Theorems · Theorem · integral transforms

MeasureTheory.hasFDerivAt_convolution_right_with_param

∀ {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] {f : G → E} [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} (L : E →L[𝕜] E' →L[𝕜] F)
  {g : P → G → E'} {s : Set P} {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 1 (↿g) (s ×ˢ Set.univ) →
            ∀ (q₀ : P × G),
              q₀.1 ∈ s →
                HasFDerivAt (fun q => MeasureTheory.convolution f (g q.1) L μ q.2)
                  (MeasureTheory.convolution f (fun x => fderiv 𝕜 ↿g (q₀.1, x)) (ContinuousLinearMap.precompR (P × G) L)
                    μ q₀.2)
                  q₀

The derivative of the convolution f * g is given by f * Dg, when f is locally integrable and g is C^1 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).

Defined in

Mathlib.Analysis.Calculus.ContDiff.Convolution

Cited by

1 results in Mathlib

Foundations

Depth 264 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedAddCommGroupNormedAddCommGroupRCLikeNormedSpaceNormedSpaceNormedSpaceNormedSpaceMeasurableSpaceNormedAddCommGroupBorelSpaceNormedSpaceNormedAddCommGroupNormedSpace