internlm/Lean-Github huggingface dataset:
Theorem: le_CarlesonOperatorReal'
Proof state before:
case e_a.hf
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
  Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
    (𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
measurable₁ : Measurable fun x_1 => cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)
boundedness₁ : ∀ {y : ℝ}, r ≤ dist x y → ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ 2 ^ 2 / (2 * r)
integrable₁ : MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1})
⊢ ∀ᵐ (x_1 : ℝ) ∂MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1},
    ‖cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)‖ ≤ ?m.492969

f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
  Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
    (𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
measurable₁ : Measurable fun x_1 => cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)
boundedness₁ : ∀ {y : ℝ}, r ≤ dist x y → ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ 2 ^ 2 / (2 * r)
integrable₁ : MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1})
⊢ ℝ
Tactic:
. rw [MeasureTheory.ae_restrict_iff' annulus_measurableSet]
  apply eventually_of_forall
  intro y hy
  exact boundedness₁ hy.1.le
Proof state after:
no goals
Theorem: le_CarlesonOperatorReal'
Proof state before:
case e_a.hf
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
  Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
    (𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
measurable₁ : Measurable fun x_1 => cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)
boundedness₁ : ∀ {y : ℝ}, r ≤ dist x y → ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ 2 ^ 2 / (2 * r)
integrable₁ : MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1})
⊢ ∀ᵐ (x_1 : ℝ) ∂MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1},
    ‖cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)‖ ≤ ?m.492969
Tactic:
rw [MeasureTheory.ae_restrict_iff' annulus_measurableSet]
Proof state after:
case e_a.hf
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
  Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
    (𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
measurable₁ : Measurable fun x_1 => cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)
boundedness₁ : ∀ {y : ℝ}, r ≤ dist x y → ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ 2 ^ 2 / (2 * r)
integrable₁ : MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1})
⊢ ∀ᵐ (x_1 : ℝ),
    x_1 ∈ {y | dist x y ∈ Set.Ioo r 1} → ‖cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)‖ ≤ ?m.492969

f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
  Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
    (𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
measurable₁ : Measurable fun x_1 => cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)
boundedness₁ : ∀ {y : ℝ}, r ≤ dist x y → ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ 2 ^ 2 / (2 * r)
integrable₁ : MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1})
⊢ ℝ
Theorem: le_CarlesonOperatorReal'
Proof state before:
case e_a.hf.hp
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
  Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
    (𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
measurable₁ : Measurable fun x_1 => cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)
boundedness₁ : ∀ {y : ℝ}, r ≤ dist x y → ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ 2 ^ 2 / (2 * r)
integrable₁ : MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1})
y : ℝ
hy : y ∈ {y | dist x y ∈ Set.Ioo r 1}
⊢ ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ ?m.492969

f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
  Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
    (𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
measurable₁ : Measurable fun x_1 => cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)
boundedness₁ : ∀ {y : ℝ}, r ≤ dist x y → ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ 2 ^ 2 / (2 * r)
integrable₁ : MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1})
⊢ ℝ
Tactic:
exact boundedness₁ hy.1.le
Proof state after:
no goals
Theorem: le_CarlesonOperatorReal'
Proof state before:
case e_a.hg
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
  Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
    (𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
measurable₁ : Measurable fun x_1 => cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)
boundedness₁ : ∀ {y : ℝ}, r ≤ dist x y → ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ 2 ^ 2 / (2 * r)
integrable₁ : MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1})
⊢ MeasureTheory.Integrable (fun y => f y * (starRingEnd ℂ) (cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)))
    (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1})
Tactic:
conv => pattern ((f _) * _); rw [mul_comm]
Proof state after:
case e_a.hg
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
  Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
    (𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
measurable₁ : Measurable fun x_1 => cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)
boundedness₁ : ∀ {y : ℝ}, r ≤ dist x y → ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ 2 ^ 2 / (2 * r)
integrable₁ : MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1})
⊢ MeasureTheory.Integrable (fun y => (starRingEnd ℂ) (cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)) * f y)
    (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1})
Theorem: le_CarlesonOperatorReal'
Proof state before:
case e_a.hg.hf_bound
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
  Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
    (𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
measurable₁ : Measurable fun x_1 => cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)
boundedness₁ : ∀ {y : ℝ}, r ≤ dist x y → ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ 2 ^ 2 / (2 * r)
integrable₁ : MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1})
⊢ ∀ᵐ (x_1 : ℝ) ∂MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1},
    ‖(starRingEnd ℂ) (cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1))‖ ≤ ?m.494150

f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
  Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
    (𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
measurable₁ : Measurable fun x_1 => cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)
boundedness₁ : ∀ {y : ℝ}, r ≤ dist x y → ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ 2 ^ 2 / (2 * r)
integrable₁ : MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1})
⊢ ℝ
Tactic:
. rw [MeasureTheory.ae_restrict_iff' annulus_measurableSet]
  apply eventually_of_forall
  intro y hy
  rw [RCLike.norm_conj]
  exact boundedness₁ hy.1.le
Proof state after:
no goals
Theorem: le_CarlesonOperatorReal'
Proof state before:
case e_a.hg.hf
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
  Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
    (𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
measurable₁ : Measurable fun x_1 => cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)
boundedness₁ : ∀ {y : ℝ}, r ≤ dist x y → ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ 2 ^ 2 / (2 * r)
integrable₁ : MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1})
⊢ MeasureTheory.AEStronglyMeasurable
    (fun x_1 => (starRingEnd ℂ) (cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)))
    (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1})
Tactic:
apply Measurable.aestronglyMeasurable
Proof state after:
case e_a.hg.hf.hf
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
  Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
    (𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
measurable₁ : Measurable fun x_1 => cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)
boundedness₁ : ∀ {y : ℝ}, r ≤ dist x y → ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ 2 ^ 2 / (2 * r)
integrable₁ : MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1})
⊢ Measurable fun x_1 => (starRingEnd ℂ) (cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1))
Theorem: le_CarlesonOperatorReal'
Proof state before:
case e_a.hg.hf.hf
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
  Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
    (𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
measurable₁ : Measurable fun x_1 => cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)
boundedness₁ : ∀ {y : ℝ}, r ≤ dist x y → ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ 2 ^ 2 / (2 * r)
integrable₁ : MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1})
⊢ Measurable fun x_1 => (starRingEnd ℂ) (cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1))
Tactic:
apply Measurable.comp continuous_star.measurable measurable₁
Proof state after:
no goals
Theorem: le_CarlesonOperatorReal'
Proof state before:
case e_a.hg.hf_bound
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
  Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
    (𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
measurable₁ : Measurable fun x_1 => cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)
boundedness₁ : ∀ {y : ℝ}, r ≤ dist x y → ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ 2 ^ 2 / (2 * r)
integrable₁ : MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1})
⊢ ∀ᵐ (x_1 : ℝ) ∂MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1},
    ‖(starRingEnd ℂ) (cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1))‖ ≤ ?m.494150
Tactic:
rw [MeasureTheory.ae_restrict_iff' annulus_measurableSet]
Proof state after:
case e_a.hg.hf_bound
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
  Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
    (𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
measurable₁ : Measurable fun x_1 => cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)
boundedness₁ : ∀ {y : ℝ}, r ≤ dist x y → ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ 2 ^ 2 / (2 * r)
integrable₁ : MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1})
⊢ ∀ᵐ (x_1 : ℝ),
    x_1 ∈ {y | dist x y ∈ Set.Ioo r 1} →
      ‖(starRingEnd ℂ) (cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1))‖ ≤ ?m.494150

f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
  Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
    (𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
measurable₁ : Measurable fun x_1 => cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)
boundedness₁ : ∀ {y : ℝ}, r ≤ dist x y → ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ 2 ^ 2 / (2 * r)
integrable₁ : MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1})
⊢ ℝ
Theorem: le_CarlesonOperatorReal'
Proof state before:
case e_a.hg.hf_bound.hp
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
  Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
    (𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
measurable₁ : Measurable fun x_1 => cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)
boundedness₁ : ∀ {y : ℝ}, r ≤ dist x y → ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ 2 ^ 2 / (2 * r)
integrable₁ : MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1})
y : ℝ
hy : y ∈ {y | dist x y ∈ Set.Ioo r 1}
⊢ ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ ?m.494150

f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
  Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
    (𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
measurable₁ : Measurable fun x_1 => cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)
boundedness₁ : ∀ {y : ℝ}, r ≤ dist x y → ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ 2 ^ 2 / (2 * r)
integrable₁ : MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1})
⊢ ℝ

f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
  Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
    (𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
measurable₁ : Measurable fun x_1 => cexp (I * (-↑n * ↑x)) * K x x_1 * cexp (I * ↑n * ↑x_1)
boundedness₁ : ∀ {y : ℝ}, r ≤ dist x y → ‖cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y)‖ ≤ 2 ^ 2 / (2 * r)
integrable₁ : MeasureTheory.Integrable (fun x => f x) (MeasureTheory.volume.restrict {y | dist x y ∈ Set.Ioo r 1})
⊢ ℝ
Tactic:
exact boundedness₁ hy.1.le
Proof state after:
no goals
Theorem: le_CarlesonOperatorReal'
Proof state before:
f : ℝ → ℂ
hf : IntervalIntegrable f MeasureTheory.volume (-Real.pi) (3 * Real.pi)
N : ℕ
x : ℝ
hx : x ∈ Set.Icc 0 (2 * Real.pi)
s : ℕ → Set ℝ := fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
sdef : s = fun n => {y | dist x y ∈ Set.Ioo (1 / (↑n + 2)) 1}
hs : {y | dist x y ∈ Set.Ioo 0 1} = ⋃ n, s n
this :
  Tendsto (fun i => ∫ (y : ℝ) in s i, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)) atTop
    (𝓝 (∫ (y : ℝ) in ⋃ n, s n, f y * ↑(max (1 - |x - y|) 0) * dirichletKernel' N (x - y)))
n : ℤ
r : ℝ
rpos : 0 < r
rle1 : r < 1
⊢ ‖(∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * (cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y))) +
        ∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1},
          f y * (starRingEnd ℂ) (cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y))‖₊ ≤
    ‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1}, f y * (cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y))‖₊ +
      ‖∫ (y : ℝ) in {y | dist x y ∈ Set.Ioo r 1},
          f y * (starRingEnd ℂ) (cexp (I * (-↑n * ↑x)) * K x y * cexp (I * ↑n * ↑y))‖₊
Tactic:
apply nnnorm_add_le
Proof state after:
no goals