Lipschitz Harmonic Capacity and Bilipschitz Images of Cantor Sets

Lipschitz Harmonic Capacity and Bilipschitz Images of Cantor Sets

John Garnett, Laura Prat and Xavier Tolsa

Abstract

For bilipschitz images of Cantor sets in R^d we estimate the Lipschitz har- monic capacity and show this capacity is invariant under bilipschitz homeo- morphisms.

1 Introduction

Let Lip¹_loc(R^d) be the set of locally Lipschitz real functions on Euclidean space R^d, let E be compact subset of R^d, and let

L(E, 1) = {f ∈: Lip¹_loc : supp(∆f ) ⊂ E, ||∇f ||∞≤ 1, ∇f (∞) = 0}

be the set of locally Lipschitz functions harmonic on R^d\ E and normalized by the conditions ||∇f ||_∞ ≤ 1 and ∇f (∞) = 0. The Lipschitz harmonic capacity of E is defined by

κ(E) = sup{|h∆f, 1i| : f ∈ L(E, 1)}.

It was introduced by Paramonov [P] to study problems of C¹ approximation by harmonic functions in R^d.

If d = 2, if C \ E is simply connected, and if the Hausdorff measure Λ₂(E) = 0, then f ∈ L(E, 1) if and only if F (z) = fx− ify is a single-valued bounded analytic function on C\E which satisfies |F (z)| ≤ 1. In that case it then follows from Green’s theorem that κ(E) = 2πγ_R(E), where

γR(E) = sup{| lim

z→∞zF (z)| : F is analytic on C \ E, |F | ≤ 1, F (∞) = 0, ¯∂F real}

is the so called real analytic capacity of E. (See [P].) Now let T : R^d→ R^d be a bilipschitz homeomorphism:

A⁻¹|x − y| ≤ |T x − T y| ≤ A|x − y|. (1) This paper is concerned with the following conjecture.

Conjecture 1.1. If T is a bilipschitz homeomorphism, then κ(T (E)) ≤ C(A)κ(E),

where A is the constant in (1).

When d = 2 this conjecture was established in [T2] using the connection between analytic capacity and Menger curvature obtained in [T1]. The papers [T1] and [T2]

were preceded by two papers [MTV] and [GV] that estimated the analytic capacity of planar Cantor sets and of their bilipschitz images. The recent paper [MT] estimated the Lipschitz harmonic capacity of certain Cantor sets in R^d, and our purpose here is to establish Conjecture 1.1 for bilipschitz images of these Cantor sets. Thus in the language of fractions, this paper is to [MT] as paper [GV] was to [MTV] or paper [T2] was to [T1].

For fixed ratios λ_n such that

2^−d−1d ≤ λ_n ≤ λ₀ < 1

2, (2)

we write

σ_n = Yn k=0

λ_k, and define the sets

E =

\∞ n=0

En, En = [

|J|=n

Qⁿ_J, (3)

where J = (j1, j2, . . . , jn) is a multi-index of length n with jk ∈ {1, 2, . . . 2^d} and the Qⁿ_J are compact sets such that

Qⁿ⁺¹_(J,jn+1) ⊂ Qⁿ_J, for all n and J, and such that for all n and J,

c₁σ_n≤ diam(Qⁿ_J) ≤ c₂σ_n, (4) and

dist(Qⁿ_J, Qⁿ_K) ≥ c3σn, J 6= K. (5) for positive constants c₁, c₂, and c₃.

When Qⁿ_J is a cube with sides parallel to the coordinate axes and side-length σn

and

{Qⁿ⁺¹_(J,jn+1) ⊂ Qⁿ_J : j_n+1= 1, . . . , 2^d}

consists of the 2^d corner subcubes of Qⁿ_J, the set defined by (3) is the Cantor set studied in [MT], and a set E is the bilipschitz image of such a Cantor set if and only if E satisfies (3), (4), and (5). Write

θ_n = 2^−nd σ_n^d−1 and θ(Q) = θ_n if Q = Qⁿ_J. Note that by (2),

θ_n+1 ≤ θ_n. For Cantor sets it was proved in [MT] that

C⁻¹³X^∞

n=0

θ_n²´₋¹

2 ≤ κ(E) ≤ C³X^∞

n=0

θ_n²´₋¹

2,

where C depends only on the constant λ₀ in (2) and we extend their result to bilipschitz images of Cantor sets.

Theorem 1.2. If E is defined by (3), (4), and (5), then there is constant C = C(c₁, c₂, c₃, λ₀)

such that

C⁻¹

³X^∞

n=1

θ_n²

´₋¹

2 ≤ κ(E) ≤ C

³X^∞

n=1

θ_n²

´₋¹

2.

The proof of Theorem 1.2 follows the reasoning in [MT], but with certain changes.

In Section 2 we give some needed geometric properties of the sets E. In Section 3 we obtain L² estimates for the (truncated) Riesz transforms with respect to the probability measure p on E defined by p(Qⁿ_J) = 2^−nd. In Section 4 we derive Theorem 1.2 from the L²-estimates in section 3 by applying the dyadic T (b) Theorem of M.

Christ to a measure used in [MTV] and [MT].

2 The Geometry of E

Fix E such that (2) - (5) hold.

Lemma 2.1. There is c₄ = c₄(λ₀, c₁, c₂, c₃) such that for j = 1, 2, . . . , d, and all Qⁿ_J sup

Qⁿ_J∩E

x_j − inf

Qⁿ_J∩Ex_j ≥ c₄σ_n. (6)

Proof. Write

w = sup

Qⁿ_J∩E

x_j − inf

Qⁿ_J∩Ex_j. Let P be the hyperplane

xj = 1 2( sup

Qⁿ_J∩Exj + inf

Qⁿ_J∩Exj),

and let ˜Q^k_K be the orthogonal projection of Q^k_K onto P. If w < c3

2σn+p

then for k = n + 1, · · · , n + p, (5) and the Pythagorean Theorem give dist( ˜Q^k_J⁰, ˜Q^k_J⁰⁰) ≥

√3 2 c₃σ_k,

and there are (d−1)-dimensional balls B_J^k0 with diameter comparable to the diameter of ˜Q^k_J0 and such that

dist( ˜Q^k_J0, B^k_J0) ≤ c₄σ_k and

B^k_J∩ B_K^m = ∅, when k ≥ m.

Hence for constants c₅ > c₆ depending only on d and c₁, c₂, and c₃,

c₅σ_n^d−1 ≥ Λ_d−1³[^p

k=1

[

|K|=k

B_(J,K)^n+k ´

≥ Xp

k=1

X

|K|=k

Λ_d−1

³ B_(J,K)^n+k

´

≥ Xp

k=1

c₆2^kdσ^d−1_n+k,

and by (2) this can only happen if p ≤ ^c_c⁵₆. Thus (6) holds with c4 = c32^{d−1−d c5c6−1}. Define the probability measure p on E by p(Qⁿ_J) = 2^−nd.

Lemma 2.2. There exist c₇, c₈, and 0 < γ < 1, depending only on λ₀, c₁, c₂, and c₃ such that for j = 1, 2, . . . , d, there exist c₇2ⁿ disjoint slabs of the form

Sk = {ak ≤ xj ≤ bk} such that b_k− a_k ≤ c₇σ_n, p(S

S_k) ≥ c₈, but p(S_k) < c₇γⁿ.

Proof. Condition (4) implies that there exist disjoint slabs S_k satisfying all the conditions of the lemma except possibly p(S_k) ≤ c₇γⁿ. However, by Lemma 2.1 there exists m₀ such that if m ≤ n − m₀, then for each Q^m_J at most 2^d− 1 cubes Q^m+1_K ⊂ Q^m_J can meet S_k. Hence the number of Qⁿ_L with Qⁿ_L∩ S_k 6= ∅ does not exceed (2^d− 1)^(n−m0)2^dm0 and p(S_k) ≤ (1 − 2^−d)^n−m0 ≤ c₇γⁿ.

3 The L

Estimate

Let E satisfy properties (2) - (5). For x ∈ E we define Qⁿ_x = Qⁿ_J to be the unique Qⁿ_J such that x ∈ Qⁿ_J. If f ∈ L²(p) and j = 1, 2, . . . , d, we define the truncated Riesz transform as

R^j_Nf (x) = Z

y /∈Q^Nx

Kj(y − x)f (y)dp(y),

where K_j(y − x) = (y − x)_j

|y − x|^d. By (5) it is clear that kR^j_Nk_L²_(p) < ∞.

Theorem 3.1. Let 0 < α < 1 and let G ⊂ E be a closed set such that p(G) > α.

There are constants C₁(α) and C₂, both depending on λ₀, c₁, c₂ and c₃, such that for all N big enough,

C₁

³X^N

n=0

θ_n²

´¹

2 ≤ kR_N^j k_L²_(G,p) ≤ C₂

³X^N

n=0

θ_n²

´¹

2. (7)

To begin we prove the upper bound in (7). Since the norm kR^j_Nk_L²_(G,p)increases with G we may assume G = E, which also means C2 does not depend on α. The proof of the upper bound in (7) follows the paper [MT], but for convenience we repeat their argument. By the T (1)-Theorem for spaces of homogeneous type

kR^j_Nk_L²_(p) ≤ C sup

n≤N

sup

|J|=n

p(Qⁿ_J)

σ^d−1_n + C sup

n≤N

sup

|J|=n

kR^j_N(χ_Qⁿ_J)k_L²_(Qⁿ_J,p) p(Qⁿ_J)¹² .

Therefore the upper bound in (7) will be an immediate consequence of the following two lemmas. For convenience we fix j, write K(y − x) = K_j(y − x), and define

R_mf (x) = Z

Q^m_x\Q^m+1x

K_j(y − x)f (y)dp(y).

Lemma 3.2. If n ≤ m, there is c₇ such that

kRmχQⁿ_Jk_L²_(Qⁿ_J,p) ≤ c7θmp(Qⁿ_J)¹² Proof. For y ∈ Q^m_x \ Q^m+1_x , (5) gives

|K(y − x)| ≤ 1 c^d−1₃ σ_m+1^d−1 . Hence by (2)

|R_mχ_Qⁿ_J| ≤ 2^d c^d−1₃ θ_m,

and

||R_mχ_Qⁿ_J||_L²_(p) ≤ 2^d

c₃^d−1θ_mp(Qⁿ_J)¹².

Lemma 3.3. There is a constant C depending only on λ0, c1, c2 and c3 such that for all N > n and all J,

kR_N^j χQⁿ_Jk²_L²_(Qⁿ

J,p) ≤ C XN k=n

θ²_kp(Qⁿ_J).

Proof. Fix j = 1, . . . , d, then for x ∈ Qⁿ_J

R^j_NχQⁿ_J(x) =

N −1X

m=n

RmχQⁿ_J(x).

We claim that for m 6= k,

¯¯

¯ Z

R_mχ_Qⁿ_JR_kχ_Qⁿ_Jdp

¯¯

¯ ≤ C2^−|m−k|kR_mχ_Qⁿ_Jk_L²_(p)kR_kχ_Qⁿ_Jk_L²_(p). (8) Accepting (8) for the moment, we conclude that

kR^j_Nχ_Qⁿ_Jk²_L2(Qⁿ_J) = k

N −1X

m=n

R_mχ_Qⁿ_Jk²

=

N −1X

m=n

kRmχQⁿ_Jk²+ 2 X

n≤k<m≤N −1

hRmχQⁿ_J, RkχQⁿ_Ji

≤ C

N −1X

m=n

kR_mχ_Qⁿ_Jk²,

so that Lemma 3.2 gives the right inequality in (7).

To prove (8) assume n ≤ k < m ≤ N − 1. Then because the kernel K is odd, Z

Q^m_K

R_mχ_Qⁿ_J(x)dp(x) =X

r6=q

Z

Q^m+1_(K,r)

Z

Q^m+1_(K,q)

K(x − y)dp(y)dp(x) = 0,

so that for any x^m_K ∈ Q^m_K, Z

Q^m_K

R_mχ_Qⁿ_J(x)R_kχ_Qⁿ_J(x)dp(x) = Z

Q^m_K

R_mχ_Qⁿ_J(x)(R_kχ_Qⁿ_J(x) − R_kχ_Qⁿ_J(x^m_K))dp(x).

But when x ∈ Q^m_K, (4), (5) and (2) give

|R_kχ_Qⁿ_J(x) − R_kχ_Qⁿ_J(x^m_K)| ≤ Cσmp(Q^k_x)

σ^d_k ≤ Cθ_kσm

σ_k ≤ C2^−(m−k)θ_k. Hence using Lemma 3.2

| Z

R_mχ_Qⁿ_JR_kχ_Qⁿ_Jdp| ≤ C2^−(m−k)θ_kkR_mχ_Qⁿ_Jk_L¹_(Qⁿ_J,p)

≤ C2^−(m−k)θkp(Qⁿ_J)¹²kRmχQⁿ_Jk_L²_(p)

≤ C2^−(m−k)kR_mχ_Qⁿ_Jk_L²_(p)kR_kχ_Qⁿ_Jk_L²_(p) and (8) holds.

The proof of the lower bound in (7) also follows [MT] but with two alterations because G 6= E and because the sets Qⁿ_J may be incongruent. When Q = Qⁿ_J we also write n = n(Q), Q ∈ D_n, and θ(Q) = θ_n.

Let 0 < δ < 1, fix G and define B(δ) = {Q ∈S

nD_n : p(G ∩ Q) < δp(Q)}.

Lemma 3.4. Assume δ < α and p(G) ≥ α.

(a) Then for all n,

p(G \ [

Dn∩B(δ)

Q^J_n) ≥ p(G \ [

B(δ)

Q) ≥ α − δ.

(b) For N₀ ∈ N there exists M(N₀) such that whenever Q /∈ B(δ), there exist Q⁰ ⊂ Q with n(Q⁰) ≤ n(Q) + M such that for all Q⁰⁰⊂ Q⁰ with n(Q⁰⁰) ≤ n(Q⁰) + N₀

Q⁰⁰ ∈ B(/ δ 2).

Proof. To prove (a) let {Qj} be a family of maximal cubes in B(δ), note that p(G ∩ [

B(δ)

Q) ≤X

p(G ∩ Q_j) ≤ δp(E) = δ

and subtract this quantity from p(G).

To prove (b) fix N₀ and suppose (b) is false for N₀, δ, Q and M = 0. Write n = n(Q). Then there is Q1 ⊂ Q with n(Q1) ≤ n + N0 and Q1 ∈ B(^δ₂). Set F₁ = {Q₁}. Then p(Q \ Q₁) ≤ (1 − 2^−N0d)p(Q) = βp(Q). Now assume (b) is also false for N₀, δ, Q and M = N₀ and write Q \ Q₁ =S

{Q⁰ : n(Q⁰) = n(Q₁), Q⁰ 6= Q₁}.

Then for each Q⁰ 6= Q1 with n(Q⁰) = n(Q1) there is Q2 ⊂ Q⁰ with n(Q2) ≤ 2N0 and

Q₂ ∈ B(^δ₂). Set F₂ = {Q₂}. Then p(Q \S

F1∪F2Q_j) ≤ β²p(Q). Further assume (b) is false for N0, δ, Q and M = 2N0 and repeat the above construction in each Q⁰ \ Q2. After m steps we obtain families F_j of cubes Q_j ∈ B(₂^δ) such that ∪F_j is disjoint and

p(Q \ [m j=1

[

Fj

Qj) ≤ β^mp(Q)

and for β^m < ^δ₂ we obtain p(Q ∩ G) ≤ ^δ₂P_m

j=1

P

Fjp(Qj) + β^mp(Q) < δp(Q), which is a contradiction. We conclude that (b) holds for M = mN₀.

We will later fix δ = ^α₂. But for any δ < α we say Q⁰ ∈ G^∗(δ) if Q⁰ satisfies conclusion (b) of Lemma 3.4 for N0 and δ. Then by parts (b) and (a) of Lemma 3.4 we have:

Lemma 3.5. If p(G) ≥ α then X

G^∗(^δ₂)

θ(Q⁰)²p(Q⁰∩ G) ≥ C(M) X

Q /∈B(δ)

θ(Q)²p(Q ∩ G) ≥ C(M, α)X θ_n².

Now let A be a large constant. As in [MT], for R ∈ D we will define a family Stop(R) of “stopping cubes” Q ⊂ R. We say Q ∈ Stop₀(R) if Q ⊂ R and Q /∈ B(^δ₂), and if

infQ

¯¯

¯ Z

G∩(R\Q)

K(y − x)dp(y)

¯¯

¯ ≥ Aθ(R).

We further say Q ∈ Stop₁(R) if Q ⊂ R and Q /∈ B(^δ₂), if θ(Q) ≤ ηθ(R) for constant η to be chosen below, if n(Q) ≥ n(R) + N1 for constant N1 to be chosen below, and if

P ∈ Stop₀(R) ⇒ n(P ) ≥ n(Q).

Then define

Stop(R) = {Q ∈ Stop₀(R) ∪ Stop₁(R) : Q is maximal}.

Notice that by the construction either Stop(R) ⊂ Stop₀(R) or Stop(R) ⊂ Stop₁(R).

Inductively we define Stop¹(P ) = Stop(P ) and Stop^k(P ) =[

{Stop(Q) : Q ∈ Stop^k−1(P )}, Top = {P₀} ∪[

k≥1

Stop^k(P₀), and

P^stp= [

Stop(P )

Q, where P0 is the unique cube in D0.

Remark. The constants N₀, N₁, A, η are chosen as follows. First we take δ = α/2, then N1 is fixed in Lemma 3.7, then η and A in the proof of Lemma 3.8, and N0

which depends on A, η, δ in the proof of Lemma 3.6.

Lemma 3.6. Assume p(G) ≥ α, and take δ = ^α₂. If N₀ = N₀(A, η, δ) is sufficiently large, then for all Q ∈ G^∗(₂^δ) there exists a cube P ⊂ Q such that P ∈ Top and n(P ) ≤ n(Q) + N₀.

Proof. Let Q ∈ G^∗(^δ₂) and let R be the smallest cube R ∈ Top such that Q ⊂ R. We assume the conclusion of the lemma is false for Q. Thus Q /∈ Top, and Q /∈ Stop(R).

Hence by definition there is x₀ ∈ Q such that

¯¯

¯ Z

G∩R\Q

K(y − x₀)dp(y)

¯¯

¯ ≤ Aθ(R).

Then for x ∈ Q (5) gives

¯¯

¯ Z

G∩R\Q

(K(y − x) − K(y − x₀))dp

¯¯

¯ ≤ Cσn(Q) n(Q)−1X

k=n(R)

θ_k

σ_k ≤ C₁θ(R) so that

sup

Q

¯¯

¯ Z

G∩R\Q

K(y − x)dp(y)

¯¯

¯ ≤ (A + C1)θ(R). (9) Take x^∗ ∈ Q ∩ E with x^∗_j = inf_Qx_j and let Q^∗ be that Q^∗ ⊂ Q such that x^∗ ∈ Q^∗ and n(Q^∗) = n(Q) + N₀. Then by Lemma 2.1 there is a constant n₀ such that

K(y − x^∗) ≥ c σ^d−1_n

if y ∈ Qⁿ_J ⊂ (Q \ Q^∗) and n ≤ n(Q^∗) − n0. Because θn+1 ≤ θnand because we assume the lemma is false for Q, we also have θ(Qⁿ_J) ≥ ηθ(R) for every such Qⁿ_J. Hence by

(5) Z

G∩Q\Q^∗

K(y − x^∗)dp(y) ≥ (N₀− n₀)ηδ 2θ(R) and by the proof of (9),

infQ^∗

Z

G∩Q\Q^∗

K(y − x)dp(y) ≥ ((N0− n0)ηδ

2 − C)θ(R). (10) Taking N₀ = N₀(A) sufficiently large and comparing (10) with (9) we conclude that Q^∗ ∈ Stop₀(R), which is a contradiction.

Note that by Lemma 3.5 and Lemma 3.6 we have for all P , XN

n=0

θ²_n≤ C(α) XN n=0

X

Dn\B(δ)

θ(Q)²p(Q) ≤ C⁰(α)X

Top

θ(P )²p(G ∩ P ). (11)

We define

K_P1(x) = X

Q∈Stop(P )

χ_G∩Q(x) Z

G∩P \Q

K(y − x)dp(y)

+ χ_{G∩P \P}^stp(x) Z

G∩P \Q^N(x)

K(y − x)dp(y).

By construction

χ_GR_N1 =X

Top

K_P1 and

kR_N1k²_L2(G) =X

Top

kK_P1k²_L2(G)+ X

P,Q∈Top,P 6=Q

hK_P1, K_Q1i_L²_(G).

Lemma 3.7. If N₁ is chosen big enough, then for all P ∈ Top,

kK_P1k²_L2(G)≥ C⁻¹θ(P )²p(G ∩ P ), (12) where C = C(α), and

kK_P1k²_L2(G) ≥ A²θ(P )²p(G ∩ P^stp0), (13) where

P^stp0 =[n

Q : Q ∈ Stop(P ) ∩ Stop₀(P ) o

. Lemma 3.8.

X

P,Q∈Top,P 6=Q

|hK_P1, K_Q1i_L²_(G)| ≤ C(A⁻¹+ c(η))X

Top

kK_P1k²_L2(G), (14)

with c(η) → 0 as η → 0.

Assuming Lemma 3.7 and Lemma 3.8 for the moment, we see that if A is large and η is small, then

kR_N1k²_L2(G) ≥ C⁻¹X

Top

θ(P )²p(G ∩ P )

and then the lower bound in (7) follows from inequality (11).

To prove Lemma 3.7, first note that (13) follows from the definitions of Stop₀(P ) and Stop(P ). To prove (12), recall that K = K_j for some 1 ≤ j ≤ d. We apply Lemma 2.2 to P with γⁿ ∼ α to obtain sets S₁ ⊂ P and S₂ ⊂ P such that

sup

S1

x_j = a < inf

S2

x_j

and

Min(p(G ∩ S1), p(G ∩ S2)) ≥ c(α)p(P ).

We may assume that S1, S2 are much bigger that any stopping cube of P , because if there exists some Q ∈ Stop₀(P ) with size similar to S₁ or S₂, then (12) follows from (13); and if we choose N₁ big enough, any cube Q ∈ Stop₁(P ) will be much smaller that S1, S2. Then we get

¯¯ Z

S2∩G

K_Pχ_S1(x)dp(x)¯

¯ ≥ C⁻¹p(S₂∩ G) p(S₁ ∩ G) diam(P )^d−1. Set

E1 = P ∩ {xj ≤ a} and E2 = P ∩ {xj > a}.

By its definition,

K_P1 = χ_G(x)X

k

χ_Qk(x) Z

G∩P \Qk

K(y − x)dp(y)

where {Q_k} is a cover of P by disjoint cubes from D. We also have K_P1(x) = χ_G(x)X

i=1,2

X

k

χ_Qk(x) Z

G∩Ei\Qk

K(y − x)dp(y)

≡ K_Pχ_E1(x) + K_Pχ_E2(x).

Write Q_k= Q(x) when x ∈ Q_k and note that

y /∈ Q(x) ⇐⇒ x /∈ Q(y).

Hence by the antisymmetry K(y − x) = −K(x − y) we have Z

G∩E2

K_Pχ_E2(x)dp(x) = 0.

Therefore by the construction of E₁ and E₂, (p(G ∩ E2))^1/2kKP1k_L²_(G) ≥ ¯

¯ Z

G∩E2

KP1(x)dp(x)¯

¯

= ¯

¯ Z

G∩E2

K_Pχ_E1(x)dp(x)¯

¯

≥ p(G ∩ E₂)c(α)p(G ∩ P ) diam(P )^d−1 ,

which is (12). Notice that the last inequality holds because S1, S2 ahve size smaller than stopping square

To prove Lemma 3.8 we again follow [MT]. Suppose P 6= Q ∈ Top and Q ⊂ P.

Let PQ ∈ Stop(P ) be such that Q ⊂ PQ ⊂ P. By the antisymmetry of K we have R

Q∩GK_Q1dp = 0 so that

¯¯

¯ Z

Q∩G

KQ1(x)KP1(x)dp

¯¯

¯ =

¯¯

¯ Z

Q∩G

KQ1(x)(KP1(x) − KP1(xQ))dp(x)

¯¯

¯

≤ kK_Q1k_L¹_(Q)sup

Q

|K_P1(x) − K_P1(x_Q)|, where xQ is a fixed point from Q. But for any x ∈ Q, standard estimates yield

¯¯K_P1(x) − K_P1(x_Q)¯

¯ ≤ Z

G∩P \PQ

|K(y − x) − K(y − x_Q)|dp(y)

≤ C diam(Q) Z

G∩P \PQ

dp(y)

|x − y|^d

≤ C diam(Q) X

PQ⊂R⊂P

θ(R) diam(R).

Assume first that P_Q∈ Stop₀(P ). Since θ(R) ≤ θ(P ) in the last sum, we get

¯¯K_P1(x) − K_P1(x_Q)¯

¯ ≤ C diam(Q) diam(PQ)θ(P ).

Hence by (13)

¯¯hK_P1, KQ1i_L²_(G,p)¯

¯ ≤ C A

diam(Q) diam(P_Q)

µ p(G ∩ Q) p(G ∩ P^stp0)

¶_1/2

kKQ1k_L²_(G)kKP1k_L²_(G), when P_Q ∈ Stop₀(P ).

Consider now the case P_Q ∈ Stop₁(P ). This means that θ(P_Q) ≤ ηθ(P ). It is easy to check that this implies that

diam(Q) X

PQ⊂R⊂P

θ(R)

diam(R) ≤ c(η) diam(Q)

diam(P_Q)θ(P ) with c(η) → 0 as η → 0.

(See Lemma 3.6 in [MT] for a similar argument). So we get

¯¯hK_P1, KQ1i_L²_(G,p)¯

¯ ≤ c(η) diam(Q)

diam(P_Q)kKQ1k_L²_(G)kKP1k_L²_(G). Thus (14) follows from Schur’s lemma.

4 Lipschitz harmonic capacity

In this section we will prove Theorem 1.2. We will assume that each cube Qⁿ_J in the definition of the Cantor set E (see (3)) contains a closed ball B_Jⁿ such that

c⁰₁σ_n ≤ diam(B_Jⁿ).

This assumption comes for free from the definition of E in Section 1. Indeed, one easily deduces that there exists a family of balls B_Jⁿ centered at Qⁿ_J such that

c⁰₁σ_n≤ diam(B_Jⁿ) ≤ c⁰₂σ_n, and

dist(Bⁿ_J, B_Kⁿ) ≥ c⁰₃σ_n, J 6= K.

Then if one replaces the cubes Qⁿ_J in the definition of E by the sets Q˜ⁿ_J = [

Q^m_K⊂Qⁿ_J

(Q^m_K ∪ B_K^m),

E does not change.

Given a real Radon measure µ and f ∈ L¹(µ), let R_µ,²(f dµ)(x) =

Z

|y−x|>²

y − x

|y − x|^df (y)dµ(y)

be the (truncated) (d − 1)-Riesz transform of f ∈ L¹(µ) with respect to the measure µ and set kRµk_L²_(µ)= sup_²>0kRµ,²k_L²_(µ).

As in [MT], we need to introduce the following capacity of the sets E_N: κ_p(E_N) = sup{α : 0 ≤ α ≤ 1, kR_αµNk_L²_(αµN)≤ 1},

where µ_N is a probability measure on E_N such that µ_N(Q^N_J) = 2^{−N d}. The L² estimates from the previous section yield the following lemma.

Lemma 4.1.

κ_p(E_N) ≈

³X^N

n=1

θ²_n

´_−1/2 .

Proof. By Theorem 3.1 we have

kR_αµNk_L²_(αµN)= αkR_µNk_L²_(µN) ≈ α

³X^N

n=1

θ²_n

´_1/2 . The lemma follows because the sum above is ≥ 2^−d.

We will prove the following:

Lemma 4.2. There exists an absolute constant C₀ such that for all N ∈ N we have

κ(EN) ≤ C0κp(EN). (15)

Notice that Theorem 1.2 follows from Lemma 4.2 and

κ(E_N) ≥ κ₊(E_N) ≥ C⁻¹κ_p(E_N), (16) where

κ₊(E) = sup{|h∆f, 1i| : f ∈ L(E, 1), ∆f = µ ∈ M₊(E)}

and M₊(E) is the set of positive Borel measures supported on E. The first inequality in (16) is just a consequence of the definitions of κ and κ₊ and the second inequality follows from a well known method that dualizes a weak (1,1) inequality (see Theorem 23 in [Ch2] and Theorem 2.2 in [MTV]. The original proof is from [DØ]).

In [Vo] it is shown that the capacities κ and κ₊ are comparable for all subsets of R^d, but we do not use that deep result.

For any s > 0, we write Λs and Λ^∞_s for the s-dimensional Hausdorff measure and the s-dimensional Hausdorff content, respectively.

Proof. The arguments are similar to those in [MTV] and [MT], but a little more involved because our Cantor sets are not homogeneous. Also, instead of using the local T (b)-Theorem of M. Christ, we will run a stopping time argument in the spirit of [Ch1] and then use a dyadic T (b)-Theorem (see Theorem 20 in [Ch1]).

We set

S_n= θ₁²+ θ₂²+ · · · + θ_n².

Without loss of generality we can assume that for each N > 1 there exists 1 ≤ M <

N such that

S_M ≤ S_N

2 < S_{M +1}. (17)

Otherwise ^S₂^N < S₁ and by Lemma 4.1 it follows that κ_p(E_N) ≥ C⁻¹λ^d−1₁ . By [P] we have

κ(E_N) ≤ κ(E₁) ≤ CΛ^∞_d−1(E₁) ≤ Cλ^d−1₁ ,

and if C₀ is chosen big enough the conclusion of the lemma will follow in this case.

Assuming (17), we will now prove (15) by induction on N. For N = 1 (15) holds clearly. The induction hypothesis is

κ(E_n) ≤ C₀κ_p(E_n), for 0 < n < N, where the precise value of C₀ is to be determined later.

Notice that for n ≥ 0, (Q^N_K ∩ E)n is the n−th generation of the Cantor set Q^N_K ∩ E, i.e. the union of 2^nd sets Q^n+N_J satisfying properties (4) and (5) with n replaced by n + N. Let J^∗ be the multi-index of length M such that

κ((Q^M_J^∗∩ E)N −M) = max

|J|=Mκ((Q^M_J ∩ E)N −M).

We distinguish two cases.

Case 1: For some absolute constant A0 to be determined below, κ((Q^M_J∗∩ E)_{N −M}) ≥ A₀2^{−M d}κ(E_N),

By the induction hypothesis (applied to (Q^M_J^∗∩ E)N −M) and by Lemma 4.1 we have that

κ(E_N) ≤ A⁻¹₀ 2^{M d}κ((Q^M_J∗∩ E)_{N −M}) ≤ A⁻¹₀ 2^{M d}C₀κ_p((Q^M_J∗∩ E)_{N −M})

≤ A⁻¹₀ C₀C2^{M d}

³^{N −M}X

n=1

³ 2^−dn σ_{M +n}^d−1

´₂´_−1/2

= A⁻¹₀ C₀C

³ X^N

n=M +1

θ_n²

´_−1/2 . Now by using that S_M ≤ S_N/2 is equivalent toP_N

n=1θ_n² ≤ 2P_N

n=M +1θ²_nand Lemma 4.1 again, we obtain that

κ(E_N) ≤ 2^1/2A⁻¹₀ C₀C

³X^N

n=1

θ²_n

´_−1/2

≤ CA⁻¹₀ C₀κ_p(E_N).

Hence if A₀ = C, we obtain (15).

Case 2: For the same constant A0,

κ((Q^M_J∗∩ E)_{N −M}) ≤ A₀2^{−M d}κ(E_N). (18) Then if θ_{M +1}² > S_M, S_{M +1} = S_M + θ²_{M +1}≈ θ²_{M +1}. Therefore

κ_p(E_{M +1}) ≈ S_{M +1}^−1/2 ≈ θ⁻¹_{M +1}≥ CΛ^∞_d−1(E_{M +1}).

Hence by (17),

κ(E_N) ≤ κ(E_{M +1}) ≤ CΛ^∞_d−1(E_{M +1}) ≤ Cκ_p(E_{M +1}) ≈ κ_p(E_N), which is (15) if C₀ is chosen big enough.

On the other hand, if θ²_{M +1} ≤ SM, then SM +1 ≈ SM ≈ SN. Recall that we are assuming that each cube Q^M_J contains some ball B_J^M with comparable diameter.

Moreover, we may suppose that all the balls B^M_J , J = 1, . . . , 2^{M d}, have the same diameter dM. We set

E˜_M = [

|J|=M

B_J^M. We consider now the measure

σ = κ(E_N)µ⁰_M,