Theorems · Theorem · functional analysis

TemperedDistribution.memSobolev_iff_exists_smulLeftCLM_fourier

∀ {E : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : NormedAddCommGroup F]
  [inst_2 : InnerProductSpace ℝ E] [inst_3 : FiniteDimensional ℝ E] [inst_4 : MeasurableSpace E] [inst_5 : BorelSpace E]
  [inst_6 : InnerProductSpace ℂ F] [inst_7 : CompleteSpace F] {s : ℝ} {f : TemperedDistribution E F},
  TemperedDistribution.MemSobolev s 2 f ↔
    ∃ f',
      (TemperedDistribution.smulLeftCLM F fun x => ↑((1 + ‖x‖ ^ 2) ^ (s / 2))) (FourierTransform.fourier f) =
        MeasureTheory.Lp.toTemperedDistribution f'

A tempered distribution belongs to the Sobolev space of order s and p = 2 if and only if its Fourier transform multiplied by (1 + ‖x‖ ^ 2) ^ (s / 2) is in Lp.

Defined in

Mathlib.Analysis.Distribution.Sobolev

Cited by

2 results in Mathlib

Foundations

Depth 313 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedAddCommGroupInnerProductSpaceFiniteDimensionalMeasurableSpaceBorelSpaceInnerProductSpaceCompleteSpace