Theorems · Theorem · abstract harmonic analysis

MeasureTheory.continuousOn_convolution_right_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] {f : G → E} [inst_3 : NontriviallyNormedField 𝕜]
  [inst_4 : NormedSpace 𝕜 E] [inst_5 : NormedSpace 𝕜 E'] [inst_6 : NormedSpace 𝕜 F] (L : E →L[𝕜] E' →L[𝕜] F)
  [inst_7 : MeasurableSpace G] {μ : MeasureTheory.Measure G} [inst_8 : NormedSpace ℝ F] [inst_9 : AddGroup G]
  [inst_10 : TopologicalSpace G] [IsTopologicalAddGroup G] [BorelSpace G] [inst_13 : TopologicalSpace P] {s : Set P}
  {v : P → G},
  ContinuousOn v s →
    ∀ {g : P → G → E'} {k : Set G},
      IsCompact k →
        (∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0) →
          MeasureTheory.LocallyIntegrable f μ →
            ContinuousOn (↿g) (s ×ˢ Set.univ) → ContinuousOn (fun x => MeasureTheory.convolution f (g x) L μ (v x)) s

The convolution f * g is continuous if f is locally integrable and g is continuous 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 compositions with an additional continuous map.

Defined in

Mathlib.Analysis.Convolution

Cited by

1 results in Mathlib

Foundations

Depth 262 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedAddCommGroupNormedAddCommGroupNontriviallyNormedFieldNormedSpaceNormedSpaceNormedSpaceMeasurableSpaceNormedSpaceAddGroupTopologicalSpaceIsTopologicalAddGroupBorelSpaceTopologicalSpace