Project-declaredLean 4.32.0 · mathlib@81a5d257c8e4

Holder On With of i Hol ENorm ne top

HolderOnWith.of_iHolENorm_ne_top

harmonic analysis Fourier analysis measure theory holder norm

Plain-language statement

If the project’s inhomogeneous $t$ -Hölder norm of $\varphi$ on the ball $B(z,R)$ is finite and $t\ge0$ , then $\varphi$ is $t$ -Hölder on that ball. A valid Hölder constant is the finite normalized norm divided by $R^t$ .

Exact Lean statement

lemma HolderOnWith.of_iHolENorm_ne_top (ht : 0 ≤ t) (hφ : iHolENorm φ z R t ≠ ⊤) :
    HolderOnWith (iHolNNNorm φ z R t / R.toNNReal ^ t) t.toNNReal φ (ball z R)

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

lemma HolderOnWith.of_iHolENorm_ne_top (ht : 0 ≤ t) (hφ : iHolENorm φ z R t ≠ ⊤) :    HolderOnWith (iHolNNNorm φ z R t / R.toNNReal ^ t) t.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) ^ t * (‖φ x - φ y‖ₑ / (edist x y) ^ t) ≤ iHolENorm φ z R t := calc    _ ≤ (ENNReal.ofReal R) ^ t *        ⨆ (x ∈ ball z R) (y ∈ ball z R) (_ : x ≠ y), (‖φ x - φ y‖ₑ / (edist x y) ^ t) := 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]), iHolNNNorm, coe_toNNReal hφ,    ← ENNReal.div_le_iff_le_mul (by simp [hne]) (by simp [edist_ne_top]),    ENNReal.le_div_iff_mul_le (by simp [hR]) (by simp)]  apply this.trans_eq'  rw [ENNReal.coe_rpow_of_ne_zero (by simp [hR]), Real.coe_toNNReal t ht, ENNReal.ofReal, mul_comm]

Project: Carleson formalization
License: Apache-2.0
Commit: 74ef907d6bdb
Source: Carleson/HolderNorm.lean:46-65

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Related declarations

Project-declaredLean 4.32.0

Adjoint Carleson adjoint

adjointCarleson_adjoint

Plain-language statement

adjointCarleson is the adjoint of carlesonOn.

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.

View proof record

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