On the analytic capacity and curvature of some cantor sets with non-σ-finite length

Publicacions Matem`atiques, Vol 40 (1996), 195–204. ON THE ANALYTIC CAPACITY AND CURVATURE OF SOME CANTOR SETS WITH NON-σ-FINITE LENGTH Pertti Mattila Abstract We show that if a Cantor set Eas considered by Garnett in [G2] has positive Hausdorff h-measure for a non-decreasing function h satisfying 1 0r−3h(r)2dr < ∞, then the analytic capacity of Eis positive. Our tool will be the Menger three-point curvature and Melnikov’s identity relating it to the Cauchy kernel. We shall also prove some related more general results. 1. Introduction. In this paper we shall study the analytic capacity of some Cantor sets with non-σ-finite length. The analytic capacity of a compact set Ein the complex plane Cis defined as γ(E) = sup f lim z→∞ |zf(z)| where the supremum is taken over all analytic functions f:C\E→C with |f|≤1 and limz→∞ f(z) = 0. Then γ(E) = 0 if and only if Eis removable for bounded analytic functions. For this and other properties of the analytic capacity, see e.g. [G2] and [M]. Let Λhbe the Hausdorff measure generated by a non-decreasing function h:[0,∞)→[0,∞), i.e., Λh(A) = lim δ↓0inf ∞  i=1 hd(Ei):A⊂ ∞  i=1 Ei,d(Ei)≤δ. Here d(E) denotes the diameter of E.Forh(r)=h1(r)=r, a classical result of Painlev´e tells us that Λh1(E) = 0 implies γ(E) = 0. Some of the sets Ewith 0 <Λh1(E)<∞have zero and some positive analytic capacity, and although a complete characterization within this class is 196 P. Mattila lacking, pretty much is known, see [G2], [M] and [MMV]. It seems that some kind of rectifiability properties are required in order that a set of positive and finite length could have positive analytic capacity. To the other direction we have an easy classical result saying that positive Newtonian capacity implies positive analytic capacity, cf. [G2] or [M]. The Newtonian capacity of Ecan be defined as C1(E) = sup µ(E) : spt µ⊂E, |x−y|−1dµy ≤1 for x∈C where the supremum is taken over all (non-negative) Borel measures µ, and spt µstands for the support of µ. It is fairly easy to see (cf. [G2, p. 73]) that for non-decreasing functions h:[0,∞)→[0,∞) and Borel sets E⊂C, 1  0 r−2h(r)dr < ∞and Λh(E)>0 implies C1(E)>0, whence also γ(E)>0. There are also functions hwith 1 0r−2h(r)dr = ∞for which Λh(E)>0 implies C1(E)>0, see [E]. As in [G2] we shall consider Cantor sets E(λ)⊂C=R2for nonincreasing sequences (λn) with 0 <λ n<1/2. Each E(λ) is a product set E(λ)=K(λ)×K(λ) where K(λ)⊂Ris the linear Cantor set constructed as follows. Let K0=[0,1], K1=[0,λ 1]∪[1 −λ1,1], and at each stage n,Knis obtained from Kn−1by replacing each component of Kn−1by its two endmost intervals of length λntimes the length of the component. Then K(λ)= ∞  n=1 Kn where Knis a union of 2nintervals of length (1.1) σn=λ1·...·λn. Thus E(λ)=K(λ)×K(λ)= ∞  n=1 4n  j=1 En,j where each En,j is a square of side-length σn. Let h:[0,∞)→[0,∞) be a non-decreasing function with (1.2) h(σn)=4 −nfor n=1,2,... . Analytic capacity and curvature of Cantor sets 197 Then the Hausdorff h-measure Λhof E(λ) is positive and finite, and moreover there are positive and finite constants cand dsuch that (1.3) ch(r)≤ΛhE(λ)∩B(z,r)≤dh(r) for z∈E(λ),0<r≤2, see e.g. [G2]or[M]. Here B(z,r) is the closed disc with centre zand radius r. Moreover, C1E(λ)>0 if and only if 1 0r−2h(r)dr < ∞, see [G2]. In this paper we shall prove that if 1  0 r−3h(r)2dr < ∞ then E(λ) has positive analytic capacity. I believe that also the converse holds but I have not been able to prove it. Note that if h(r)≤rfor r>0, the case which is really of the main interest for us, then 1 0r−2h(r)dr =1 0r−1h(r)dr/r < ∞implies 1 0r−3h(r)2dr =1 0r−1h(r)2dr/r < ∞, but not vice versa. So it is easy to construct sequences (λn) such that for hrelated to (λn)as above 1  0 r−3h(r)2dr < ∞but 1  0 r−2h(r)dr =∞. Thus many of the sets E(λ)haveγ(E)>0 but C1(E)=0. Garnett considered the analytic capacity of the sets E(λ)in[G2]. He claimed there that γE(λ)>0 if and only if C1E(λ)>0. However, Eiderman, see the Commentary in [I2], found a mistake in the proof and the result of this paper shows that also the statement of Garnett’s result was false. Earlier in [G1] Garnett, and independently Ivanov, see [I1], proved for E(λ) with λn=1/4 for all n(which is the case 0 < Λh1E(λ)<∞) that γE(λ)= 0. Using this fact Ivanov showed in [I2] that there exists λfor which γE(λ)= 0 and yet the corresponding hsatisfies limr↓0h(r)/r =0. 2. The Menger and Melnikov curvatures. As in [MV] and [MMV] our method will be based on the so-called Menger curvature c(x, y, z) of the triple x,y,z∈C, and its relation to the Cauchy kernel first discovered by Melnikov in [Me]. By definition, c(x, y, z) is the reciprocal of the radius of the circle passing through x,yand z;c(x, y, z)=0 if and only if x,yand zare collinear. By elementary geometry, (2.1) c(x, y, z)= 2d(z,Lx,y) |x−z||y−z| 198 P. Mattila where Lx,y is the line through xand y, and d(z,A) denotes the distance of zfrom the set A. Melnikov [Me] introduced the curvature c2(µ)= c(x, y, z)2dµx dµy dµz for any Borel measure µand found that it is closely related to the Cauchy transform of µ; we return to this in Section 3. We shall first prove the following general inequality. 2.2. Theorem. Let h:[0,∞)→[0,∞)be a non-decreasing function with ∞ 0r−3h(r)2dr < ∞.Ifµis a finite Borel measure on Csuch that µB(z,r)≤h(r)for z∈Cand r>0, then c2(µ)≤12µ(C) ∞  0 r−3h(r)2dr. Proof: Note that h(r)/r2≤82r rt−3h(t)2dt, whence limr↓0h(r)/r= 0. Set A=(x, y, z)∈C3:|x−y|≤|x−z|and |x−y|≤|y−z|. Then by (2.1) and the fact d(z,Lx,y)≤|x−z|, c2(µ)≤3 A c(x, y, z)2dµx dµy dµz ≤12  Ad(z,Lx,y) |x−z||y−z|2 dµx dµy dµz ≤12   B(y,|y−z|) |y−z|−2dµx dµy dµz =12 µB(y,|y−z|) |y−z|2dµz dµy =12∞  0 µy(r) r2dµy(r)dµy where µy(r)=µB(y,r) and we are using Riemann-Stieltjes integration. Integrating by parts and using the facts µy(r)/r ≤h(r)/r →0asr→0 and µy(r)/r ≤µ(C)/r →0asr→∞,wehave c2(µ)≤12 ∞  0 µy(r)2 r3dr dµy ≤12µ(C) ∞  0 h(r)2 r3dr. Analytic capacity and curvature of Cantor sets 199 The argument in the above proof is rather precise except that the converse of the inequality d(z,Lx,y)≤|x−z|may fail, even with constants, very badly. But if µis at no scales concentrated near lines, then such a converse inequality holds for many triples (x, y, z), and we can obtain a converse for Theorem 2.2. We do this now for the Cantor sets E(λ). We denote by µAthe restriction of the measure µto the set A. 2.3. Theorem. Let λnbe a non-increasing sequence with 0<λ n< 1/2and let hbe related to it as in (1.1) and (1.2). Then c2ΛhE(λ)= E(λ)3 c(x, y, z)2dΛhxdΛhydΛhz<∞ if and only if 1 0r−3h(r)2dr < ∞. Proof: That 1 0r−3h(r)2dr < ∞implies c2ΛhE(λ)<∞follows immediately from Theorem 2.2 and (1.3). To prove the converse, let for y,z∈E(λ), C(y,z)=x∈E(λ):d(z,Lx,y)≥|x−z|/4. Note that by the construction of E(λ) there is a constant c1such that for all y,z∈E(λ) with y=zand for all 0 <r≤|y−z|/2 µC(y,z)∩B(y,2r)\B(y,r)≥c1µB(y,2r)\B(y,r) with µdenoting the restriction of Λhto E(λ). Hence c2(µ)= 2d(z,Lx,y) |x−z||y−z|2 dµx dµy dµz ≥ ∞  i=1  C(y,z)∩B(y,21−i|y−z|)\B(y,2−i|y−z|) 2d(z,Lx,y) |x−z||y−z|)2 dµx dµy dµz ≥1 4 ∞  i=1 µC(y,z)∩B(y,21−i|y−z|)\B(y,2−i|y−z|) |y−z|2dµy dµz ≥c1 4 µB(y,|y−z|) |y−z|2dµy dµz. If lim infr↓0h(r)/r = 0, we have as in the proof of Theorem 2.2, recalling also (1.3), that c2(µ)≥c1 4∞  0µB(y,r)2 r3dr dµy ≥c2c1 4ΛhE(λ) 1  0 h(r)2 r3dr, 200 P. Mattila which proves the theorem in this case. If lim infr↓0h(r)/r > 0 there is c2>0 such that h(r)≥c2rfor 0 <r≤2. Then the above estimate yields by (1.3) and the change of variable r=1/t, c2(µ)≥c1 4 µB(y,|y−z|) |y−z|2dµy dµz ≥c1c2c 4 1 |y−z|dµy dµz =(c1c2c/4) ∞  0 µy:|y−z|−1>t dt dµz =(c1c2c/4) ∞  0 r−2µB(z,r)dr dµz ≥(c1c2c2/4) 1  0 r−2h(r)dr ΛhE(λ)=∞, and the theorem follows also in this case. 2.4. Remark. Theorem 2.2 yields also immediately that  (E(λ)∩B(a,r))3 c(x, y, z)2dΛhxdΛhydΛhz≤12dh(r) ∞  0 t−3h(t)2dt for a∈E(λ) and 0 <r≤1. Moreover, an inspection of the proofs of Theorems 2.2 and 2.3 shows that this triple integral is comparable to h(r) r  0 t−3h(t)2dt. 3. Cauchy integral and analytic capacity. The following identity relating the Menger curvature to the Cauchy kernel 1/z was found by Melnikov in [Me], see also [MV]: for z1,z2,z3∈C,zi=zjfor i=j, (3.1) c(z1,z 2,z 3)2= σ 1 (zσ(1) −zσ(3))(zσ(2) −zσ(3)) where the sum is over all six permutations σof {1,2,3}. Analytic capacity and curvature of Cantor sets 201 Let µbe a finite Borel measure on C. For a disc ∆ and ε>0, let Tε(∆) = (z1,z 2,z 3)∈∆3:|z1−z3|>ε,|z2−z3|>ε,|z1−z2|≤ε. Using (3.1) and Fubini’s theorem, we have, as in [MV]or[MMV], that  ∆  ∆\B(z,ε) 1 ζ−zdµζ  2 dµz ≤ ∆3 c(z1,z 2,z 3)2dµz1dµz2dµz3 + Tε(∆) 1 (z1−z3)(z2−z3)dµz1dµz2dµ3. Assuming that µsatisfies µB(z,r)≤rfor z∈C,r>0, it is easy to see that the last term is bounded by c1µ(∆) for some constant c1independent of ∆ and ε.Thus (3.2)  ∆  ∆\B(z,ε) (ζ−z)−1dµζ  2 dµz ≤ ∆3 c(z1,z 2,z 3)2dµz1dµz2dµz3+c1µ(∆). We now assume that h:[0,∞)→[0,∞) is a non-decreasing function such that (3.3) ∞  0 r−3h(r)2dr < ∞, and that µsatisfies (3.4) µB(z,r)≤h(r) for z∈C,r>0, and the doubling condition with some c2<∞, (3.5) µB(z,2r)≤c2µB(z,r) for z∈spt µ, r > 0. 202 P. Mattila Then we have by (3.2) and Theorem 2.2 (3.6)  ∆  ∆\B(z,ε) (ζ−z)−1dµζ  2 dµz ≤c3µ∆ for all discs ∆ and all ε>0, where c3is a constant independent of ∆ and ε. This means that the operators Cµ,ε :g→  |ζ−z|>ε g(ζ) ζ−zdµζ are bounded in L2(µ), uniformly with respect to ε.Asin[MV] one can see this in two ways: either by observing that (3.6) implies that Cµ,ε(1) ∈BMO, uniformly in ε, and appealing to the T(1)-theorem. Here BMO means the BMO with respect to the “dyadic” cube system constructed by Christ in [C2] as a generalization of the system of David, see [D, pp. 91–96]. Or one can first check that the estimates above imply that Cµ,ε maps boundedly, and uniformly in ε,L∞(µ) to BMO(µ) and the atomic H1(µ)toL1(µ), and then use interpolation. For the validity of the T(1)-theorem in this setting, see [C1, p. 94] or [D, pp. 47–48]. The interpolation can be proven by the method of [J, Section 3.III]. Thus we have the following theorem. 3.7. Theorem. Suppose that hand µsatisfy the conditions (3.3)- (3.5). Then the operators Cµ,ε :L2(µ)→L2(µ)are bounded, uniformly with respect to ε. From this we could deduce with standard methods as in [C1] that γ(E)>0ifE⊂Cis a compact set supporting a non-zero Borel measure µfor which (3.3)-(3.5) hold. However, using a recent result of Melnikov we can get this without the doubling condition (3.5), and also without using the L2-boundedness. Namely, Melnikov proved in [Me, Theorem 3, p. 829] that γ(E)>0 provided Esupports µsuch that µB(z,r)≤rfor all z∈C,r>0 and c2(µ)<∞. Combining this with Theorem 2.2 we have 3.8. Theorem. Let E⊂Cbe a compact set such that there are a non-zero Borel measure µon Eand a non-decreasing function h: [0,∞)→[0,∞)satisfying (3.3) and (3.4). Then γ(E)>0. We now return to the Cantor sets E(λ) for non-increasing sequences λ=(λn), 0 <λ n<1/2. We may assume λn≥1/4 for all nbecause otherwise γE(λ)= 0. Letting hbe as in (1.1) and (1.2) and µ= ΛhE(λ), the condition (3.4) is satisfied. Thus we have by Theorem 3.8. Analytic capacity and curvature of Cantor sets 203 3.9. Theorem. If the function hrelated to the sequence λby (1.1) and (1.2) satisfies 1 0r−3h(r)2dr < ∞, then γE(λ)>0. 3.10. Question. Is the converse valid, that is, does γE(λ)>0 imply 1 0r−3h(r)2dr < ∞? I believe that there would be good chances to prove this if the following were true: if f:C\E(λ)→Cis bounded and analytic with f(∞) = 0, then there is a bounded Borel function ϕ:E(λ)→Csuch that f(z)=  E(λ) ϕ(ζ) ζ−zdΛhζ,z∈C\E(λ). This is well-known if h(r)=rfor r>0, but in the general case I don’t even know whether fcan be represented as a Cauchy transform of some complex measure. Acknowledgement. I would like to thank Vladimir Eiderman, Mark Melnikov and Joan Verdera for useful discussions. References [C1] M. Christ,“Lectures on Singular Integral Operators,” Regional Conference Series in Mathematics 77, Amer. Math. Soc., 1990. [C2] M. Christ,AT(b) theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math. 60/61 (1990), 601–628. [D] G. David,“Wavelets and Singular Integrals on Curves and Surfaces,” Lecture Notes in Math. 1465, Springer-Verlag, 1991. [E] V. Ya. Eiderman, On the comparision of Hausdorff measure and capacity, Algebra i Analiz 3(6) (1991), 174–189 (Russian); English translation in St. Petersburg Math. J. 3(6) (1992), 1367–1381. [G1] J. Garnett, Positive length but zero analytic capacity, Proc. Amer. Math. Soc. 21 (1970), 696–699. [G2] J. Garnett,“Analytic Capacity and Measure,” Lecture Notes in Math. 297, Springer-Verlag, 1972. [I1] L. D. Ivanov,“Variations of Sets and Functions,” Nauka, 1975 (Russian). [I2] L. D. Ivanov, On sets of analytic capacity zero, in “Linear and Complex Analysis Problem Book 3, Part II,” ed. V. P. Havin and N. K. Nikolski, Lecture Notes in Math. 1574, Springer-Verlag, 1994, pp. 150–153.