Theorems · Theorem · functional analysis

TemperedDistribution.MemSobolev.fourier_memL1

∀ {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},
  ↑(Module.finrank ℝ E) < 2 * s →
    ∀ {f : TemperedDistribution E F},
      TemperedDistribution.MemSobolev s 2 f →
        ∃ v, FourierTransform.fourier f = MeasureTheory.Lp.toTemperedDistribution v

The Fourier transform of a Sobolev function of order s with s > d / 2 can be represented by a L1 function. This is the main calculation of the Sobolev embedding theorem.

Defined in

Mathlib.Analysis.Distribution.Sobolev

Cited by

0 results in Mathlib

Foundations

Depth 314 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedAddCommGroupInnerProductSpaceFiniteDimensionalMeasurableSpaceBorelSpaceInnerProductSpaceCompleteSpace