Theorems · Theorem · integral transforms

MeasureTheory.contDiffOn_convolution_left_with_param_comp

∀ {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} {P : Type uP} [inst : NormedAddCommGroup E]
  [inst_1 : NormedAddCommGroup E'] [inst_2 : NormedAddCommGroup F] [inst_3 : RCLike 𝕜] [inst_4 : NormedSpace 𝕜 E]
  [inst_5 : NormedSpace 𝕜 E'] [inst_6 : NormedSpace ℝ F] [inst_7 : NormedSpace 𝕜 F] [inst_8 : MeasurableSpace G]
  [inst_9 : NormedAddCommGroup G] [BorelSpace G] [inst_11 : NormedSpace 𝕜 G] [inst_12 : NormedAddCommGroup P]
  [inst_13 : NormedSpace 𝕜 P] {μ : MeasureTheory.Measure G} [μ.IsAddLeftInvariant] [μ.IsNegInvariant]
  (L : E' →L[𝕜] E →L[𝕜] F) {s : Set P} {n : ℕ∞} {v : P → G},
  ContDiffOn 𝕜 (↑n) v s →
    ∀ {f : G → E} {g : P → G → E'} {k : Set G},
      IsOpen s →
        IsCompact k →
          (∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0) →
            MeasureTheory.LocallyIntegrable f μ →
              ContDiffOn 𝕜 (↑n) (↿g) (s ×ˢ Set.univ) →
                ContDiffOn 𝕜 (↑n) (fun x => MeasureTheory.convolution (g x) f L μ (v x)) s

The convolution g * f is C^n when f is locally integrable and g is C^n and compactly supported. Version where g depends on an additional parameter in an open subset s of a parameter space P (and the compact support k is independent of the parameter in s), given in terms of composition with additional C^n functions.

Defined in

Mathlib.Analysis.Calculus.ContDiff.Convolution

Cited by

0 results in Mathlib

Foundations

Depth 268 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedAddCommGroupNormedAddCommGroupRCLikeNormedSpaceNormedSpaceNormedSpaceNormedSpaceMeasurableSpaceNormedAddCommGroupBorelSpaceNormedSpaceNormedAddCommGroupNormedSpaceMeasureTheory.Measure.IsAddLeftInvariantMeasureTheory.Measure.IsNegInvariant