Theorems · Theorem · functional analysis

MeasureTheory.eLpNorm_le_eLpNorm_fderiv_of_le

∀ {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] [inst_8 : FiniteDimensional ℝ F] {u : E → F} {s : Set E},
  ContDiff ℝ 1 u →
    Function.support u ⊆ s →
      ∀ {p q : NNReal},
        1 ≤ p →
          p < ↑(Module.finrank ℝ E) →
            ↑p⁻¹ - (↑(Module.finrank ℝ E))⁻¹ ≤ (↑q)⁻¹ →
              Bornology.IsBounded s →
                MeasureTheory.eLpNorm u (↑q) μ ≤
                  ↑(MeasureTheory.eLpNormLESNormFDerivOfLeConst F μ s p q) * MeasureTheory.eLpNorm (fderiv ℝ u) (↑p) μ

The Gagliardo-Nirenberg-Sobolev inequality. Let u be a continuously differentiable function u supported in a bounded set s in a normed space E of finite dimension n, equipped with Haar measure, and let 1 < p < n and 0 < q ≤ (p⁻¹ - (finrank ℝ E : ℝ)⁻¹)⁻¹. Then the L^q norm of u is bounded above by a constant times the Lᵖ norm of the Fréchet derivative of u. Note: The codomain of u needs to be a finite-dimensional normed space.

Defined in

Mathlib.Analysis.FunctionalSpaces.SobolevInequality

Cited by

1 results in Mathlib

Foundations

Depth 289 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceMeasurableSpaceBorelSpaceFiniteDimensionalMeasureTheory.Measure.IsAddHaarMeasureFiniteDimensional