only [norm_mul, Complex.norm_real, Real.norm_eq_abs,
abs_of_pos (expandingSobolevWeight_pos a k L hL n)] using
congrArg norm (weight_mul_expandingFourierCoefficient a k L hL f n)
theorem norm_expandingFourierCoefficient_le (a k L : ℝ)
(ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (hL : 1 ≤ L)
(f : FourierL2) (n : frequencyLattice) :
‖expandingFourierCoefficient a k L f n‖ ≤ expandingEmbeddingBound a k * ‖f‖ := by
have hs := summable_norm_expandingFourierCoefficient a k L ha ha1 hk hL f
have hn := hs.sum_le_tsum {n} (fun j _ => norm_nonneg (expandingFourierCoefficient a k L f j))
have hn' : ‖expandingFourierCoefficient a k L f n‖ ≤
∑' j, ‖expandingFourierCoefficient a k L f j‖ := by simpa using hn
exact hn'.trans
(tsum_norm_expandingFourierCoefficient_le a k L ha ha1 hk hL f)
/-- The physical coefficients of the product are the discrete convolution. -/
noncomputable def expandingProductCoefficient (a k L : ℝ) (f g : FourierL2)
(n : frequencyLattice) : ℂ :=
∑' m, expandingFourierCoefficient a k L f m * expandingFourierCoefficient a k L g (n - m)
theorem summable_norm_expandingProductTerms (a k L : ℝ) (ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (hL : 1 ≤ L) (f g : FourierL2)
(n : frequencyLattice) :
Summable (fun m => ‖expandingFourierCoefficient a k L f m‖ *
‖expandingFourierCoefficient a k L g (n - m)‖) := by
apply Summable.of_nonneg_of_le (fun m => mul_nonneg (norm_nonneg _) (norm_nonneg _))
(fun m => mul_le_mul_of_nonneg_left (norm_expandingFourierCoefficient_le a k L ha ha1 hk hL g _)
(norm_nonneg _))
exact (summable_norm_expandingFourierCoefficient a k L ha ha1 hk hL f).mul_right (expandingEmbeddingBound a k * ‖g‖)
/-- The two unweighted Young convolutions which dominate the weighted product. -/
noncomputable def expandingProductMajorant (a k L : ℝ) (f g : FourierL2) : FourierL2 :=
Complex.ofReal (Real.sqrt ((4 : ℝ) ^ k)) •
(fourierConvolution (fun m => (‖expandingFourierCoefficient a k L f m‖ : ℂ)) (fourierAbsolute g) +
fourierConvolution (fun m => (‖expandingFourierCoefficient a k L g m‖ : ℂ)) (fourierAbsolute f))
theorem expandingProductMajorant_norm_apply (a k L : ℝ) (ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (hL : 1 ≤ L) (f g : FourierL2)
(n : frequencyLattice) :
‖expandingProductMajorant a k L f g n‖ = Real.sqrt (4 ^ k) *
((∑' m, ‖expandingFourierCoefficient a k L f m‖ * ‖g (n - m)‖) +
∑' m, ‖expandingFourierCoefficient a k L g m‖ * ‖f (n - m)‖) := by
simp only [expandingProductMajorant, lp.coeFn_smul, Pi.smul_apply, lp.coeFn_add,
Pi.add_apply, smul_eq_mul]
rw [absolute_convolution_apply _ (summable_norm_expandingFourierCoefficient a k L ha ha1 hk hL f),
absolute_convolution_apply _ (summable_norm_expandingFourierCoefficient a k L ha ha1 hk hL g),
← Complex.ofReal_add, ← Complex.ofReal_mul, Complex.norm_real, Real.norm_eq_abs]
apply abs_of_nonneg
exact mul_nonneg (Real.sqrt_nonneg _)
(add_nonneg (tsum_nonneg (fun m => mul_nonneg (norm_nonneg _) (norm_nonneg _)))
(tsum_nonneg (fun m => mul_nonneg (norm_nonneg _) (norm_nonneg _))))
lemma expanding_weighted_product_term_bound (a k L : ℝ) (ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (hL : 1 ≤ L) (f g : FourierL2)
(n m : frequencyLattice) :
expandingSobolevWeight a k L n *
(‖expandingFourierCoefficient a k L f m‖ * ‖expandingFourierCoefficient a k L g (n - m)‖) ≤
Real.sqrt (4 ^ k) * (‖f m‖ * ‖expandingFourierCoefficient a k L g (n - m)‖ +
‖expandingFourierCoefficient a k L f m‖ * ‖g (n - m)‖) := by
have hw := expandingSobolevWeight_add_le a k L ha ha1 hk hL m (n - m)
have hn : m + (n - m) = n := by abel
rw [hn] at hw
calc
_ ≤ (Real.sqrt (4 ^ k) * (expandingSobolevWeight a k L m + expandingSobolevWeight a k L (n - m))) *
(‖expandingFourierCoefficient a k L f m‖ * ‖expandingFourierCoefficient a k L g (n - m)‖) :=
mul_le_mul_of_nonneg_right hw (mul_nonneg (norm_nonneg _) (norm_nonneg _))
_ = _ := by
rw [← weight_mul_norm_expandingFourierCoefficient a k L hL f m,
← weight_mul_norm_expandingFourierCoefficient a k L hL g (n - m)]
ring
/-- The weighted convolution is bounded pointwise by an ℓ² vector. -/
theorem weighted_expandingProduct_le_majorant (a k L : ℝ) (ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (hL : 1 ≤ L) (f g : FourierL2)
(n : frequencyLattice) :
‖(expandingSobolevWeight a k L n : ℂ) * expandingProductCoefficient a k L f g n‖ ≤
‖expandingProductMajorant a k L f g n‖ := by
let A := fun m : frequencyLattice => ‖f m‖ * ‖expandingFourierCoefficient a k L g (n - m)‖
let B := fun m : frequencyLattice => ‖expandingFourierCoefficient a k L f m‖ * ‖g (n - m)‖
have hA : Summable A := by
apply Summable.of_nonneg_of_le (fun m => mul_nonneg (norm_nonneg _) (norm_nonneg _))
(fun m => mul_le_mul_of_nonneg_right
(lp.norm_apply_le_norm (by norm_num : (2 : ℝ≥0∞) ≠ 0) f m) (norm_nonneg _))
exact ((Equiv.subLeft n).summable_iff.mpr
(summable_norm_expandingFourierCoefficient a k L ha ha1 hk hL g)).mul_left ‖f‖
have hB : Summable B := by
apply Summable.of_nonneg_of_le (fun m => mul_nonneg (norm_nonneg _) (norm_nonneg _))
(fun m => mul_le_mul_of_nonneg_left
(lp.norm_apply_le_norm (by norm_num : (2 : ℝ≥0∞) ≠ 0) g (n - m)) (norm_nonneg _))
exact (summable_norm_expandingFourierCoefficient a k L ha ha1 hk hL f).mul_right ‖g‖
have hp := summable_norm_expandingProductTerms a k L ha ha1 hk hL f g n
have hswap : (∑' m, A m) =
∑' m, ‖expandingFourierCoefficient a k L g m‖ * ‖f (n - m)‖ := by
rw [← (Equiv.subLeft n).tsum_eq A]
simp only [A, Equiv.subLeft_apply, sub_sub_cancel, mul_comm]
rw [norm_mul, Complex.norm_real, Real.norm_eq_abs,
abs_of_pos (expandingSobolevWeight_pos a k L hL n), expandingProductMajorant_norm_apply a k L ha ha1 hk hL]
calc
_ ≤ expandingSobolevWeight a k L n *
∑' m, ‖expandingFourierCoefficient a k L f m‖ * ‖expandingFourierCoefficient a k L g (n - m)‖ := by
apply mul_le_mul_of_nonneg_left _ (expandingSobolevWeight_pos a k L hL n).le
simpa only [expandingProductCoefficient, norm_mul] using
(norm_tsum_le_tsum_norm
(f := fun m => expandingFourierCoefficient a k L f m * expandingFourierCoefficient a k L g (n - m))
(by simpa only [norm_mul] using hp))
_ = ∑' m, expandingSobolevWeight a k L n *
(‖expandingFourierCoefficient a k L f m‖ * ‖expandingFourierCoefficient a k L g (n - m)‖) :=
by rw [tsum_mul_left]
_ ≤ ∑' m, Real.sqrt (4 ^ k) * (A m + B m) :=
(hp.mul_left _).tsum_le_tsum (fun m => expanding_weighted_product_term_bound a k L ha ha1 hk hL f g n m)
((hA.add hB).mul_left _)
_ = _ := by rw [tsum_mul_left, hA.tsum_add hB, hswap]; ring
/-- Multiplication closes with the exact expanding-torus Fourier weight. -/
noncomputable def expandingProduct (a k L : ℝ) (ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (hL : 1 ≤ L) (f g : FourierL2) : FourierL2 :=
⟨fun n => (expandingSobolevWeight a k L n : ℂ) * expandingProductCoefficient a k L f g n,
(lp.memℓp (expandingProductMajorant a k L f g)).mono'
(weighted_expandingProduct_le_majorant a k L ha ha1 hk hL f g)⟩
theorem expandingProduct_coefficient (a k L : ℝ) (ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (hL : 1 ≤ L) (f g : FourierL2)
(n : frequencyLattice) :
expandingFourierCoefficient a k L (expandingProduct a k L ha ha1 hk hL f g) n =
expandingProductCoefficient a k L f g n := by
apply mul_left_cancel₀ (show (expandingSobolevWeight a k L n : ℂ) ≠ 0 from
Complex.ofReal_ne_zero.mpr (expandingSobolevWeight_pos a k L hL n).ne')
exact weight_mul_expandingFourierCoefficient a k L hL (expandingProduct a k L ha ha1 hk hL f g) n
/-- A quantitative norm bound for the two-Young-convolution majorant. -/
theorem expandingProductMajorant_norm_le (a k L : ℝ) (ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (hL : 1 ≤ L) (f g : FourierL2) :
‖expandingProductMajorant a k L f g‖ ≤
2 * Real.sqrt (4 ^ k) * expandingEmbeddingBound a k * ‖f‖ * ‖g‖ := by
have hf := summable_norm_expandingFourierCoefficient a k L ha ha1 hk hL f
have hg := summable_norm_expandingFourierCoefficient a k L ha ha1 hk hL g
have hcf := fourierConvolution_norm_le (fun m => (‖expandingFourierCoefficient a k L f m‖ : ℂ))
(by simpa using hf) (fourierAbsolute g)
have hcg := fourierConvolution_norm_le (fun m => (‖expandingFourierCoefficient a k L g m‖ : ℂ))
(by simpa using hg) (fourierAbsolute f)
simp only [Complex.norm_real, Real.norm_eq_abs, abs_norm, fourierAbsolute_norm] at hcf hcg
have hC : 0 ≤ Real.sqrt ((4 : ℝ) ^ k) := Real.sqrt_nonneg _
calc
_ = Real.sqrt (4 ^ k) * ‖fourierConvolution
(fun m => (‖expandingFourierCoefficient a k L f m‖ : ℂ)) (fourierAbsolute g) +
fourierConvolution (fun m => (‖expandingFourierCoefficient a k L g m‖ : ℂ))
(fourierAbsolute f)‖ := by
rw [expandingProductMajorant, norm_smul, Complex.norm_real, Real.norm_eq_abs,
abs_of_nonneg hC]
_ ≤ Real.sqrt (4 ^ k) *
(‖fourierConvolution (fun m => (‖expandingFourierCoefficient a k L f m‖ : ℂ))
(fourierAbsolute g)‖ +
‖fourierConvolution (fun m => (‖expandingFourierCoefficient a k L g m‖ : ℂ))
(fourierAbsolute f)‖) := mul_le_mul_of_nonneg_left (norm_add_le _ _) hC
_ ≤ Real.sqrt (4 ^ k) * ((∑' m, ‖expandingFourierCoefficient a k L f m‖) * ‖g‖ +
(∑' m, ‖expandingFourierCoefficient a k L g m‖) * ‖f‖) :=
mul_le_mul_of_nonneg_left (add_le_add hcf hcg) hC
_ ≤ Real.sqrt (4 ^ k) * ((expandingEmbeddingBound a k * ‖f‖) * ‖g‖ +
(expandingEmbeddingBound a k * ‖g‖) * ‖f‖) := by
gcongr
· exact tsum_norm_expandingFourierCoefficient_le a k L ha ha1 hk hL f
· exact tsum_norm_expandingFourierCoefficient_le a k L ha ha1 hk hL g
_ = _ := by ring
/-- The `Y_L` multiplication constant is independent of `L ≥ 1`. -/
theorem expandingProduct_norm_le (a k L : ℝ) (ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (hL : 1 ≤ L) (f g : FourierL2) :
‖expandingProduct a k L ha ha1 hk hL f g‖ ≤
2 * Real.sqrt (4 ^ k) * expandingEmbeddingBound a k * ‖f‖ * ‖g‖ := by
exact (lp.norm_mono (by norm_num : (2 : ℝ≥0∞) ≠ 0)
(weighted_expandingProduct_le_majorant a k L ha ha1 hk hL f g)).trans
(expandingProductMajorant_norm_le a k L ha ha1 hk hL f g)
end DefocusingNLS
end OAI
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