The Vitali covering lemma for sections of Monge–Ampère solutions #
Let E be a real inner product space of finite dimension n ≥ 1 with an additive Haar measure
μ, and let u be convex on an open set Ω with λ μ ≤ MA_u ≤ Λ μ on Ω, where MA_u is the
Aleksandrov Monge–Ampère measure of u and 0 < λ. Write
S(x, h) = {y ∈ Ω | u y < u x + ℓ (y - x) + h} for the section of u at x with a supporting
slope ℓ and height h, and θ = 1 + C Λ / λ for the engulfing constant of
TauCeti.exists_convexSection_subset_convexSection.
In Caffarelli's regularity theory, sections play the role of balls. This file proves the two properties of balls that a covering argument uses.
- Swallowing. If two sections
S(x, h)andS(y, k)meet andh ≤ 2 k, thenS(x, h) ⊆ S(y, 2 θ² k). Applying the engulfing property atx, atyand at a common pointz, with a supporting slope atzgiven byConvexOn.exists_forall_add_inner_le, givesS(x, h) ⊆ S(z, θ h) ⊆ S(z, 2 θ k) ⊆ S(y, 2 θ² k). - Doubling. For
K ≥ 1,μ(S(x, K h)) ≤ √(C K^n Λ / λ) μ(S(x, h)), from the volume estimatesλ μ(S(x, K h))² ≤ C (K h)^nandhⁿ ≤ C Λ μ(S(x, h))².
Every application of engulfing needs a section of larger height whose closure lies inside Ω.
This localization is a hypothesis here: all sections of height H centred in a set D are
bounded with closure in Ω, and the sections of the family lie in D, with heights small
compared with H. This is the situation of the interior theory of strictly convex solutions,
whose sections of small height centred in a compact subset of Ω are compactly contained in
Ω; that localization property is not proved here.
With the abstract Vitali lemma Vitali.exists_disjoint_subfamily_covering_enlargement,
swallowing gives the Vitali covering lemma for sections: from every family of such sections
one can extract a countable disjoint subfamily such that each section of the family lies in the
2 θ²-enlargement of a section of the subfamily. With doubling, the union of the family then has
measure at most a constant multiple, depending only on E, μ and Λ / λ, of the measure of
the union of the disjoint subfamily.
Main statements #
TauCeti.exists_convexSection_subset_convexSection_of_nonempty_inter— intersecting sections of comparable heights swallow each other;TauCeti.exists_measure_convexSection_le_sqrt_mul— the doubling property of sections;TauCeti.exists_countable_pairwiseDisjoint_convexSection— the Vitali covering lemma for sections;TauCeti.exists_countable_pairwiseDisjoint_convexSection_measure_le— its measure form.
References #
- L. A. Caffarelli, Some regularity properties of solutions of Monge Ampère equation, Comm. Pure Appl. Math. 44 (1991), 965–969.
- L. A. Caffarelli, Interior
W^{2,p}estimates for solutions of the Monge–Ampère equation, Ann. of Math. 131 (1990), 135–150, for covering arguments with sections. - C. E. Gutiérrez, The Monge–Ampère Equation, 2nd ed., Progress in Nonlinear Differential
Equations and Their Applications 89, Birkhäuser, 2016, §3.3 (engulfing) and the covering
lemmas for sections used in its
W^{2,p}estimates. - A. Figalli, The Monge–Ampère Equation and Its Applications, Zurich Lectures in Advanced Mathematics, EMS, 2017, Chapter 4.
Doubling of sections. There is a constant C, depending only on E and μ, with the
following property. Let u be convex on an open set Ω, with λ μ ≤ MA_u ≤ Λ μ on Ω for the
Monge–Ampère measure of u extended by ⊤ off Ω and 0 < λ, let ℓ be a supporting slope of
u at x ∈ Ω, and let K ≥ 1. If the section S(x, K h) is bounded with closure inside Ω,
then μ(S(x, K h)) ≤ √(C K^n Λ / λ) μ(S(x, h)), where n = finrank ℝ E.
Intersecting sections swallow each other. There is a constant C, depending only on E
and μ, with the following property. Let u be convex on an open set Ω, with
λ μ ≤ MA_u ≤ Λ μ on Ω for the Monge–Ampère measure of u extended by ⊤ off Ω and
0 < λ, and write θ = 1 + C Λ / λ. Suppose that every section of height H centred at a point
of D, with any supporting slope, is bounded with closure inside Ω. Let ℓ and m be
supporting slopes of u at x, y ∈ Ω, with S(x, h) ⊆ D and y ∈ D. If S(x, h) meets
S(y, k), h ≤ 2 k and 4 θ k ≤ H, then S(x, h) ⊆ S(y, 2 θ² k).
The Vitali covering lemma for sections. There is a constant C, depending only on E
and μ, with the following property. Let u be convex on an open set Ω, with
λ μ ≤ MA_u ≤ Λ μ on Ω for the Monge–Ampère measure of u extended by ⊤ off Ω and
0 < λ, and write θ = 1 + C Λ / λ. Suppose that every section of height H centred at a point
of D, with any supporting slope, is bounded with closure inside Ω. Consider a family of
sections S i = S(c i, h i), i ∈ t, with centres c i ∈ Ω, supporting slopes ℓ i, positive
heights with 4 θ h i ≤ H, and S i ⊆ D. Then there is a countable subfamily s ⊆ t of pairwise
disjoint sections such that every S i with i ∈ t lies in S(c j, 2 θ² h j) for some
j ∈ s.
The Vitali covering lemma for sections, measure form. There are constants C and A,
depending only on E and μ, with the following property. Let u be convex on an open set Ω,
with λ μ ≤ MA_u ≤ Λ μ on Ω for the Monge–Ampère measure of u extended by ⊤ off Ω and
0 < λ, and write θ = 1 + C Λ / λ. Suppose that every section of height H centred at a point
of D, with any supporting slope, is bounded with closure inside Ω. Consider a family of
sections S i = S(c i, h i), i ∈ t, with centres c i ∈ Ω, supporting slopes ℓ i, positive
heights with 4 θ h i ≤ H and 2 θ² h i ≤ H, and S i ⊆ D. Then there is a countable subfamily
s ⊆ t of pairwise disjoint sections whose union has measure at least a fixed fraction of the
measure of the union of the whole family:
μ (⋃ i ∈ t, S i) ≤ √(A (2 θ²)ⁿ Λ / λ) μ (⋃ j ∈ s, S j), where n = finrank ℝ E.