Theorems · Theorem · functional analysis

MeasureTheory.eLpNorm_le_eLpNorm_fderiv_one

∀ {F : Type u_3} [inst : NormedAddCommGroup F] [inst_1 : NormedSpace ℝ F] {E : Type u_4} [inst_2 : NormedAddCommGroup E]
  [inst_3 : NormedSpace ℝ E] [inst_4 : MeasurableSpace E] [inst_5 : BorelSpace E] [inst_6 : FiniteDimensional ℝ E]
  (μ : MeasureTheory.Measure E) [inst_7 : μ.IsAddHaarMeasure] {u : E → F},
  ContDiff ℝ 1 u →
    HasCompactSupport u →
      ∀ {p : NNReal},
        (↑(Module.finrank ℝ E)).HolderConjugate p →
          MeasureTheory.eLpNorm u (↑p) μ ≤
            ↑(MeasureTheory.eLpNormLESNormFDerivOneConst μ ↑p) * MeasureTheory.eLpNorm (fderiv ℝ u) 1 μ

The Gagliardo-Nirenberg-Sobolev inequality. Let u be a continuously differentiable compactly-supported function u on a normed space E of finite dimension n ≥ 2, equipped with Haar measure. Then the Lᵖ norm of u, where p := n / (n - 1), is bounded above by a constant times the L¹ norm of the Fréchet derivative of u.

Defined in

Mathlib.Analysis.FunctionalSpaces.SobolevInequality

Cited by

1 results in Mathlib

Foundations

Depth 281 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceMeasurableSpaceBorelSpaceFiniteDimensionalMeasureTheory.Measure.IsAddHaarMeasure