Theorems · Theorem · functional analysis

MeasureTheory.Lp.toTemperedDistribution_smul_eq

∀ {E : Type u_3} {F : Type u_4} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [inst_2 : NormedAddCommGroup F]
  [inst_3 : NormedSpace ℂ F] [inst_4 : CompleteSpace F] [inst_5 : MeasurableSpace E] [inst_6 : BorelSpace E]
  {μ : MeasureTheory.Measure E} [hμ : μ.HasTemperateGrowth] {p q r : ENNReal} [inst_7 : p.HolderTriple q r]
  [inst_8 : Fact (1 ≤ q)] [inst_9 : Fact (1 ≤ r)] {g : E → ℂ},
  Function.HasTemperateGrowth g →
    ∀ (hg₂ : MeasureTheory.MemLp g p μ) (f : ↥(MeasureTheory.Lp F q μ)),
      MeasureTheory.Lp.toTemperedDistribution (MeasureTheory.MemLp.toLp g hg₂ • f) =
        (TemperedDistribution.smulLeftCLM F g) (MeasureTheory.Lp.toTemperedDistribution f)

Coercion of the product of two Lp functions to a tempered distribution is equal to the left multiplication if the left factor is a function of temperate growth.

Defined in

Mathlib.Analysis.Distribution.TemperedDistribution

Cited by

2 results in Mathlib

Foundations

Depth 255 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceCompleteSpaceMeasurableSpaceBorelSpaceMeasureTheory.Measure.HasTemperateGrowthENNReal.HolderTripleFactFact