Calderon-Zygmund capacities and Wolff potentials on Cantor sets

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by  Xavier Tolsa

CALDERO´N-ZYGMUND CAPACITIES AND WOLFF
POTENTIALS ON CANTOR SETS
0
1
XAVIER TOLSA
0
2
n Abstract. We showthat,forsomeCantorsetsinRd,the capacityγs associated
Ja tothes-dimensionalRieszkernelx/|x|s+1 iscomparabletothecapacityC˙23(d−s),23
from non linear potential theory. It is an open problem to show that, when s is
8 positive and non integer,they arecomparablefor allcompactsets in Rd. We also
1
discuss other open questions in the area.

]
A
C
. 1. Introduction
h
t
a In the first part of this paper we show that, for some Cantor sets in Rd, the
m
capacity γ associated to the s-dimensional Riesz kernel x/|x|s+1 is comparable to
s
1 [ the capacity C˙32(d−s),32 from non linear potential theory. It is an open problem to
show that, when s is a positive and non integer, they are comparable for all compact
v
sets in Rd. In the last part of the paper, we discuss other related open questions.
6
8 To state our results in detail we need to introduce some notation. For 0 < s < d,
9
the s−dimensional Riesz kernel is defined by
2
. x
1 Ks(x) = , x ∈ Rd, x 6= 0.
0 |x|s+1
0
1 Notice that this is a vectorial kernel. The s−dimensional Riesz transform (or
v: s−Riesz transform) of a real Radon measure ν with compact support is
i
X
Rsν(x) = Ks(y −x)dν(y), x 6∈ supp(ν).
r Z
a
Although the preceding integral converges a.e. with respect to Lebesgue measure,
the convergence may fail for x ∈ supp(ν). This is the reason why one considers the
truncated s−Riesz transform of ν, which is defined as
Rsν(x) = Ks(y −x)dν(y), x ∈ Rd, ε > 0.
ε
Z
|y−x|>ε
These definitions also make sense if one consider distributions instead of measures.
Given a compactly supported distribution T, set
Rs(T) = Ks ∗T
Partially supported by grant MTM2007-62817(Spain) and 2009-SGR-420(Catalonia). Part of
thispaperwasdoneduringtheattendancetotheresearchsemester“HarmonicAnalysis,Geometric
Measure Theory and QuasiconformalMappings” at the CRM (Barcelona), in 2009.
1
2 X.TOLSA
(in the principal value sense for s = d), and analogously
Rs(T) = Ks ∗T,
ε ε
where Ks(x) = χ x/|x|s+1.
ε |x|>ε
Given a positive Radon measure with compact support and a function f ∈ L1(µ),
we consider the operators Rs(f) := Rs(f dµ) and Rs (f) := Rs(f dµ). We say that
µ µ,ε ε
Rs is bounded on L2(µ) if Rs is bounded on L2(µ) uniformly in ε > 0, and we set
µ µ,ε
kRsk = supkRs k .
µ L2(µ)→L2(µ) µ,ε L2(µ)→L2(µ)
ε>0
Given a compact set E ⊂ Rd, the capacity γ of E is
s
(1.1) γ (E) = sup|hT,1i|,
s
where the supremum is taken over all distributions T supported on E such that
kRs(T)k ≤ 1. Following [Vol03], we call γ the s-dimensional Caldero´n-
L∞(Rd) s
Zygmund capacity. The case s = d − 1 is particularly relevant: γ coincides
d−1
with the capacity κ introduced by Paramonov [Par93] in order to study problems
of C1 approximation by harmonic functions in Rd (the reader should notice that κ
is called κ′ in [Par93]). When d = 2 and s = 1, z/|z|s+1 coincides with the com-
plex conjugate of the Cauchy kernel 1/z. Thus, if one allows T to be a complex
distribution in the supremum above, then γ is the analytic capacity.
1
If we restrict the supremum in (1.1) to distributions T given by positive Radon
measures supported on E, we obtain the capacities γ . Clearly, we have γ (E) ≥
s,+ s
γ (E). Ontheotherhand, theoppositeinequalityalsoholds(uptoamultiplicative
s,+
absolute constant c ):
s
γ (E) ≤ c γ (E).
s s s,+
This was first shown for s = 1,d = 2 by the author [Tol03], and it was extended to
the case s = d−1 by Volberg [Vol03]. For other values of s, this can be proved by
combining the techniques from [Vol03] with others from [MPV05].
Now we turn to non linear potential theory. Given α > 0 and 1 < p < ∞ with
0 < αp < 2, the capacity C˙ of E ⊂ Rd is defined as
α,p
C˙ (E) = supµ(E)p,
α,p
µ
where the supremum runs over all positive measures µ supported on E such that
1
I (µ)(x) = dµ(x)
α
Z |x−y|2−α
satisfies kIα(µ)kp′ ≤ 1, where as usual p′ = p/(p−1).
For our purposes, the characterization of C˙ in terms of Wolff potentials is more
α,p
useful than its definition above. Consider
∞ µ(B(x,r)) p′−1 dr
W˙ µ (x) = .
α,p Z (cid:18) r2−αp (cid:19) r
0
CALDERO´N-ZYGMUND CAPACITIES AND WOLFF POTENTIALS 3
A well known theorem of Wolff asserts that
(1.2) C˙ (E) ≈ supµ(E),
α,p
µ
wherethesupremumistakenoverallmeasuresµsupportedonE suchthatW˙ µ (x) ≤
α,p
1 for all x ∈ E (see [AH96, Chapter 4], for instance). The notation A ≈ B means
that there is an absolute constant c > 0, or depending on d and s at most, such that
c−1A ≤ B ≤ cB.
Mateu, Prat and Verdera showed in [MPV05] that if 0 < s < 1, then
˙
γ (E) ≈ C (E).
s 2(d−s),3
3 2
Notice that for the indices 2(d−s), 3, one has
3 2
∞ µ(B(x,r)) 2 dr
W˙ µ (x) = .
32(d−s),32 Z0 (cid:18) rs (cid:19) r
When s = 1 and d = 2, from the characterization of γ in terms of curvature of
1,+
measures, one easily gets γ (E) & C˙ (E). Using analogous arguments (involving a
1 2,3
3 2
symmetrization of the kernel and the T(1) theorem), in [ENV08] it has been shown
that this also holds for all indices 0 < s < d:
γ (E) & C˙ (E),
s 2(d−s),3
3 2
for any compact set E ⊂ Rd. The opposite inequality is false when s is integer (for
instance, if E is contained in an s-plane and has positive s-dimensional Hausdorff
measure, then γ (E) > 0, but C˙ (E) = 0). When 0 < s < d is non integer, it
s 2(d−s),3
3 2
is an open problem to prove (or disprove) that
γ (E) . C˙ (E).
s 2(d−s),3
3 2
See Section 6 for more details and related questions.
˙
In the present paper we show that the comparability γ (E) ≈ C (E) holds
s 2(d−s),3
3 2
for some Cantor sets E ⊂ Rd, which are are defined as follows. Given a sequence
λ = (λ )∞ , 0 ≤ λ < 1/2, we construct E by the following algorithm. Consider
n n=1 n
the unit cube Q0 = [0,1]d. At the first step we take 2d closed cubes inside Q0, of
side length ℓ = λ , with sides parallel to the coordinate axes, such that each cube
1 1
contains a vertex of Q0. At the second step 2 we apply the preceding procedure
to each of the 2d cubes produced at step 1, but now using the proportion factor
λ . Then we obtain 22d cubes of side length ℓ = λ λ . Proceeding inductively, we
2 2 1 2
have at the n−th step 2nd cubes Qn, 1 ≤ j ≤ 2nd, of side length ℓ = n λ . We
j n j=1 j
consider Q
2nd
E = E(λ ,...,λ ) = Qn,
n 1 n j
j[=1
4 X.TOLSA
and we define the Cantor set associated to λ = (λ )∞ as
n n=1
∞
E = E(λ) = E .
n
n\=1
For example, if lim ℓ /2−nd/s = 1, then the Hausdorff dimension of E(λ) is s. If
n→∞ n
moreover ℓ = 2−nd/s for each n, then 0 < Hs(E(λ)) < ∞, where Hs stands for the
n
s−dimensional Hausdorff measure. In the planar case (d = 2), This class of Cantor
sets first appeared in [Gar72] (as far as we know), and its study has played a very
important role in the last advances concerning analytic capacity.
Our result reads as follows.
Theorem 1.1. Assume that, for all n, 0 < λ ≤ τ < 1. Denote θ = 2−nd/ℓs. For
n 0 2 n n
any N = 1,2,... we have
N −1/2
γ (E ) ≈ C˙ (E ) ≈ θ2 ,
s N 32(d−s),23 N (cid:18) n(cid:19)
Xn=1
where the constants involved in the relationship ≈ depend on d, s and τ , but not on
0
N.
Observe that if µ is for the probability measure on E given by µ = Ld|EN ,
N Ld(EN)
where Ld stands for the Lebesgue measure in Rd, then θ = µ(Qn)/ℓs. So θ is the
n j n n
s-dimensional density of µ on a cube from the n-th generation.
−1/2
Showing that C˙ (E ) ≈ N θ2 is not difficult, using the charac-
2(d−s),3 N n=1 n
3 2 (cid:16) (cid:17)
˙ P
terization of C in terms of Wolff’s potentials (see Section 2). The difficult
2(d−s),3
3 2
part of the theorem consists in showing that
N
−1/2
(1.3) γ (E ) ≈ θ2 .
s N n
(cid:16)Xn=1 (cid:17)
The main step in proving this result consists in estimating the L2(µ) norm of the
s-dimensional Riesz transform Rs.
µ
Let us remark that (1.3) has been proved for analytic capacity (s = 1, d = 2)
`
in [MTV03] (using previous results from Mattila [Mat96] and Eiderman [E˘ıd98]).
The arguments in [MTV03] (as well as the ones in [Mat96] and Eiderman [E`˘ıd98])
rely heavily on the relationship between the Cauchy transform and curvature of
measures. See [Mel95] and [MV95] for more details on this relationship.
In the case s = d − 1, the comparability (1.3) was proved by Mateu and the
author [MT04] under the additional assumption that λ ≥ 2−d/s for all n, which
n
is equivalent to saying that the sequence {θ } is non increasing. It is not difficult
n
to show that the arguments in [MT04] extend to all indices 0 < s < d. However,
getting rid of the assumption λ ≥ 2−d/s is much more delicate. This is what we
n
carry out in this paper.
CALDERO´N-ZYGMUND CAPACITIES AND WOLFF POTENTIALS 5
Letusalsomentionthatin[GPT06]itwasshownthattheestimate(1.3)alsoholds
if one replaces E by some bilipschitz image of itself, also under the assumption
N
λ ≥ 2−d/s. On the other hand, recently in [ENV08] some examples of random
n
Cantor sets where the comparability γ ≈ C˙ holds have been studied.
s 2(d−s),3
3 2
The plan of the paper is the following. In Section 2 we show that C˙ (E ) ≈
2(d−s),3 N
3 2
−1/2
N θ2 . The proof of (1.3) is contained in Sections 3, 4, and 5. In the
n=1 n
(cid:16) (cid:17)
finPal Section 6 we discuss open problems in connection with Caldero´n-Zygmund
capacities, Riesz transforms, and Wolff potentials.
Throughout all the paper, the letters c,C will stand for a absolute constants
(which may depend ond and s) that may change at different occurrences. Constants
with subscripts, such as C , will retain their values, in general.
1
−1/2
2. Proof of C˙ (E ) ≈ N θ2
2(d−s),3 N n=1 n
3 2 (cid:16) (cid:17)
P
The proof of this result is essentially contained in [AH96, Section 5.3]. However,
for the reader’s convenience we give a simple and almost self-contained proof.
Recall that µ stands for the probability measure on E defined by µ = Ld|EN .
N Ld(EN)
Given x ∈ E , let Qn(x) denote the cube Qn from the n-th generation in the
N j
construction of E that contains x, so that ℓ(Qn(x)) = ℓ is its side length. It is
N n
straightforward to check that for all x ∈ E ,
N
∞ µ(B(x,r)) 2 dr µ(Qn(x)) 2
W˙ µ (x) = ≈ = θ2.
32(d−s),23 Z0 (cid:18) rs (cid:19) r Xn≥0(cid:18)ℓ(Qn(x))s(cid:19) Xn≥0 n
Thus, if we consider the measure
−1/2
ν = θ2 µ,
(cid:18) n(cid:19)
Xn≥0
we have W˙ ν (x) . 1 for all x ∈ E . From (1.2) we infer that
2(d−s),3 N
3 2
−1/2
C˙ (E ) & ν(E ) = θ2 .
23(d−s),23 N N (cid:18) n(cid:19)
X
n≥0
To prove the converse inequality, we recall that given any Borel measure σ on Rd,
for any capacity C˙ ,
α,p
σ(Rd)
C˙ {x ∈ Rd : Wσ (x) > λ} ≤ c , for all λ > 0.
α,p α,p α,p λp−1
(cid:0) (cid:1)
6 X.TOLSA
˙
See Proposition 6.3.12 of [AH96]. If we apply this estimate to C , σ = µ, and
2(d−s),3
3 2
λ ≈ θ2, we get
n≥0 n
P
1
C˙ (E ) ≤ C˙ {x ∈ Rd : Wσ (x) > λ} . .
32(d−s),23 N 32(d−s),23 23(d−s),32 1/2
(cid:0) (cid:1) θ2
n≥0 n
(cid:16) (cid:17)
P
−1/2
3. Preliminaries for the proof of γ (E ) ≈ N θ2
s N n=1 n
(cid:16) (cid:17)
P
To simplify notation, to denote the s-dimensional Riesz transform of µ we will
write Rµ instead of Rsµ, and also K(x) instead of Ks(x) = x/|x|s+1. Moreover,
k · k stands for the L2(µ) norm.
Arguing as in [MT04, Lemma 4.2], it turns out that the estimate
N
−1/2
(3.1) γ (E ) ≈ θ2
s N n
(cid:16)Xn=1 (cid:17)
follows from the next result.
Theorem 3.1. Let µ be the preceding probability measure supported on E . We
N
have
N
kRµk2 ≈ θ2.
j
Xj=0
Wewillskiptheargumentsthatshowthat(3.1)canbededucedfromthistheorem,
which the interested reader can find in the aforementioned reference.
Sections 4 and 5 of this paper are devoted to the proof of Theorem 3.1. In the
remaining part of the current section, we introduce some additional notation that
we will use below, and we prove a technical estimate.
Denote ∆ = Qn : n ≥ 0, 1 ≤ j ≤ 2nd , where the Qn’s are the cubes which ap-
j j
pear in the cons(cid:8)truction of the E(λ). Le(cid:9)t ∆ be the family of cubes in ∆ from the
n
e
n-th generation. That is, ∆ = {Qn}2nd. For a fixed N ≥ 1, we set ∆ = N ∆
n j j=1 n=1 n
(so EN is constructed using the cubes from ∆N). S
Given a cube Q ⊂ Rd, we set
µ(Q)
θ(Q) := ,
ℓ(Q)s
i.e.θ(Q)istheaverages-dimensionaldensity ofµoverQ. Thusθ = θ(Q)ifQ ∈ ∆ .
n n
Given a cube Q ∈ ∆ and a function f ∈ L1 (µ), we define
loc
1
S f(x) = f dµ χ (x).
Q Q
µ(Q) Z
CALDERO´N-ZYGMUND CAPACITIES AND WOLFF POTENTIALS 7
Also, for 0 ≤ j ≤ N, we set S f = S f. If we denote by F(Q) the cubes
j Q∈∆j Q
from ∆ which are sons of Q, we set P
D f(x) = S f(x)−S f(x),
Q P Q
X
P∈F(Q)
and for 0 ≤ j ≤ N we denote D f = D f = S f −S f.
j Q∈∆j Q j+1 j
Let ∆0 = ∆ \ ∆ . Notice that thPe functions D f and D f are orthogonal for
N Q P
P 6= Q. If f dµ = 0, then
R
N−1
S f = D f = D f,
N j Q
Xj=0 QX∈∆0
and thus
kfk2 ≥ kS fk2 = kD fk2.
N Q
QX∈∆0
In particular, if we take f = Rµ, by antisymmetry Rµdµ = 0, and thus
R
(3.2) kRµk2 ≥ kS (Rµ)k2 = kD (Rµ)k2.
N Q
QX∈∆0
Given cubes Q,R ∈ ∆, we denote
ℓ(Q) ℓ(Q)
(3.3) p(Q) := θ(P) , p(Q,R) := θ(P) .
ℓ(P) ℓ(P)
X X
P∈∆:Q⊂P P∈∆:Q⊂P⊂R
For 0 ≤ j ≤ N, we denote p := p(Q), for Q ∈ ∆ .
j j
Lemma 3.2. Let Q ∈ ∆ and x,x′ ∈ Q. Let Q the parent of Q. Then we have
ℓ(Q)
R(χ µ)(x)−R(χ µ)(bx′) ≤ C p(Q).
Rd\Q Rd\Q 1
ℓ(Q)
(cid:12) (cid:12)
(cid:12) (cid:12) b
Thus,
b
R(χ µ)(x)−R(χ µ)(x′) ≤ C p(Q) ≤ C p(Q).
Rd\Q Rd\Q 1 2
(cid:12) (cid:12)
Proof. We hav(cid:12)e (cid:12) b
R(χ µ)(x)−R(χ µ)(x′)
Rd\Q Rd\Q
(cid:12) (cid:12)
(cid:12) ≤ (cid:12) |K(x−y)−K(x′ −y)|dµ(y)
Z
Rd\Q
1
≤ C|x−x′| dµ(y)
Z |x−y|s+1
Rd\Q
µ(P) ℓ(Q)
≤ C|x−x′| ≤ C p(Q).
ℓ(P)s+1 ℓ(Q)
P∈∆X:Q(P
b
b (cid:3)
8 X.TOLSA
4. Proof of kRµk2 . N θ2.
j=0 j
P
Lemma 4.1. If Q ∈ ∆0 and P is a son of Q, then
(4.1) |S (Rµ)−S (Rµ)| . p(Q).
P Q
As a consequence,
(4.2) kD (Rµ)k2 . p(Q)2µ(Q).
Q
Proof. It is clear that (4.2) follows from (4.1). To prove (4.1), we use the antisym-
metry of the kernel K(x):
S (Rµ)−S (Rµ) = S (R(χ µ))−S (R(χ µ))
P Q P Rd\P Q Rd\Q
(4.3) = S (R(χ µ))+S (R(χ µ))−S (R(χ µ)).
P Q\P P Rd\Q Q Rd\Q
From Lemma 3.2 it follows that
|S (R(χ µ))−S (R(χ µ))| . p(Q).
P Rd\Q Q Rd\Q
To estimate S (R(χ µ)) we take into account that dist(Q∩E \P,P) ≈ ℓ(Q),
P Q\P N
and so for every x ∈ P,
µ(Q)
|R(χ µ)(x)| . = θ(Q) ≤ p(Q).
Q\P ℓ(Q)s
From the preceding estimates and (4.3), we get (4.1). (cid:3)
Lemma 4.2. We have
N−1 N
kS (Rµ)k2 . θ2 and kRµk2 . θ2.
N j j
Xj=0 Xj=0
Proof. By (3.2) and Lemma (4.1),
N−1
kS (Rµ)k2 = kD (Rµ)k2 . p(Q)2µ(Q) = p2.
N Q j
QX∈∆0 QX∈∆0 Xj=0
On the other hand, by Lemma 3.2, for each Q ∈ ∆ and x ∈ Q,
N
|S (Rµ)−R(χ µ)(x)| = |S (Rχ µ))−R(χ µ)(x)| . p(Q).
Q Rd\Q N Rd\Q Rd\Q
Using also that
kχ R(χ µ)k ≤ θ(Q)µ(Q)1/2,
Q Q
CALDERO´N-ZYGMUND CAPACITIES AND WOLFF POTENTIALS 9
we obtain
kRµk2 = kχ R(µ)k2 ≤ 2 kχ R(χ µ)k2 +kχ R(χ µ)k2
Q Q Q Q Rd\Q
X X (cid:16) (cid:17)
Q∈∆N Q∈∆N
≤ 2 kχ R(χ µ)k2 +kR(χ µ)−S (Rχ µ))k2 +kS (Rµ)k2
Q Q Rd\Q N Rd\Q N
X (cid:16) (cid:17)
Q∈∆N
. θ(Q)2µ(Q)+ p(Q)2µ(Q)+ p(Q)2µ(Q)
QX∈∆N QX∈∆N QX∈∆0
N
. p(Q)2µ(Q) = p2.
j
QX∈∆ Xj=0
It only remains to show that M p2 . M θ2 both for M = N −1 and M = N.
j=0 j j=0 j
This follows easily from the dPefinition ofPpj and Cauchy-Schwartz:
M M j 2 M j j
ℓ ℓ ℓ
p2 = θ j ≤ θ2 j j
j (cid:18) k ℓ (cid:19) (cid:18) k ℓ (cid:19)(cid:18) ℓ (cid:19)
Xj=0 Xj=0 Xk=0 k Xj=0 Xk=0 k Xk=0 k
M j M M M
ℓ ℓ
(4.4) ≤ 2 θ2 j = 2 θ2 j ≤ 4 θ2.
k ℓ k ℓ k
Xj=0 Xk=0 k Xk=0 Xj=k k Xk=0
(cid:3)
5. Proof of kRµk2 & N θ2
j=0 j
P
5.1. The main lemma. The main lemma to prove the estimate
N
(5.1) kRµk2 & θ2
j
Xj=0
is the following.
Lemma 5.1. We have
N−1
(5.2) kD (Rµ)k2 & θ2.
Q j
QX∈∆0 Xj=0
Let us see how one deduces (5.1) from the preceding inequality.
Proof of (5.1) using Lemma 5.1. From (3.2) and (5.2) we infer that
N−1
(5.3) kRµk2 ≥ kS (Rµ)k2 ≥ C−1 θ2.
N 3 j
Xj=0
So we only have to show that kRµk2 & θ2 .
N
10 X.TOLSA
Consider Q ∈ ∆ and x ∈ Q. We split Rµ(x) as follows:
N
Rµ(x) = R(χ µ)(x)+R(χ µ)(x)
Q Rd\Q
= R(χ µ)(x)+S (Rµ)(x)+ R(χ µ)(x)−S (Rµ)(x) .
Q N Rd\Q N
(cid:0) (cid:1)
So we get
kRµk ≥ χ R(χ µ) −kS (Rµ)k
Q Q N
(cid:13) X (cid:13)
(cid:13)Q∈∆N (cid:13)
(cid:13) (cid:13)
(5.4) − χ R(χ µ)−S (Rµ) .
Q Rd\Q N
(cid:13) X (cid:13)
(cid:13)Q∈∆N (cid:13)
(cid:13) (cid:13)
It is easy to check that
χ R(χ µ) ≥ C−1θ .
Q Q 4 N
(cid:13) X (cid:13)
(cid:13)Q∈∆N (cid:13)
(cid:13) (cid:13)
To deal with S (Rµ) we simply use the fact that
N
kS (Rµ)k ≤ kRµk.
N
On the other hand, by Lemma 3.2, if x ∈ Q ∈ ∆ ,
N
|R(χ µ)(x)−S (Rµ)(x)| = |R(χ µ)(x)−S (R(χ µ)(x)| . p .
Rd\Q N Rd\Q Q Rd\Q N−1
By Cauchy-Schwartz, it follows easily that p ≤ C N−1θ2 1/2. Then we deduce
N−1 j=0 j
(cid:0)P (cid:1)
N−1
2
χ R(χ µ)−S (Rµ) ≤ C θ2.
Q Rd\Q N j
(cid:13)(cid:13)QX∈∆N (cid:13)(cid:13) Xj=0
(cid:13) (cid:13)
Then, by (5.4) and the estimates above, we get
N−1
kRµk ≥ C−1θ −kRµk−C θ2 1/2.
4 n 5 j
(cid:0)Xj=0 (cid:1)
From (5.3), we infer that
N−1
1/2
C−1θ ≤ 2kRµk+C θ2 ≤ 2kRµk+C1/2C kRµk,
4 n 5 j 3 5
(cid:16)Xj=0 (cid:17)
and thus the lemma follows. (cid:3)
5.2. The stopping scales and the intervals I . To prove Lemma 5.1 we need
k
to define some stopping scales on the squares from ∆. Let B be some big constant
(say, B > 100) to be fixed below. We proceed by induction to define a subset
Stop := {s ,...,s } ⊂ {0,1,...,N}. First we set s = 0. If, for some k ≥ 0, s
0 m 0 k
has already been defined and s < N −1, then s is the least integer i > s which
k k+1 k
verifies at least one of the following conditions:
(a) i = N, or