Project-declaredLean 4.32.0 · mathlib@81a5d257c8e4

Lipschitz On With of i Lip ENorm ne top

LipschitzOnWith.of_iLipENorm_ne_top

harmonic analysis Fourier analysis measure theory lipschitz norm

Plain-language statement

If the project’s inhomogeneous Lipschitz norm of $\varphi$ on the ball $B(z,R)$ is finite, then $\varphi$ is Lipschitz on that ball. A valid Lipschitz constant is the finite normalized norm divided by $R$ .

Exact Lean statement

lemma LipschitzOnWith.of_iLipENorm_ne_top (hφ : iLipENorm φ z R ≠ ⊤) :
    LipschitzOnWith (iLipNNNorm φ z R / R.toNNReal) φ (ball z R)

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

lemma LipschitzOnWith.of_iLipENorm_ne_top (hφ : iLipENorm φ z R ≠ ⊤) :    LipschitzOnWith (iLipNNNorm φ z R / R.toNNReal) φ (ball z R) := by  intro x hx y hy  have hR : 0 < R := by    simp only [mem_ball] at hx    apply dist_nonneg.trans_lt hx  rcases eq_or_ne x y with rfl | hne  · simp  have : (ENNReal.ofReal R) * (‖φ x - φ y‖ₑ / edist x y) ≤ iLipENorm φ z R := calc      (ENNReal.ofReal R) * (‖φ x - φ y‖ₑ / (edist x y))    _ ≤ (ENNReal.ofReal R) *        ⨆ (x ∈ ball z R) (y ∈ ball z R) (_ : x ≠ y), (‖φ x - φ y‖ₑ / edist x y) := by      gcongr      simp only [ne_eq, le_iSup_iff, iSup_le_iff]      tauto    _ ≤ _ := le_add_self  rw [edist_eq_enorm_sub, ENNReal.coe_div (by simp [hR]), iLipNNNorm, coe_toNNReal hφ]  rw [← ENNReal.div_le_iff_le_mul]; rotate_left  · have : edist x y ≠ 0 := by simp [hne]    simp [this]  · simp [edist_ne_top]  rw [ENNReal.le_div_iff_mul_le]; rotate_left  · simp [hR]  · simp  convert this using 1  simp only [ENNReal.ofReal, mul_comm]

Project: Carleson formalization
License: Apache-2.0
Commit: 74ef907d6bdb
Source: Carleson/LipschitzNorm.lean:75-100

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Related declarations

Project-declaredLean 4.32.0

Adjoint Carleson adjoint

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Plain-language statement

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harmonic analysisFourier analysismeasure theory

Source project: Carleson formalization

Person-level attribution pending.

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Project-declaredLean 4.32.0

Ae tendsto zero of distribution le

ae_tendsto_zero_of_distribution_le

Plain-language statement

Suppose that, for every error threshold $\delta>0$ and every measure tolerance $\varepsilon>0$ , one can choose $N_0$ so that the set where $\sup_{N>N_0}\lVert f(x)-F_N(x)\rVert$ exceeds $\delta$ has measure at most $\varepsilon$ . Then $F_N(x)$ converges to $f(x)$ for almost every $x$ .

harmonic analysisFourier analysismeasure theory

Source project: Carleson formalization

Person-level attribution pending.

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Project-declaredLean 4.32.0

Antichain operator

antichain_operator

Plain-language statement

For an antichain $\mathfrak{A}$ of pairwise incomparable tiles, and measurable functions $f$ and $g$ bounded by the indicators of $F$ and $G$ , the pairing of $g$ with the Carleson sum over $\mathfrak{A}$ is controlled by the $L^2$ norms of $f$ and $g$ and by positive powers of the two tile-density parameters. Concretely, the bound is

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harmonic analysisFourier analysismeasure theory

Source project: Carleson formalization

Person-level attribution pending.

View proof record