/-- 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 a subset `s` of
a parameter space `P` (and the compact support `k` is independent of the parameter in `s`). -/
theorem continuousOn_convolution_right_with_param {g : P → G → E'} {s : Set P} {k : Set G}
    (hk : IsCompact k) (hgs : ∀ p, ∀ x, p ∈ s → x ∉ k → g p x = 0)
    (hf : LocallyIntegrable f μ) (hg : ContinuousOn (↿g) (s ×ˢ univ)) :
    ContinuousOn (fun q : P × G => (f ⋆[L, μ] g q.1) q.2) (s ×ˢ univ) := by
  /- First get rid of the case where the space is not locally compact. Then `g` vanishes everywhere
  and the conclusion is trivial. -/
  by_cases H : ∀ p ∈ s, ∀ x, g p x = 0
  · apply (continuousOn_const (c := 0)).congr
    rintro ⟨p, x⟩ ⟨hp, -⟩
    apply integral_eq_zero_of_ae (eventually_of_forall (fun y ↦ ?_))
    simp [H p hp _]
  have : LocallyCompactSpace G := by
    push_neg at H
    rcases H with ⟨p, hp, x, hx⟩
    have A : support (g p) ⊆ k := support_subset_iff'.2 (fun y hy ↦ hgs p y hp hy)
    have B : Continuous (g p) := by
      refine hg.comp_continuous (continuous_const.prod_mk continuous_id') fun x => ?_
      simpa only [prod_mk_mem_set_prod_eq, mem_univ, and_true] using hp
    rcases eq_zero_or_locallyCompactSpace_of_support_subset_isCompact_of_addGroup hk A B with H|H
    · simp [H] at hx
    · exact H
  /- Since `G` is locally compact, one may thicken `k` a little bit into a larger compact set
  `(-k) + t`, outside of which all functions that appear in the convolution vanish. Then we can
  apply a continuity statement for integrals depending on a parameter, with respect to
  locally integrable functions and compactly supported continuous functions. -/
  rintro ⟨q₀, x₀⟩ ⟨hq₀, -⟩
  obtain ⟨t, t_comp, ht⟩ : ∃ t, IsCompact t ∧ t ∈ 𝓝 x₀ := exists_compact_mem_nhds x₀
  let k' : Set G := (-k) +ᵥ t
  have k'_comp : IsCompact k' := IsCompact.vadd_set hk.neg t_comp
  let g' : (P × G) → G → E' := fun p x ↦ g p.1 (p.2 - x)
  let s' : Set (P × G) := s ×ˢ t
  have A : ContinuousOn g'.uncurry (s' ×ˢ univ) := by
    have : g'.uncurry = g.uncurry ∘ (fun w ↦ (w.1.1, w.1.2 - w.2)) := by ext y; rfl
    rw [this]
    refine hg.comp (continuous_fst.fst.prod_mk (continuous_fst.snd.sub
      continuous_snd)).continuousOn ?_
    simp (config := {contextual := true}) [MapsTo]
  have B : ContinuousOn (fun a ↦ ∫ x, L (f x) (g' a x) ∂μ) s' := by
    apply continuousOn_integral_bilinear_of_locally_integrable_of_compact_support L k'_comp A _
      (hf.integrableOn_isCompact k'_comp)
    rintro ⟨p, x⟩ y ⟨hp, hx⟩ hy
    apply hgs p _ hp
    contrapose! hy
    exact ⟨y - x, x, by simpa using hy, hx, by simp⟩
  apply ContinuousWithinAt.mono_of_mem (B (q₀, x₀) ⟨hq₀, mem_of_mem_nhds ht⟩)
  

Proof state:
case neg.mk.intro.intro.intro
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✝¹⁵ : NormedAddCommGroup E
inst✝¹⁴ : NormedAddCommGroup E'
inst✝¹³ : NormedAddCommGroup E''
inst✝¹² : NormedAddCommGroup F
f f' : G → E
g✝ g'✝ : G → E'
x x' : G
y y' : E
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : NormedSpace 𝕜 E'
inst✝⁸ : NormedSpace 𝕜 E''
inst✝⁷ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
inst✝⁶ : MeasurableSpace G
μ ν : Measure G
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : AddGroup G
inst✝³ : TopologicalSpace G
inst✝² : TopologicalAddGroup G
inst✝¹ : BorelSpace G
inst✝ : TopologicalSpace P
g : P → G → E'
s : Set P
k : Set G
hk : IsCompact k
hgs : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0
hf : LocallyIntegrable f
hg : ContinuousOn (↿g) (s ×ˢ univ)
H : ¬∀ p ∈ s, ∀ (x : G), g p x = 0
this : LocallyCompactSpace G
q₀ : P
x₀ : G
hq₀ : (q₀, x₀).1 ∈ s
t : Set G
t_comp : IsCompact t
ht : t ∈ 𝓝 x₀
k' : Set G := -k +ᵥ t
k'_comp : IsCompact k'
g' : P × G → G → E' := fun p x => g p.1 (p.2 - x)
s' : Set (P × G) := s ×ˢ t
A : ContinuousOn (uncurry g') (s' ×ˢ univ)
B : ContinuousOn (fun a => ∫ (x : G), (L (f x)) (g' a x) ∂μ) s'
⊢ s' ∈ 𝓝[s ×ˢ univ] (q₀, x₀)

Next tactic:
exact mem_nhdsWithin_prod_iff.2 ⟨s, self_mem_nhdsWithin, t, nhdsWithin_le_nhds ht, Subset.rfl⟩