simp [f.hf, l1, add_mul, Real.fourierIntegral_deriv f.hf' l5 f.hf'', Real.fourierIntegral_deriv f.hf l4 f.hf']
f : W21
u : ℝ
l1 : Integrable (fun x => (↑π ^ 2)⁻¹ * 4⁻¹ * deriv (deriv f.toFun) x) volume
l4 : Differentiable ℝ f.toFun
l5 : Differentiable ℝ (deriv f.toFun)
⊢ (1 + ↑u ^ 2) * 𝓕 f.toFun u = 𝓕 (fun u => f.toFun u - 1 / (4 * ↑π ^ 2) * deriv^[2] f.toFun u) u
f : W21
u : ℝ
l1 : Integrable (fun x => (↑π ^ 2)⁻¹ * 4⁻¹ * deriv (deriv f.toFun) x) volume
l4 : Differentiable ℝ f.toFun
l5 : Differentiable ℝ (deriv f.toFun)
⊢ 𝓕 f.toFun u + ↑u ^ 2 * 𝓕 f.toFun u =
    𝓕 (fun x => f.toFun x) u - (↑π ^ 2)⁻¹ * 4⁻¹ * (2 * ↑π * I * ↑u * (2 * ↑π * I * ↑u * 𝓕 f.toFun u))