Project-declaredLean 4.32.0 · mathlib@81a5d257c8e4

D Lp Norm prod le

MeasureTheory.dLpNorm_prod_le

additive combinatorics arithmetic progressions Fourier analysis discrete

Plain-language statement

Hölder's inequality, finitary case.

Exact Lean statement

lemma dLpNorm_prod_le {ι : Type*} {s : Finset ι} (hs : s.Nonempty) {p : ι → ℝ≥0} (hp : ∀ i, p i ≠ 0)
    (q : ℝ≥0) (hpq : ∑ i ∈ s, ((p i)⁻¹ : ℝ≥0∞) = (q : ℝ≥0∞)⁻¹) (f : ι → α → 𝕜) :
    ‖∏ i ∈ s, f i‖_[q] ≤ ∏ i ∈ s, ‖f i‖_[p i]

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

lemma dLpNorm_prod_le {ι : Type*} {s : Finset ι} (hs : s.Nonempty) {p : ι → ℝ≥0} (hp : ∀ i, p i ≠ 0)    (q : ℝ≥0) (hpq : ∑ i ∈ s, ((p i)⁻¹ : ℝ≥0∞) = (q : ℝ≥0∞)⁻¹) (f : ι → α → 𝕜) :    ‖∏ i ∈ s, f i‖_[q] ≤ ∏ i ∈ s, ‖f i‖_[p i] := by  induction s using Finset.cons_induction generalizing q with  | empty => cases not_nonempty_empty hs  | cons i s hi ih =>    obtain rfl | hs := s.eq_empty_or_nonempty    · simp only [sum_cons, sum_empty, add_zero, inv_inj] at hpq      simp [← hpq]    simp_rw [prod_cons]    rw [sum_cons, ← inv_inv (∑ _ ∈ _, _)] at hpq    have : ENNReal.HolderTriple (p i) ↑(∑ i ∈ s, (p i)⁻¹)⁻¹ q := ⟨by      simpa [ENNReal.coe_inv (by simpa [hp] :        (∑ j ∈ s, (p j)⁻¹ : ℝ≥0) ≠ 0), ENNReal.coe_inv, hp] using hpq⟩    grw [dLpNorm_mul_le (p i) ↑(∑ i ∈ s, (p i)⁻¹)⁻¹, ih hs]    rw [← ENNReal.coe_inv, inv_inv]    · push_cast      congr! with i      exact (ENNReal.coe_inv <| hp _).symm    · simpa [hp]

Project: Arithmetic Progressions Almost Periodicity
License: Apache-2.0
Commit: afafc42a5326
Source: APAP/Prereqs/Inner/Hoelder/Discrete.lean:112-131

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Project-declaredLean 4.32.0

Add Dissociated boring Energy le

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Project documentation

If a finite set $s$ is additively dissociated, then its order- $n$ additive energy is at most $C^n n^n |s|^n$ , where $C$ is the project's Chang constant. This is the quantitative dissociated-set estimate used in the proof of Chang's lemma.

additive combinatoricsarithmetic progressionsFourier analysis

Source project: Arithmetic Progressions Almost Periodicity

Person-level attribution pending.

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Project-declaredLean 4.32.0

Linfty almost periodicity

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Project documentation

An $L^\infty$ almost-periodicity theorem. Under the small-growth hypothesis $\sigma[A,S]\le K$ , and for nonempty finite sets $B,C$ , there is a set of translations $T$ with $|T|\ge K^{-4096\lceil\mathcal L(|C|/|B|)\rceil/\varepsilon^2}|S|$ . Every $t\in T$ changes the normalized convolution $\mu_A*1_B*\mu_C$ by at most $\varepsilon$ in $L^\infty$ .

additive combinatoricsarithmetic progressionsFourier analysis

Source project: Arithmetic Progressions Almost Periodicity

Person-level attribution pending.

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Project-declaredLean 4.32.0

Linfty almost periodicity boosted

AlmostPeriodicity.linfty_almost_periodicity_boosted

Plain-language statement

A boosted $L^\infty$ almost-periodicity estimate. Under $\sigma[A,S]\le K$ , it finds a large set $T$ , with the stated lower bound $|T|\ge K^{-4096\lceil\mathcal L(|C|/|B|)\rceil k^2/\varepsilon^2}|S|$ , such that averaging the target convolution against the $k$ -fold convolution of $\mu_T$ changes it by at most $\varepsilon$ in $L^\infty$ .

additive combinatoricsarithmetic progressionsFourier analysis

Source project: Arithmetic Progressions Almost Periodicity

Person-level attribution pending.

View proof record