Proof of Theorem 1 - Approximation via maximal functions

Approximation via maximal functions

3. Proof of Theorem 1

s and

c^L₃C₂^Ldϕ(x, y) < 4c²_LC₂^Ls ≤ 32γ₂(n)s, so it is enough to check that

32γ₂(n) ≤ ρ(n) 2 − 1, but this is true thanks to the definition of ρ(n) in (3.11).

We can now conclude the proof. Let x, y ∈ U_ϕ(0, s) \ J_θ^ϕ and r = d_ϕ(x, y). Then

|ϕ(x) − ϕ(y)| ≤ |ϕ(x) − (ϕ)_x,c^L

3r| + |(ϕ)_x,c^L

3r− (ϕ)_y,c^L

3r| + |ϕ(y) − (ϕ)_y,c^L

3r|

≤2(c^L₃)²ⁿ⁺²+ 2²ⁿ⁺³c^L₃ c^L₂ c^L₁

C₁^L(C₂^L)²ⁿ⁺¹θr

= C(n, L) d_ϕ(x, y)

and (3.13) follows.

3. Proof of Theorem 3.1

In this section, we prove Theorem 3.1 following the ideas outlined in [24, 25]. As we already did for the proof of Theorem 2.2, up to replacing E with its blow-up E_p₀_,r and, correspondingly, ϕ with ϕ_r = ¹_rϕ ◦ δ_r, we can simplify Theorem 3.1 to the following statement.

Theorem 3.7. Let n ≥ 2 and α ∈ (0,¹₂). There exist positive constants C₂(n), ε₂(α, n) and k₂ = k₂(n) with the following property. Let E ⊂ Hⁿbe a (Λ⁰, r⁰₀)-minimizer of H-perimeter in C_k₂ with

Λ⁰ = Λr, r₀⁰ = r₀

r > k₂, Λ⁰r⁰₀ ≤ 1, 0 ∈ ∂E, e(k₂) ≤ ε₂(α, n).

Let ϕ : W → R be a suitably chosen approximation given by Theorem 2.2. Then there exists a set K ⊂ D₁ such that

(3.14) L²ⁿ(D₁ \ K) ≤ C₂(n) e(k₂)^1−2α, (3.15) gr(ϕ|_K) = ∂E ∩K ∗ (−1, 1), (3.16) Lip_H(ϕ|_K) ≤ C₂(n) e(k₂)^α.

We need some preliminaries. The following result is an easy consequence of Cauchy–

Schwarz inequality, see [25, Lemma A.1] and [24, Proposition 2.1].

3. PROOF OF THEOREM 3.1 27

Lemma 3.8. Let W ⊂ W be an open set and let ϕ : W → R be an L-intrinsic Lipschitz function. For any Borel set A ⊂⊂ W , we have

(3.17)

Proof. Let A ⊂⊂ W be fixed. Then, by the general area formula (1.28),

and (3.17) follows squaring both sides.

The following lemma compares the distance d_ϕ with the distance of points of the graph of an intrinsic Lipschitz function ϕ, see [15, Proposition 3.6].

Lemma 3.9. Let W ⊂ W be an open set and let ϕ : W → R be an intrinsic Lipschitz function. Then, for all x ∈ W , r > 0 and 0 < C < 1/(1 + Lip_H(ϕ)), we have

(3.18) Uϕ(x, Cr) ⊂ πBr(Φ(x)) ∩ gr(ϕ)⊂ Uϕ(x, r), where U_ϕ(x, r) is as in (3.5) and Φ(x) = x ∗ ϕ(x)e₁.

Finally, the following result compares the distance d_ϕ with the distance d_∞ in W . Its proof easily follows from Definition 1.15 and is left to the reader.

Lemma 3.10. Let W ⊂ W be an open set and let ϕ : W → R be a bounded intrinsic Lipschitz function. Then, for all x ∈ W and r > 0, we have

U_ϕ(x, r) ⊂ D_R(x) and D_r(x) ⊂ U_ϕ(x, R), where R = r + 2kϕk^1/2_L∞(W )r^1/2.

Proof of Theorem 3.7. The proof is divided in three steps.

Step 1: construction of ϕ, K and proof of (3.15). Let α ∈ (0,¹₂) be fixed. We assume ε₂(n, α) ≤ ε₁(n) and k₂ > 642. We let ϕ : W → R be a (1;₆₄₂^k² )-representative of Theorem 2.2, see Remark 2.5. Choosing ε₂(n, α) sufficiently small, by (2.3) we can assume that sup_W|ϕ| < 1.

28 3. APPROXIMATION VIA MAXIMAL FUNCTIONS

2n+1 as in Theorem 1.2. We let the non-negative measure µ on D_k2

642

We now prove (3.15). Since ϕ is a (1;₆₄₂^k² )-representative of Theorem 2.2, by Re-mark 2.5 it is enough to prove that K ∈ A(1;₆₄₂^k² ). To this end, let us fix p ∈ M ∩ D₁∗ I and q ∈ M ∩ K ∗ I. We proceed as in Steps 1 of the proof of Theorem 2.4. Indeed, by (1.13) in Lemma 1.10 we have

(3.20) |

h

(ξ)| < 1 ∀ξ ∈ C_k2

and, by (1.10), we can estimate e(₃₂₁^k² ) ≤ 321²ⁿ⁺¹e(k₂) ≤ 321²ⁿ⁺¹ε₂(n, α) ≤ ω(n,¹₂,_k¹

3. PROOF OF THEOREM 3.1 29

for any ξ ∈ C₁ and 0 < s < ₆₄₂^k² − 1, we can estimate e(q, s) = 1

s²ⁿ⁺¹

Cs(q)∩∂E

|νE − ν|²_g 2 d|µ_E|

≤ 1

s²ⁿ⁺¹

∂E∩D2s(π(q))∗I

|ν_E − ν|²_g 2 d|µ_E|

≤ 2²ⁿ⁺¹κ_n sup

0<ρ<₆₄₂^k2−kπ(q)k_∞

1 κ_nρ²ⁿ⁺¹

∂E∩Dρ(π(q))∗I

|ν_E − ν|²_g 2 d|µ_E|

≤ 2²ⁿκ_nM µ(π(q)) ≤ 2²ⁿκ_nη

for any 0 < s < ₁₂₈₄^k² , where κ_n = L²ⁿ(D₁) as in (1.4).

We consider the blow-up of E at scale d_C(p, q) centred at q, that is, F = E_q,d_C_(p,q). By Remark 1.4, F is a (Λ⁰⁰, r⁰⁰₀)-perimeter minimizer in (Ck2)_q,d_C_(p,q), with

Λ⁰⁰= Λ⁰d_C(p, q), r⁰⁰₀ = r⁰₀

d_C(p, q) > 1.

Now

C₁₆ ⊂ (C_k₂)_q,d_C_(p,q), Λ⁰⁰r₀⁰⁰≤ 1, 0 ∈ ∂F and, by (1.11) and by definition of M₀,

e(F, 0, 16, ν) = e(E, q, 16d_C(p, q), ν) ≤ 2²ⁿκ_nη, since we can choose k2 > 82176. Therefore, provided we assume

2²ⁿκ_nη ≤ ε₀(n), by Theorem 1.8 we have

supⁿ

h

(ξ) : ξ ∈ C₁∩ ∂F^o≤ C(n)η^2(2n+1)¹ , where C(n) is a dimensional constant. In particular, choosing

ξ = 1

d_C(p, q)q⁻¹∗ p ∈ C₁∩ ∂F, we get

(3.22) |

h

^(q⁻¹∗ p)| ≤ C(n)η^2(2n+1)¹ d_C(p, q).

We now set

L⁰(n, η) = C(n)η^2(2n+1)¹

and we choose η so small that L⁰(n, η) ≤ L(n), where L(n) < 1 is as in (2.11). Then, by (3.22), we conclude that d_C(p, q) = kπ(q⁻¹∗ p)k_∞ and we get

(3.23) |

h

^(q⁻¹∗ p)| ≤ L(n)kπ(q⁻¹∗ p)k_∞ for all p ∈ M ∩ D₁∗ I, q ∈ M ∩ K ∗ I, so K ∈ A(1;₆₄₂^k² ). Thus, by (3.20) and (3.23), equality (3.15) follows.

30 3. APPROXIMATION VIA MAXIMAL FUNCTIONS

Step 2: proof of (3.14). We now apply Lemma 3.3 with s = ₂₅₆₈^k² and measure µ as defined in (3.19). By Theorem 2.2, we have

µ(D_k2_/642) = 2 to apply Lemma 3.3, we need to check that

µ(D_k2_/642) ≤ η

5²ⁿ⁺¹κ_n k₂ 2568

!2n+1

By (3.24), this follows if we assume that

ε₂(n, α) ≤

We remark that this condition on ε₂(n, α) is the only one that depends also on the parameter α. Thus, by (3.3) in Lemma 3.3 and by (3.24), we conclude that

L²ⁿ(D₁\ K) = L²ⁿ(J_η ∩ D₁) ≤ 5²ⁿ⁺¹

Step 3: proof of (3.16). By Lemma 3.8 and by Proposition 1.17, we have

µ_ϕ(A)² =

3. PROOF OF THEOREM 3.1 31

for all Borel sets A ⊂ D₁, where C(n) is a dimensional constant. Moreover, for any x ∈ K and 4r < ₆₄₂^k² − kxk_∞, by (3.18) in Lemma 3.9, by (1.8) and by (3.21), we have

We now apply Lemma 3.6. We choose the parameter s > 0 in Lemma 3.6 such that D₁ ⊂ U_ϕ(0, s) and U_ϕ(0, ρ(n)s) ⊂ D_k₂,

where ρ(n) is the dimensional constant defined in (3.11). Since Lip_H(ϕ) ≤ L(n) < 1, where L(n) is the dimensional constant defined in (2.11), possibly choosing ε₂(n, α) smaller, we can directly assume that L ≤ `(n) as in (3.8). In particular, the constant c(n, L) appearing in (3.13) of Lemma 3.6, is controlled from above by a dimensional constant. Since sup_W|ϕ| < 1, by Lemma 3.10 we can choose s = 3 provided that we also choose

where C⁰(n) is a dimensional constant. Now we can choose k₂ > 7704ρ(n) + 642,

so that 12ρ(n) ≤ ₆₄₂^k² − kxk_∞ for any x ∈ D₁. Therefore, for any x ∈ K, we get [µ_ϕ](x) ≤^qC⁰(n) η = C⁰⁰(n) e(k₂)^α,

32 3. APPROXIMATION VIA MAXIMAL FUNCTIONS

where C⁰⁰(n) is a positive dimensional constant. Thus K ⊂ U_ϕ(0, 3) \ J_θ^ϕ, where J_θ^ϕ is as in (3.12) and θ = C⁰⁰(n) e(k₂)^α. Therefore, by (3.13) in Lemma 3.6, we conclude that

|ϕ(x) − ϕ(y)| ≤ C(n) e(k₂)^αd_ϕ(x, y) for all x, y ∈ K.

This proves (3.16) and the proof of Theorem 3.7 is complete. 4. Proof of Corollary 3.2

In this section, we prove Corollary 3.2. As we already did for the proof of The-orem 3.1, up to replacing E with its blow-up E_p₀_,r and, correspondingly, ϕ with ϕ_r = ¹_rϕ ◦ δ_r, we can simplify Corollary 3.2 to the following statement.

Corollary 3.11. Let n ≥ 2 and α ∈ (0,¹₂). There exist positive constants C₃(n), ε₃(α, n) and k₃ = k₃(n) with the following property. Let E ⊂ Hⁿbe a (Λ⁰, r⁰₀)-minimizer of H-perimeter in C_k₃ with

Λ⁰ = Λr, r₀⁰ = r₀

r > k₃, Λ⁰r⁰₀ ≤ 1, 0 ∈ ∂E, e(k₃) ≤ ε₃(α, n).

Then there exist a set K ⊂ D₁ and an intrinsic Lipschitz function ϕ : W → R with the following properties:

L²ⁿ(D₁ \ K) ≤ C₃(n) e(k₃)^1−2α,

(3.26) gr(ϕ|_K) = ∂E ∩ K ∗ (−1, 1),

(3.27) S²ⁿ⁺¹(∂E 4 gr(ϕ)) ∩ C₁≤ C₃(n) e(k₃)^1−2α, Lip_H(ϕ) ≤ C₃(n) e(k₃)^α,

(3.28)

|∇^ϕϕ|² dL²ⁿ≤ C₃(n) e(k₃).

Proof. Let α ∈ (0,¹₂) be fixed and assume that ε₃(n, α) ≤ ε₂(n, α) and k₃ = k₂. Let K and ϕ be as in Theorem 3.7. Recall that, by construction, Lip_H(ϕ) < 1 and sup_W|ϕ| < 1. Moreover, by (3.16), we have

Lip_H(ϕ|_K) ≤ C₂(n) e(k₂)^α.

Thus, according to Proposition 1.14, choosing ε₃(n, α) ≤ ε₂(n, α) sufficiently small, we can extend ϕ outside K to the whole W in such a way that sup_W|ϕ| < 1 and

Lip_H(ϕ) ≤ C(n) e(k₃)^α,

where C(n) is a dimensional constant. Thus we only need to prove (3.27) and (3.28).

We prove (3.27). Let J = D₁\ K, I = (−1, 1) and note that, by (3.26), we have S²ⁿ⁺¹(∂E 4 gr(ϕ)) ∩ C₁= S²ⁿ⁺¹(∂E 4 gr(ϕ)) ∩ J ∗ I

= S²ⁿ⁺¹(∂E \ gr(ϕ)) ∩ J ∗ I+ S²ⁿ⁺¹(gr(ϕ) \ ∂E) ∩ J ∗ I

≤ S²ⁿ⁺¹(∂E ∩ J ∗ I) + S²ⁿ⁺¹(gr(ϕ) ∩ J ∗ I).

4. PROOF OF COROLLARY 3.2 33

On the one hand, by definition of excess 1.9 and by equality (1.16) in Lemma 1.11, we have

thus, by (1.10) and by (3.14), we can estimate

(3.30) S²ⁿ⁺¹(∂E ∩ J ∗ I) ≤ δ(n)⁻¹k₃²ⁿ⁺¹e(k₃) + C₂(n) e(k₃)^1−2α ≤ C(n) e(k₃)^1−2α, where C(n) is a dimensional constant. On the other hand, by the area formula (1.27), we have

thus, by Proposition 1.17 and again by (3.14), we can estimate S²ⁿ⁺¹(gr(ϕ) ∩ J ∗ I) ≤ C(n) e(k₃)^1−2α,

where C(n) is a dimensional constant. Combining (3.29) with (3.30) and (3.31), we prove (3.27). On the one hand, by Proposition 1.17 and by (3.15), we have

where C(n) and C⁰(n) are dimensional constants. On the other hand, again by Propo-sition 1.17 and by (3.16), we have

|∇^ϕϕ|² dL²ⁿ ≤ k∇^ϕϕk²_L∞(D1)L²ⁿ(J )

≤ C(n) Lip_H(ϕ)²L²ⁿ(J ) ≤ C⁰(n) e(k₃).

(3.34)

Combining (3.32) with (3.33) and (3.34), we prove (3.28).

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