mellin_hasDerivAt_of_isBigO_rpow — Mathlib

English

If f is locally integrable and satisfies suitable Big-O bounds, then mellin f has a Mellin transform of log-t•f, and Mellin has a derivative at s equal to that transform.

Русский

Если f локально интегрируема и удовлетворяет подходящим условиям Big-O, то у mellin f есть преобразование лог t • f, и имеется производная по s, равная этому преобразованию.

LaTeX

$$$\text{MellinConvergent}(f,s) \;\land\; \text{HasDerivAt}(\operatorname{mellin}f, \operatorname{mellin}(\log\cdot f)s, s).$$$

Lean4

/-- Suppose `f` is locally integrable on `(0, ∞)`, is `O(x ^ (-a))` as `x → ∞`, and is
`O(x ^ (-b))` as `x → 0`. Then its Mellin transform is differentiable on the domain `b < re s < a`,
with derivative equal to the Mellin transform of `log • f`. -/
theorem mellin_hasDerivAt_of_isBigO_rpow [NormedSpace ℂ E] {a b : ℝ} {f : ℝ → E} {s : ℂ}
    (hfc : LocallyIntegrableOn f (Ioi 0)) (hf_top : f =O[atTop] (· ^ (-a))) (hs_top : s.re < a)
    (hf_bot : f =O[𝓝[>] 0] (· ^ (-b))) (hs_bot : b < s.re) :
    MellinConvergent (fun t => log t • f t) s ∧ HasDerivAt (mellin f) (mellin (fun t => log t • f t) s) s :=
  by
  set F : ℂ → ℝ → E := fun (z : ℂ) (t : ℝ) => (t : ℂ) ^ (z - 1) • f t
  set F' : ℂ → ℝ → E := fun (z : ℂ) (t : ℝ) => ((t : ℂ) ^ (z - 1) * log t) • f t
  obtain ⟨v, hv0, hv1, hv2⟩ : ∃ v : ℝ, 0 < v ∧ v < s.re - b ∧ v < a - s.re :=
    by
    obtain ⟨w, hw1, hw2⟩ := exists_between (sub_pos.mpr hs_top)
    obtain ⟨w', hw1', hw2'⟩ := exists_between (sub_pos.mpr hs_bot)
    exact ⟨min w w', lt_min hw1 hw1', (min_le_right _ _).trans_lt hw2', (min_le_left _ _).trans_lt hw2⟩
  let bound : ℝ → ℝ := fun t : ℝ => (t ^ (s.re + v - 1) + t ^ (s.re - v - 1)) * |log t| * ‖f t‖
  have h1 : ∀ᶠ z : ℂ in 𝓝 s, AEStronglyMeasurable (F z) (volume.restrict <| Ioi 0) :=
    by
    refine Eventually.of_forall fun z => AEStronglyMeasurable.smul ?_ hfc.aestronglyMeasurable
    refine ContinuousOn.aestronglyMeasurable ?_ measurableSet_Ioi
    refine continuousOn_of_forall_continuousAt fun t ht => ?_
    exact continuousAt_ofReal_cpow_const _ _ (Or.inr <| ne_of_gt ht)
  have h2 : IntegrableOn (F s) (Ioi (0 : ℝ)) := by exact mellinConvergent_of_isBigO_rpow hfc hf_top hs_top hf_bot hs_bot
  have h3 : AEStronglyMeasurable (F' s) (volume.restrict <| Ioi 0) :=
    by
    apply LocallyIntegrableOn.aestronglyMeasurable
    refine
      hfc.continuousOn_smul isOpen_Ioi.isLocallyClosed ((continuousOn_of_forall_continuousAt fun t ht => ?_).mul ?_)
    · exact continuousAt_ofReal_cpow_const _ _ (Or.inr <| ne_of_gt ht)
    · refine continuous_ofReal.comp_continuousOn ?_
      exact continuousOn_log.mono (subset_compl_singleton_iff.mpr notMem_Ioi_self)
  have h4 : ∀ᵐ t : ℝ ∂volume.restrict (Ioi 0), ∀ z : ℂ, z ∈ Metric.ball s v → ‖F' z t‖ ≤ bound t :=
    by
    refine (ae_restrict_mem measurableSet_Ioi).mono fun t ht z hz => ?_
    simp_rw [F', bound, norm_smul, norm_mul, norm_real, mul_assoc, norm_eq_abs]
    gcongr
    rw [norm_cpow_eq_rpow_re_of_pos ht]
    rcases le_or_gt 1 t with h | h
    · refine le_add_of_le_of_nonneg (rpow_le_rpow_of_exponent_le h ?_) (rpow_nonneg (zero_le_one.trans h) _)
      rw [sub_re, one_re, sub_le_sub_iff_right]
      rw [mem_ball_iff_norm] at hz
      have hz' := (re_le_norm _).trans hz.le
      rwa [sub_re, sub_le_iff_le_add'] at hz'
    · refine le_add_of_nonneg_of_le (rpow_pos_of_pos ht _).le (rpow_le_rpow_of_exponent_ge ht h.le ?_)
      rw [sub_re, one_re, sub_le_iff_le_add, sub_add_cancel]
      rw [mem_ball_iff_norm'] at hz
      have hz' := (re_le_norm _).trans hz.le
      rwa [sub_re, sub_le_iff_le_add, ← sub_le_iff_le_add'] at hz'
  have h5 : IntegrableOn bound (Ioi 0) := by
    simp_rw [bound, add_mul, mul_assoc]
    suffices ∀ {j : ℝ}, b < j → j < a → IntegrableOn (fun t : ℝ => t ^ (j - 1) * (|log t| * ‖f t‖)) (Ioi 0) volume
      by
      refine Integrable.add (this ?_ ?_) (this ?_ ?_)
      all_goals linarith
    · intro j hj hj'
      obtain ⟨w, hw1, hw2⟩ := exists_between hj
      obtain ⟨w', hw1', hw2'⟩ := exists_between hj'
      refine mellin_convergent_of_isBigO_scalar ?_ ?_ hw1' ?_ hw2
      · simp_rw [mul_comm]
        refine hfc.norm.mul_continuousOn ?_ isOpen_Ioi.isLocallyClosed
        refine Continuous.comp_continuousOn _root_.continuous_abs (continuousOn_log.mono ?_)
        exact subset_compl_singleton_iff.mpr notMem_Ioi_self
      · refine (isBigO_rpow_top_log_smul hw2' hf_top).norm_left.congr_left fun t ↦ ?_
        simp only [norm_smul, Real.norm_eq_abs]
      · refine (isBigO_rpow_zero_log_smul hw1 hf_bot).norm_left.congr_left fun t ↦ ?_
        simp only [norm_smul, Real.norm_eq_abs]
  have h6 :
    ∀ᵐ t : ℝ ∂volume.restrict (Ioi 0), ∀ y : ℂ, y ∈ Metric.ball s v → HasDerivAt (fun z : ℂ => F z t) (F' y t) y :=
    by
    refine (ae_restrict_mem measurableSet_Ioi).mono fun t ht y _ => ?_
    have ht' : (t : ℂ) ≠ 0 := ofReal_ne_zero.mpr (ne_of_gt ht)
    have u1 : HasDerivAt (fun z : ℂ => (t : ℂ) ^ (z - 1)) (t ^ (y - 1) * log t) y :=
      by
      convert ((hasDerivAt_id' y).sub_const 1).const_cpow (Or.inl ht') using 1
      rw [ofReal_log (le_of_lt ht)]
      ring
    exact u1.smul_const (f t)
  have main := hasDerivAt_integral_of_dominated_loc_of_deriv_le hv0 h1 h2 h3 h4 h5 h6
  simpa only [F', mul_smul] using main

Похожие утверждения