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namespace OAI
/-! Local L2 restriction and bounded varying multipliers. -/
namespace BFSSLocalL2
noncomputable section
open MeasureTheory Set Filter Topology
variable {E F : Type*} [MeasurableSpace E]
[NormedAddCommGroup F] [NormedSpace ℝ F]
variable (μ : Measure E) (s : Set E)
def restrictLinear : Lp F 2 μ →ₗ[ℝ] Lp F 2 (μ.restrict s) where
toFun f := ((Lp.memLp f).restrict s).toLp f
map_add' f g := by
apply Lp.ext
filter_upwards [MemLp.coeFn_toLp ((Lp.memLp (f+g)).restrict s),
MemLp.coeFn_toLp ((Lp.memLp f).restrict s), MemLp.coeFn_toLp ((Lp.memLp g).restrict s),
Lp.coeFn_add (((Lp.memLp f).restrict s).toLp f) (((Lp.memLp g).restrict s).toLp g),
ae_restrict_of_ae (Lp.coeFn_add f g)] with x hfg hf hg ha hb
simp only [Pi.add_apply,hf,hg] at ha hb
exact hfg.trans (hb.trans ha.symm)
map_smul' c f := by
apply Lp.ext
filter_upwards [MemLp.coeFn_toLp ((Lp.memLp (c • f)).restrict s),
MemLp.coeFn_toLp ((Lp.memLp f).restrict s),
Lp.coeFn_smul c (((Lp.memLp f).restrict s).toLp f),
ae_restrict_of_ae (Lp.coeFn_smul c f)] with x hcf hf ha hb
simp only [Pi.smul_apply,hf] at ha hb
exact hcf.trans (hb.trans ha.symm)
lemma restrictLinear_norm_le (f : Lp F 2 μ) : ‖restrictLinear μ s f‖ ≤ ‖f‖ := by
change ‖((Lp.memLp f).restrict s).toLp f‖ ≤ ‖f‖
rw [Lp.norm_toLp,Lp.norm_def]
exact ENNReal.toReal_mono (Lp.eLpNorm_ne_top f) (eLpNorm_mono_measure _ (Measure.restrict_le_self))
def restrict : Lp F 2 μ →L[ℝ] Lp F 2 (μ.restrict s) :=
(restrictLinear μ s).mkContinuous 1 (fun f => by simpa using restrictLinear_norm_le μ s f)
lemma restrict_ae (f : Lp F 2 μ) : (restrict μ s f : E → F) =ᵐ[μ.restrict s] f :=
MemLp.coeFn_toLp ((Lp.memLp f).restrict s)
variable {μ s}
omit [NormedSpace ℝ F] in
lemma norm_le_of_ae_norm_le (f g : Lp F 2 μ) (h : ∀ᵐ x ∂μ, ‖f x‖ ≤ ‖g x‖) : ‖f‖ ≤ ‖g‖ := by
rw [Lp.norm_def,Lp.norm_def]
exact ENNReal.toReal_mono (Lp.eLpNorm_ne_top g)
(eLpNorm_mono_ae (Lp.memLp f).aestronglyMeasurable h)
lemma norm_le_mul_of_ae_norm_le (f g : Lp F 2 μ) (c : ℝ) (hc : 0 ≤ c)
(h : ∀ᵐ x ∂μ, ‖f x‖ ≤ c*‖g x‖) : ‖f‖ ≤ c*‖g‖ := by
rw [←abs_of_nonneg hc,←Real.norm_eq_abs,←norm_smul]
apply norm_le_of_ae_norm_le
filter_upwards [h,Lp.coeFn_smul c g] with x hx hg
rw [hg,Pi.smul_apply,norm_smul,Real.norm_eq_abs,abs_of_nonneg hc]
exact hx
end
end BFSSLocalL2
namespace BFSSLocalL2
noncomputable section
open MeasureTheory Set Filter Topology
variable {E F : Type*} [MeasurableSpace E]
[NormedAddCommGroup F] [InnerProductSpace ℝ F] {μ : Measure E}
lemma boundedMul_mem (w : E → ℝ) (hw : AEStronglyMeasurable w μ)
(B : ℝ) (hB : ∀ᵐ x ∂μ, |w x| ≤ B) (f : Lp F 2 μ) :
MemLp (fun x => w x • f x) 2 μ := by
apply (Lp.memLp f).of_le_mul (hw.smul (Lp.memLp f).aestronglyMeasurable)
filter_upwards [hB] with x hx
change ‖w x • f x‖ ≤ B*‖f x‖
rw [norm_smul,Real.norm_eq_abs]
exact mul_le_mul_of_nonneg_right hx (norm_nonneg _)
def boundedMul (w : E → ℝ) (hw : AEStronglyMeasurable w μ)
(B : ℝ) (hB : ∀ᵐ x ∂μ, |w x| ≤ B) (f : Lp F 2 μ) : Lp F 2 μ :=
(boundedMul_mem w hw B hB f).toLp (fun x => w x • f x)
lemma boundedMul_ae (w : E → ℝ) (hw : AEStronglyMeasurable w μ)
(B : ℝ) (hB : ∀ᵐ x ∂μ, |w x| ≤ B) (f : Lp F 2 μ) :
(boundedMul w hw B hB f : E → F) =ᵐ[μ] fun x => w x • f x :=
MemLp.coeFn_toLp _
attribute [local irreducible] boundedMul
lemma boundedMul_dist_le (w : E → ℝ) (hw : AEStronglyMeasurable w μ)
(B : ℝ) (hB0 : 0 ≤ B) (hB : ∀ᵐ x ∂μ, |w x| ≤ B) (f g : Lp F 2 μ) :
dist (boundedMul w hw B hB f) (boundedMul w hw B hB g) ≤ B*dist f g := by
simp only [dist_eq_norm]
apply norm_le_mul_of_ae_norm_le _ (f-g) B hB0
filter_upwards [boundedMul_ae w hw B hB f, boundedMul_ae w hw B hB g,
Lp.coeFn_sub (boundedMul w hw B hB f) (boundedMul w hw B hB g),
Lp.coeFn_sub f g,hB] with x hf hg hs hfg hx
rw [hs,Pi.sub_apply,hf,hg,←smul_sub,hfg,Pi.sub_apply,norm_smul,Real.norm_eq_abs]
exact mul_le_mul_of_nonneg_right hx (norm_nonneg _)
lemma lp_norm_sq (f : Lp F 2 μ) : ‖f‖^2 = ∫ x, ‖f x‖^2 ∂μ := by
rw [←real_inner_self_eq_norm_sq,L2.inner_def]
simp only [real_inner_self_eq_norm_sq]
theorem boundedMul_fixed_tendsto (w : ℕ → E → ℝ) (hw : ∀ n, AEStronglyMeasurable (w n) μ)
(B : ℝ) (_hB0 : 0 ≤ B) (hB : ∀ n, ∀ᵐ x ∂μ, |w n x| ≤ B)
(hlim : ∀ᵐ x ∂μ, Tendsto (fun n => w n x) atTop (𝓝 1)) (u : Lp F 2 μ) :
Tendsto (fun n => boundedMul (w n) (hw n) B (hB n) u) atTop (𝓝 u) := by
have ht : Tendsto (fun n => ∫ x, ‖(w n x-1) • u x‖^2 ∂μ) atTop (𝓝 0) := by
have hu := (Lp.memLp u).integrable_norm_pow (by decide : 2 ≠ 0)
have hd := tendsto_integral_of_dominated_convergence
(F := fun n x => ‖(w n x-1) • u x‖^2) (f := fun _ => (0 : ℝ))
(fun x => (B+1)^2*‖u x‖^2)
(fun n => (((hw n).sub aestronglyMeasurable_const).smul
(Lp.memLp u).aestronglyMeasurable).norm.pow 2)
(hu.const_mul ((B+1)^2)) ?_ ?_
· simpa only [integral_zero] using hd
· intro n
filter_upwards [hB n] with x hx
rw [Real.norm_eq_abs,abs_of_nonneg (sq_nonneg _),norm_smul,Real.norm_eq_abs,mul_pow]
have hh : |w n x-1| ≤ B+1 := by
calc
|w n x-1| ≤ |w n x|+1 := by simpa only [abs_one] using abs_sub (w n x) 1
_ ≤ B+1 := by linarith
exact mul_le_mul_of_nonneg_right (pow_le_pow_left₀ (abs_nonneg _) hh 2) (sq_nonneg _)
· filter_upwards [hlim] with x hx
have hh := ((hx.sub (tendsto_const_nhds (x := (1 : ℝ)))).smul
(tendsto_const_nhds (x := u x))).norm.pow 2
simpa only [sub_self,zero_smul,norm_zero,zero_pow (by decide : 2 ≠ 0)] using hh
have he (n : ℕ) : dist (boundedMul (w n) (hw n) B (hB n) u) u ^2 =
∫ x, ‖(w n x-1) • u x‖^2 ∂μ := by
rw [dist_eq_norm,lp_norm_sq]
apply integral_congr_ae
filter_upwards [boundedMul_ae (w n) (hw n) B (hB n) u,
Lp.coeFn_sub (boundedMul (w n) (hw n) B (hB n) u) u] with x hx hs
rw [hs,Pi.sub_apply,hx,sub_smul,one_smul]
rw [tendsto_iff_dist_tendsto_zero]
have hsq : Tendsto (fun n => dist (boundedMul (w n) (hw n) B (hB n) u) u ^2)
atTop (𝓝 0) := by simpa only [he] using ht
have hr := hsq.sqrt
simpa only [Real.sqrt_sq_eq_abs,abs_of_nonneg dist_nonneg,Real.sqrt_zero] using hr
theorem boundedMul_tendsto (w : ℕ → E → ℝ) (hw : ∀ n, AEStronglyMeasurable (w n) μ)
(B : ℝ) (hB0 : 0 ≤ B) (hB : ∀ n, ∀ᵐ x ∂μ, |w n x| ≤ B)
(hlim : ∀ᵐ x ∂μ, Tendsto (fun n => w n x) atTop (𝓝 1))
(f : ℕ → Lp F 2 μ) (u : Lp F 2 μ) (hf : Tendsto f atTop (𝓝 u)) :
Tendsto (fun n => boundedMul (w n) (hw n) B (hB n) (f n)) atTop (𝓝 u) := by
let g : ℕ → Lp F 2 μ := fun n => boundedMul (w n) (hw n) B (hB n) u
let k : ℕ → Lp F 2 μ := fun n => boundedMul (w n) (hw n) B (hB n) (f n)
have hfixed : Tendsto g atTop (𝓝 u) := boundedMul_fixed_tendsto w hw B hB0 hB hlim u
have hsmall : Tendsto (fun n => B*dist (f n) u) atTop (𝓝 0) := by
simpa only [dist_self,mul_zero] using (tendsto_const_nhds (x := B)).mul (hf.dist (tendsto_const_nhds (x := u)))
have hdist (n : ℕ) : dist (k n) u ≤ B*dist (f n) u + dist (g n) u := by
have hl : dist (k n) (g n) ≤ B*dist (f n) u :=
boundedMul_dist_le (w n) (hw n) B hB0 (hB n) (f n) u
exact (dist_triangle (k n) (g n) u).trans (add_le_add hl (le_refl (dist (g n) u)))
have hsum : Tendsto (fun n => B*dist (f n) u + dist (g n) u) atTop (𝓝 0) := by
simpa only [dist_self,zero_add] using hsmall.add (hfixed.dist (tendsto_const_nhds (x := u)))
exact tendsto_iff_dist_tendsto_zero.mpr (squeeze_zero (fun _ => dist_nonneg) hdist hsum)
end
end BFSSLocalL2
end OAI
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