/-- The convolution `f * g` 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`). -/
theorem contDiffOn_convolution_right_with_param {f : G → E} {n : ℕ∞} (L : E →L[𝕜] E' →L[𝕜] F)
    {g : P → G → E'} {s : Set P} {k : Set G} (hs : IsOpen s) (hk : IsCompact k)
    (hgs : ∀ p, ∀ x, p ∈ s → x ∉ k → g p x = 0) (hf : LocallyIntegrable f μ)
    (hg : ContDiffOn 𝕜 n (↿g) (s ×ˢ univ)) :
    ContDiffOn 𝕜 n (fun q : P × G => (f ⋆[L, μ] g q.1) q.2) (s ×ˢ univ) := by
  /- The result is known when all the universes are the same, from
    `contDiffOn_convolution_right_with_param_aux`. We reduce to this situation by pushing
    everything through `ULift` continuous linear equivalences. -/
  let eG : Type max uG uE' uF uP := ULift.{max uE' uF uP} G
  borelize eG
  let eE' : Type max uE' uG uF uP := ULift.{max uG uF uP} E'
  let eF : Type max uF uG uE' uP := ULift.{max uG uE' uP} F
  let eP : Type max uP uG uE' uF := ULift.{max uG uE' uF} P
  have isoG : eG ≃L[𝕜] G := ContinuousLinearEquiv.ulift
  have isoE' : eE' ≃L[𝕜] E' := ContinuousLinearEquiv.ulift
  have isoF : eF ≃L[𝕜] F := ContinuousLinearEquiv.ulift
  have isoP : eP ≃L[𝕜] P := ContinuousLinearEquiv.ulift
  let ef := f ∘ isoG
  let eμ : MeasureTheory.Measure eG := Measure.map isoG.symm μ
  let eg : eP → eG → eE' := fun ep ex => isoE'.symm (g (isoP ep) (isoG ex))
  let eL :=
    ContinuousLinearMap.comp
      ((ContinuousLinearEquiv.arrowCongr isoE' isoF).symm : (E' →L[𝕜] F) →L[𝕜] eE' →L[𝕜] eF) L
  let R := fun q : eP × eG => (ef ⋆[eL, eμ] eg q.1) q.2
  have R_contdiff : ContDiffOn 𝕜 n R ((isoP ⁻¹' s) ×ˢ univ) := by
    have hek : IsCompact (isoG ⁻¹' k) := isoG.toHomeomorph.closedEmbedding.isCompact_preimage hk
    have hes : IsOpen (isoP ⁻¹' s) := isoP.continuous.isOpen_preimage _ hs
    refine' contDiffOn_convolution_right_with_param_aux eL hes hek _ _ _
    · intro p x hp hx
      simp only [(· ∘ ·), ContinuousLinearEquiv.prod_apply, LinearIsometryEquiv.coe_coe,
        ContinuousLinearEquiv.map_eq_zero_iff]
      exact hgs _ _ hp hx
    · apply (locallyIntegrable_map_homeomorph isoG.symm.toHomeomorph).2
      convert hf
      ext1 x
      simp only [ContinuousLinearEquiv.coe_toHomeomorph, (· ∘ ·),
        ContinuousLinearEquiv.apply_symm_apply]
    · apply isoE'.symm.contDiff.comp_contDiffOn
      apply hg.comp (isoP.prod isoG).contDiff.contDiffOn
      rintro ⟨p, x⟩ ⟨hp, -⟩
      simpa only [mem_preimage, ContinuousLinearEquiv.prod_apply, prod_mk_mem_set_prod_eq, mem_univ,
        and_true_iff] using hp
  have A : ContDiffOn 𝕜 n (isoF ∘ R ∘ (isoP.prod isoG).symm) (s ×ˢ univ) := by
    apply isoF.contDiff.comp_contDiffOn
    apply R_contdiff.comp (ContinuousLinearEquiv.contDiff _).contDiffOn
    rintro ⟨p, x⟩ ⟨hp, -⟩
    simpa only [mem_preimage, mem_prod, mem_univ, and_true_iff, ContinuousLinearEquiv.prod_symm,
      ContinuousLinearEquiv.prod_apply, ContinuousLinearEquiv.apply_symm_apply] using hp
  have : isoF ∘ R ∘ (isoP.prod isoG).symm = fun q : P × G => (f ⋆[L, μ] g q.1) q.2 := by
    apply funext
    rintro ⟨p, x⟩
    simp only [LinearIsometryEquiv.coe_coe, (· ∘ ·), ContinuousLinearEquiv.prod_symm,
      ContinuousLinearEquiv.prod_apply]
    simp only [convolution, coe_comp', ContinuousLinearEquiv.coe_coe, (· ∘ ·)]
    

Proof state:
case h.mk
𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
E'' : Type uE''
F : Type uF
F' : Type uF'
F'' : Type uF''
P : Type uP
inst✝¹⁶ : NormedAddCommGroup E
inst✝¹⁵ : NormedAddCommGroup E'
inst✝¹⁴ : NormedAddCommGroup E''
inst✝¹³ : NormedAddCommGroup F
f✝ f' : G → E
g✝ g' : G → E'
x✝ x' : G
y y' : E
inst✝¹² : IsROrC 𝕜
inst✝¹¹ : NormedSpace 𝕜 E
inst✝¹⁰ : NormedSpace 𝕜 E'
inst✝⁹ : NormedSpace 𝕜 E''
inst✝⁸ : NormedSpace ℝ F
inst✝⁷ : NormedSpace 𝕜 F
inst✝⁶ : CompleteSpace F
inst✝⁵ : MeasurableSpace G
inst✝⁴ : NormedAddCommGroup G
inst✝³ : BorelSpace G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : NormedAddCommGroup P
inst✝ : NormedSpace 𝕜 P
μ : Measure G
L✝ : E →L[𝕜] E' →L[𝕜] F
f : G → E
n : ℕ∞
L : E →L[𝕜] E' →L[𝕜] F
g : P → G → E'
s : Set P
k : Set G
hs : IsOpen s
hk : IsCompact k
hgs : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0
hf : LocallyIntegrable f
hg : ContDiffOn 𝕜 n (↿g) (s ×ˢ univ)
eG : Type (max uG uE' uF uP) := ULift.{max uE' uF uP, uG} G
this✝¹ : MeasurableSpace eG := borel eG
this✝ : BorelSpace eG
eE' : Type (max uE' uG uF uP) := ULift.{max uG uF uP, uE'} E'
eF : Type (max uF uG uE' uP) := ULift.{max uG uE' uP, uF} F
eP : Type (max uP uG uE' uF) := ULift.{max uG uE' uF, uP} P
isoG : eG ≃L[𝕜] G
isoE' : eE' ≃L[𝕜] E'
isoF : eF ≃L[𝕜] F
isoP : eP ≃L[𝕜] P
ef : eG → E := f ∘ ⇑isoG
eμ : Measure eG := Measure.map (⇑(ContinuousLinearEquiv.symm isoG)) μ
eg : eP → eG → eE' := fun ep ex => (ContinuousLinearEquiv.symm isoE') (g (isoP ep) (isoG ex))
eL : E →L[𝕜] eE' →L[𝕜] eF :=
  ContinuousLinearMap.comp (↑(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.arrowCongr isoE' isoF))) L
R : eP × eG → eF := fun q => ef ⋆[eL, q.2] eg q.1
R_contdiff : ContDiffOn 𝕜 n R ((⇑isoP ⁻¹' s) ×ˢ univ)
A : ContDiffOn 𝕜 n (⇑isoF ∘ R ∘ ⇑(ContinuousLinearEquiv.symm (ContinuousLinearEquiv.prod isoP isoG))) (s ×ˢ univ)
p : P
x : G
⊢ isoF
      (∫ (t : ULift.{max uE' uF uP, uG} G),
        ((ContinuousLinearEquiv.symm (ContinuousLinearEquiv.arrowCongr isoE' isoF)) (L (f (isoG t))))
          ((ContinuousLinearEquiv.symm isoE')
            (g (isoP ((ContinuousLinearEquiv.symm isoP) p))
              (isoG ((ContinuousLinearEquiv.symm isoG) x - t)))) ∂Measure.map (⇑(ContinuousLinearEquiv.symm isoG)) μ) =
    ∫ (t : G), (L (f t)) (g p (x - t)) ∂μ

Next tactic:
rw [ClosedEmbedding.integral_map, ← isoF.integral_comp_comm]