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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 | import OAI.MathematicalPhysics.DefocusingNLS.Linear.HomogeneousYLocalization
namespace OAI
/-! # Finite coordinates of localized torus data -/
open scoped SchwartzMap
namespace DefocusingNLS
local notation "E" => EuclideanSpace ℝ (Fin 12)
variable {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F]
noncomputable def expandingCoordinates (a k : ℝ)
(ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (χ : 𝓢(E, ℂ))
(π : HomogeneousY a k →L[ℝ] F) (L : {L : ℝ // 1 ≤ L}) : FourierL2 →L[ℝ] F :=
π.comp ((homogeneousLocalizationCLM a k L.1 ha ha1 hk L.2 χ).restrictScalars ℝ)
@[simp] theorem expandingCoordinates_apply (a k : ℝ)
(ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (χ : 𝓢(E, ℂ))
(π : HomogeneousY a k →L[ℝ] F) (L : {L : ℝ // 1 ≤ L}) (f : FourierL2) :
expandingCoordinates a k ha ha1 hk χ π L f =
π (homogeneousLocalizationCLM a k L.1 ha ha1 hk L.2 χ f) := rfl
theorem expandingCoordinates_norm_le (a k C : ℝ)
(ha : 0 < a) (ha1 : a < 1) (hk : 8 < k) (hC : 0 ≤ C) (χ : 𝓢(E, ℂ))
(π : HomogeneousY a k →L[ℝ] F)
(hCb : ∀ (L : ℝ) (hL : 1 ≤ L) (f : FourierL2),
‖homogeneousLocalizationCLM a k L ha ha1 hk hL χ f‖ ≤ C * ‖f‖)
(L : {L : ℝ // 1 ≤ L}) : ‖expandingCoordinates a k ha ha1 hk χ π L‖ ≤ ‖π‖ * C := by
apply ContinuousLinearMap.opNorm_le_bound _ (mul_nonneg (norm_nonneg _) hC)
intro f
calc
_ ≤ ‖π‖ * ‖homogeneousLocalizationCLM a k L.1 ha ha1 hk L.2 χ f‖ := π.le_opNorm _
_ ≤ ‖π‖ * (C * ‖f‖) := mul_le_mul_of_nonneg_left (hCb L.1 L.2 f) (norm_nonneg _)
_ = _ := (mul_assoc _ _ _).symm
end DefocusingNLS
end OAI
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