Project-declaredLean 4.32.0 · mathlib@81a5d257c8e4

Marcinkiewicz zygmund

Real.marcinkiewicz_zygmund

additive combinatorics arithmetic progressions Fourier analysis marcinkiewicz zygmund

Plain-language statement

The Marcinkiewicz-Zygmund inequality for real-valued functions, with a slightly easier to bound constant than Real.marcinkiewicz_zygmund'. Note that RCLike.marcinkiewicz_zygmund is another version that works for both ℝ and ℂ at the expense of a slightly worse constant.

Exact Lean statement

theorem marcinkiewicz_zygmund (hm : m ≠ 0) (f : ι → ℝ) (hf : ∀ i, ∑ a ∈ A ^^ n, f (a i) = 0) :
    ∑ a ∈ A ^^ n, (∑ i, f (a i)) ^ (2 * m) ≤
      (4 * m) ^ m * n ^ (m - 1) * ∑ a ∈ A ^^ n, ∑ i, f (a i) ^ (2 * m)

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

theorem marcinkiewicz_zygmund (hm : m ≠ 0) (f : ι → ℝ) (hf : ∀ i, ∑ a ∈ A ^^ n, f (a i) = 0) :    ∑ a ∈ A ^^ n, (∑ i, f (a i)) ^ (2 * m) ≤      (4 * m) ^ m * n ^ (m - 1) * ∑ a ∈ A ^^ n, ∑ i, f (a i) ^ (2 * m) := by  obtain _ | m := m  · simp at hm  obtain rfl | hn := n.eq_zero_or_pos  · simp  calc    ∑ a ∈ A ^^ n, (∑ i, f (a i)) ^ (2 * (m + 1))      ≤ (4 * ↑(m + 1)) ^ (m + 1) * ∑ a ∈ A ^^ n, (∑ i, f (a i) ^ 2) ^ (m + 1) :=      marcinkiewicz_zygmund' _ f hf    _ ≤ (4 * ↑(m + 1)) ^ (m + 1) * (∑ a ∈ A ^^ n, n ^ m * ∑ i, f (a i) ^ (2 * (m + 1))) := ?_    _ ≤ (4 * ↑(m + 1)) ^ (m + 1) * n ^ m * ∑ a ∈ A ^^ n, ∑ i, f (a i) ^ (2 * (m + 1)) := by      simp_rw [mul_assoc, mul_sum]; rfl  gcongr with a  rw [← div_le_iff₀' (by positivity)]  simpa only [Finset.card_fin, pow_mul] using    pow_sum_div_card_le_sum_pow (f := fun i ↦ f (a i) ^ 2) (s := univ) (fun i _ ↦ by positivity) m

Project: Arithmetic Progressions Almost Periodicity
License: Apache-2.0
Commit: afafc42a5326
Source: APAP/Prereqs/MarcinkiewiczZygmund.lean:228-245

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