Theorems · Theorem · functional analysis

MeasureTheory.MemLp.of_bilin

∀ {α : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} {m : MeasurableSpace α} [inst : NormedAddCommGroup E]
  [inst_1 : NormedAddCommGroup F] [inst_2 : NormedAddCommGroup G] {μ : MeasureTheory.Measure α} {p q r : ENNReal}
  {f : α → E} {g : α → F} (b : E → F → G) (c : NNReal),
  MeasureTheory.MemLp f p μ →
    MeasureTheory.MemLp g q μ →
      MeasureTheory.AEStronglyMeasurable (fun x => b (f x) (g x)) μ →
        (∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ c * ‖f x‖₊ * ‖g x‖₊) →
          ∀ [hpqr : p.HolderTriple q r], MeasureTheory.MemLp (fun x => b (f x) (g x)) r μ

Defined in

Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp

Cited by

1 results in Mathlib

Foundations

Depth 214 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedAddCommGroupNormedAddCommGroupENNReal.HolderTriple