Cantor sets and cyclicity in weighted Dirichlet spaces

0.15 MB·English

by  Omar El-Fallah

CANTOR SETS AND CYCLICITY IN WEIGHTED
DIRICHLET SPACES
0
1 O.EL-FALLAH1, K. KELLAY2, ANDT. RANSFORD3
0
2
Abstract. Wetreattheproblemofcharacterizingthecyclicvectorsin

n the weighted Dirichlet spaces, extending some of our earlier results in
a
theclassicalDirichletspace. TheabsenceofaCarleson-typeformulafor
J
weighted Dirichlet integrals necessitates the introduction of new tech-
7
niques.
2

]
V
C
1. Introduction
.
h
In this paper we study the weighted Dirichlet spaces D (0 ≤ α ≤ 1),
t α
a
defined by
m
1
[ D := f ∈ hol(D): D (f) := |f′(z)|2(1−|z|2)αdA(z) < ∞ .
α α
π D
1 n Z o
v Here D denotes the open unit disk, and dA is area measure on D. Clearly
4
D is a Hilbert space with respect to the norm k·k given by
6 α α
8 kfk2 := |f(0)|2+D (f).
4 α α
.
1 A classical calculation shows that, if f(z)= a zn, then
n≥0 n
0
0 kfk2 ≍ (n+1)1−Pα|a |2.
1 α n
v: Xn≥0
i NotethatD = H2 istheusualHardyspace,andD istheclassicalDirichlet
X 1 0
space (thus our labelling convention follows [1] rather than [2]).
r
a An invariant subspace of D is a closed subspace M of D such that
α α
zM ⊂ M. Givenf ∈ D ,wedenoteby[f] thesmallestinvariantsubspace
α Dα
of D containing f, namely the closure in D of {pf : p a polynomial}. We
α α
say that f is cyclic for D if [f] = D . The survey article [8] gives a
α Dα α
brief history of invariant subspaces and cyclic functions in the classical case
α= 0.
2000 Mathematics Subject Classification. 30H05, 46E20, 47A15.
Key words and phrases. Dirichlet space, weight, cyclic vector, α-capacity, Cantor set.
1. Research partially supported bya grant from Egide Volubilis (MA09209).
2. Research partially supported by grants from Egide Volubilis (MA09209) and ANR
Dynop.
3. Research partially supported by grants from NSERC (Canada), FQRNT (Qu´ebec)
and theCanada Research Chairs program.
1
2 EL–FALLAH,KELLAY,ANDRANSFORD
Our goal is to characterize the cyclic functions of D . In order to state
α
our results, we introduce the notion of α-capacity. For α ∈ [0,1), we define
the kernel function k : R+ → R∪{∞} by
α
1/tα, 0< α < 1,
k (t) :=
α
(log(1/t), α= 0.
The α-energy of a (Borel) probability measure µ on T is defined by
I (µ) := k (|ζ −ζ′|)dµ(ζ)dµ(ζ′).
α α
ZZ
A standard calculation gives
|µ(n)|2
I (µ) ≍ .
α (1+n)1−α
n≥0
X b
The α-capacity of a Borel subset E of T is defined by
C (E) := 1/inf{I (µ) :P(E)},
α α
where P(E) denotes the set of all probability measures supported on com-
pact subsets of E. In particular, C (E) > 0 if and only if there exists
α
a probability measure µ supported on a compact subset of E and having
finite α-energy. If α = 0, then C is the classical logarithmic capacity.
0
WerecallaresultduetoBeurlingandSalem–Zygmund[5,§V,Theorem3]
about radial limits of functions in the weighted Dirichlet spaces. If f ∈ D ,
α
thenf∗(ζ) := lim f(rζ)exists for allζ ∈ Toutsideasetof α-capacity zero.
r→1−
The following theorem gives two necessary conditions for cyclicity in D .
α
Theorem 1.1. Let α∈ [0,1). If f is cyclic in D , then
α
• f is an outer function,
• {ζ ∈T : f∗(ζ) = 0} is a set of α-capacity zero.
The first part is [2, Corollary 1]. For α = 0, the second part is [2,
Theorem 5], and for general α the proof is similar, the only difference being
that the logarithmic kernel k is replaced by k . We omit the details.
0 α
Ourmain result is a partial converse to this theorem. To state it, we need
to define the notion of a generalized Cantor set.
Let (a ) be a positive sequence such that a ≤ 2π and
n n≥0 0
a 1
n+1
sup < .
a 2
n≥0 n
The generalized Cantor set E associated to (a ) is constructed as follows.
n
Start with a closed arc of length a on the unit circle T. Remove an open
0
arcfromthemiddle, toleave two closed arcseach oflength a . Thenremove
1
two open arcs from their middles to leave four closed arcs each of length a .
2
After n steps, we obtain E , the union of 2n closed arcs each of length a .
n n
Finally, the generalized Cantor set is E := ∩ E .
n n
Theorem 1.2. Let α∈ [0,1) and let f ∈D . Suppose that:
α
CANTOR SETS AND CYCLICITY IN WEIGHTED DIRICHLET SPACES 3
• f is an outer function,
• |f| extends continuously to D,
• {ζ ∈ T : |f(ζ)| = 0} is contained in a generalized Cantor set E of
α-capacity zero.
Then f is cyclic for D .
α
Functions f satisfying the hypotheses exist in abundance. Indeed, any
generalized Cantor set is a so-called Carleson set, and is thus the zero set of
someouterfunctionf suchthatf andallitsderivatives extendcontinuously
to D. Moreover, it is very easy to determine which generalized Cantor sets
have α-capacity zero. More details will be given in §4.
To prove Theorem 1.2, we adopt the following strategy. In §2, using
a technique due to Korenblum, we show that [f] contains at least those
Dα
functionsg ∈D satisfying |g(z)| ≤ dist(z,E)4. Theideaisthen totake one
α
simple such g, and gradually transform it into the constant function 1 while
staying inside [f] , thereby proving that 1 ∈ [f] . This requires three
Dα Dα
tools: a general estimate for weighted Dirichlet integrals of outer functions,
some properties of generalized Cantor sets, and a regularization theorem.
These tools are developed in §§3,4,5 respectively, and all the pieces are
finally assembled in §6, to complete the proof of Theorem 1.2.
Theorem 1.2 was established for the classical Dirichlet space, α = 0, in
[7, Corollary 1.2]. The proof there followed the same general strategy, but
in several places key use was made of a formula of Carleson [4] expressing
the Dirichlet integral of an outer function f in terms of the values of |f∗|
on the unit circle. No analogue of Carleson’s formula is known in the case
0 < α < 1, and one of the main points of this note is to show how this
difficulty may be overcome.
Throughout the paper, we use the notation C(x ,...,x ) to denote a
1 n
constant that depends only on x ,...,x , where the x may be numbers,
1 n j
functions or sets. The constant may change from one line to the next.
2. Korenblum’s method
Our aim in this section is to prove the following theorem.
Theorem 2.1. Let f ∈ D be an outer function such that |f| extends con-
α
tinuously to D, and let F := {ζ ∈ T :|f(ζ)| = 0}. If g ∈D and
α
|g(z)| ≤ dist(z,F)4 (z ∈ D),
then g ∈[f] .
Dα
This theorem is a D -analogue of [7, Theorem 3.1], which was proved
α
using a technique of Korenblum. We shall use the same basic technique
here. However, the proof in [7] proceeded via a so-called fusion lemma,
which, being based on Carleson’s formula for the Dirichlet integral, is no
longer available to us here. Its place is taken by Corollary 2.3 below.
4 EL–FALLAH,KELLAY,ANDRANSFORD
We need to introduce some notation. Given an outer function f and a
Borel subset Γ of T, we define
1 ζ +z
f (z) := exp log|f∗(ζ)||dζ| (z ∈D).
Γ
2π ζ −z
(cid:16) ZΓ (cid:17)
Wewrite∂ΓandΓc fortheboundaryandcomplementofΓinTrespectively.
Lemma 2.2. Let f be a bounded outer function. For every Borel set Γ ⊂ T,
|f′(z)| ≤ C(f)(|f′(z)|+dist(z,∂Γ)−4) (z ∈ D).
Γ
Proof. Without loss of generality, we may suppose that kfk ≤ 1. Note
∞
that then kf k ≤ 1 for all Γ. Also, obviously, log|f∗|≤ 0 a.e. on T, which
Γ ∞
will help simplify some of the calculations below.
We begin by observing that
f′(z) 1 2ζ
Γ = log|f∗(ζ)||dζ| (z ∈ D),
f (z) 2π (ζ −z)2
Γ ZΓ
from which it follows easily that
2log(1/|f(0)|)
|f′(z)| ≤ (z ∈ D). (1)
Γ dist(z,Γ)2
Our aim now is to prove a similar inequality, but with ∂Γ in place of Γ.
SetG := {z ∈ D : dist(z,Γ) ≥ dist(z,Γc)2}. Clearly dist(z,Γ) ≥dist(z,∂Γ)2
for all z ∈G, so (1) implies
2log(1/|f(0)|)
|f′(z)| ≤ (z ∈ G). (2)
Γ dist(z,∂Γ)4
Now suppose that z ∈ D\G. Then dist(z,Γc)2 > dist(z,Γ) ≥ (1−|z|2)/2,
and hence
1 1−|z|2
|fΓc(z)| = exp(cid:16)2π ZΓc |ζ −z|2 log|f∗(ζ)||dζ|(cid:17) ≥ |f(0)|2.
Since obviously fΓ = f/fΓc, it follows that, for all z ∈D\G,
|f′(z)| |f(z)| |f′(z)| 1 2log(1/|f(0)|)
|fΓ′(z)| ≤ |fΓc(z)| + |fΓc(z)|2|fΓ′c(z)| ≤ |f(0)|2 + |f(0)|4 dist(z,Γc)2 ,
where once again we have used (1), this time with Γ replaced by Γc. Noting
that dist(z,Γc) ≥ dist(z,∂Γ) for all z ∈ D\G, we deduce that
|f′(z)| 1 2log(1/|f(0)|)
|f′(z)| ≤ + (z ∈ D\G). (3)
Γ |f(0)|2 |f(0)|4 dist(z,∂Γ)2
The inequalities (2) and (3) between them give the result. (cid:3)
Corollary 2.3. Let α ∈ [0,1) and f ∈ D ∩H∞ be an outer function. Then,
α
for every Borel set Γ ⊂ T and every g ∈ D satisfying |g(z)| ≤ dist(z,∂Γ)4,
α
we have
kf gk ≤ C(α,f)(1+kgk ).
Γ α α
CANTOR SETS AND CYCLICITY IN WEIGHTED DIRICHLET SPACES 5
Proof. Using Lemma 2.2, we have
|(f g)′|≤ |f′||g|+|f ||g′|≤ C(f)(|f′|+1+|g′|).
Γ Γ Γ
The conclusion follows easily from this. (cid:3)
Proof of Theorem 2.1. Let I be a connected component of T\F, say I =
(eia,eib). Let ρ > 1, and define
ψ (z) := (z−1)4/(z−ρ)4,
ρ
φ (z) := ψ (e−iaz)ψ (e−ibz).
ρ ρ ρ
The first step is to show that φρfT\I ∈ [f]Dα.
Let ǫ > 0 and set I := (ei(a+ǫ),ei(b−ǫ)) and
ǫ
φ (z) := ψ (e−i(a+ǫ)z)ψ (e−i(b−ǫ)z).
ρ,ǫ ρ ρ
By Corollary 2.3,
kφρ,ǫfT\Iǫkα ≤ C(f)kφρ,ǫkα ≤ C(f,ρ). (4)
Note that |fT\Iǫ|= |f| in a neighborhood of T\I. Since |f| does not vanish
inside I, it follows that |φρ,ǫfT\Iǫ|/|f| is bounded on T. Also f is an outer
function. Therefore, by a theorem of Aleman [1, Lemma 3.1],
φρ,ǫfT\Iǫ ∈ [f]Dα.
Using (4), we see that φρ,ǫfT\Iǫ converges weakly in Dα to φρfT\I as ǫ → 0.
Hence φρfT\I ∈[f]Dα, as claimed.
Next, we multiply by g. As g ∈ D ∩H∞, Aleman’s theorem immediately
α
yields φρfT\Ig ∈ [f]Dα. Using the fact that |g(z)| ≤ dist(z,F)4, it is easy
to check that kφ gk remains bounded as ρ → 1. By Corollary 2.3 again,
ρ α
kφρfT\Igkα is uniformly bounded, and φρfT\Ig converges weakly to fT\Ig.
Hence fT\Ig ∈ [f]Dα.
Now let (I ) be the complete set of components of T \ F, and set
j j≥1
Jn := ∪n1Ij. An argument similar to that above gives fT\Jng ∈ [f]Dα for
all n. Moreover, kfT\Jngkα is uniformly bounded. Thus fT\Jng converges
weakly to g, and so finally g ∈ [f] . (cid:3)
Dα
3. Estimates for weighted Dirichlet integrals
Thefollowing resultwillactasapartialsubstituteforCarleson’s formula.
Theorem 3.1. Let α ∈ [0,1), and let h : T → R be a positive measurable
function such that, for every arc I ⊂ T,
1
h(ζ)|dζ| ≥ |I|α. (5)
|I|
ZI
If f is an outer function, then
1 (|f∗(ζ)|2−|f∗(ζ′)|2)(log|f∗(ζ)|−log|f∗(ζ′)|)
D (f)≤ (h(ζ)+h(ζ′))|dζ||dζ′|.
α π T2 |ζ −ζ′|2
ZZ
6 EL–FALLAH,KELLAY,ANDRANSFORD
Proof. Given z ∈ D, let I bethe arc of T with midpointz/|z| andarclength
z
|I |= 2(1−|z|2). For ζ ∈I , we have |ζ −z| ≤ 2(1−|z|2), and so
z z
1−|z|2 1 1
P(z,ζ) := ≥ = .
|ζ −z|2 4(1−|z|2) 2|I |
z
Hence, using (5), we have
1 1 1
P(z,ζ)h(ζ)|dζ| ≥ h(ζ)|dζ| ≥ |I |α ≥ (1−|z|2)α.
z
ZT 2|Iz| ZIz 2 2
Therefore, by Fubini’s theorem,
1
D (f):= |f′(z)|2(1−|z|2)αdA(z) ≤ 2 D (f)h(ζ)|dζ|,
α ζ
π D T
Z Z
where
1
D (f):= |f′(z)|2P(z,ζ)dA(z) (ζ ∈T).
ζ
π D
Z
Now D (f) is the so-called local Dirichlet of integral of f at ζ, which was
ζ
studied in detail by Richter and Sundbergin [9]. In particular, they showed
that, if f is an outer function, then
1 |f∗(ζ)|2−|f∗(ζ′)|2−2|f∗(ζ′)|2log|f∗(ζ)/f∗(ζ′)|
D (f)= |dζ′|.
ζ 2π T |ζ −ζ′|2
Z
Substituting this into the preceding estimate for D (f), and noting the
α
obvious fact that h(ζ) ≤ h(ζ)+h(ζ′), we deduce that D (f) is majorized by
α
1 |f∗(ζ)|2−|f∗(ζ′)|2 −2|f∗(ζ′)|2log|f∗(ζ)/f∗(ζ′)|
(h(ζ)+h(ζ′))|dζ′||dζ|.
π T |ζ −ζ′|2
ZZ
Exchanging the roles of ζ and ζ′, we see that D (f) is likewise majorized by
α
1 |f∗(ζ′)|2−|f∗(ζ)|2−2|f∗(ζ)|2log|f∗(ζ′)/f∗(ζ)|
(h(ζ′)+h(ζ))|dζ||dζ′|.
π T |ζ′−ζ|2
ZZ
Taking the average of these last two estimates, we obtain the inequality in
the statement of the theorem. (cid:3)
We are going to apply this result with h(ζ) := Cd(ζ,E)α, where C is a
constant, d denotes arclength distance on T, and E is a closed subset of T.
Condition (5) thus becomes
1
d(ζ,E)α|dζ|≥ C−1|I|α for all arcs I ⊂T. (6)
|I|
ZI
A set E which satisfies this condition for some α,C is called a K-set (af-
ter Kotochigov). K-sets arise as the interpolation sets for certain function
spaces, and have several other interesting properties. We refer to [3, §1] and
[6, §3] for more details. In particular, if E satisfies (6), then it has measure
zero and logd(ζ,E) ∈L1(T).
CANTOR SETS AND CYCLICITY IN WEIGHTED DIRICHLET SPACES 7
Theorem 3.2. Let α ∈ (0,1), let E be a closed subset of T satisfying (6),
and let w : [0,2π] → R+ be an increasing function such that t 7→ ω(tγ) is
concave for some γ > 2/(1−α). Let f be the outer function satisfying
w
|f∗(ζ)| = w(d(ζ,E)) a.e on T.
w
Then
D (f ) ≤ C(α,γ,E) w′(d(ζ,E))2d(ζ,E)1+α|dζ|. (7)
α w
T
Z
In particular f ∈ D if the last integral is finite.
w α
Proof. The proof is largely similar to that of [7, Theorem 4.1], so we give
just a sketch, concentrating on those parts where the two proofs differ.
We begin by remarking that the concavity condition on w easily implies
that |logw(d(ζ,E))| ≤ C(w)|logd(ζ,E)|, so logw(d(ζ,E)) ∈ L1(T) and the
definition of f makes sense.
w
By Theorem 3.1, we have
(w2(δ)−w2(δ′))(logw(δ)−logw(δ′))
D (f )≤ C(α,E) (δα+δ′α)|dζ||dζ′|,
α w T2 |ζ −ζ′|2
ZZ
where we have written δ := d(ζ,E) and δ′ := d(ζ′,E).
Let (I ) be the connected components of T\E, and set
j
N (t) := 2 1 (0 < t ≤ π).
E {|Ij|>2t}
j
X
Then, for every measurable function Ω : [0,π] → R+, we have
π
Ω(d(ζ,E))|dζ| = Ω(t)N (t)dt.
E
ZT Z0
In particular, as in [7], it follows that
(w2(δ)−w2(δ′))(logw(δ)−logw(δ′))
(δα+δ′α)|dζ||dζ′|
T2 |ζ −ζ′|2
ZZ
π π (w2(s+t)−w2(t))(logw(s+t)−logw(t))
≤ C(α) (s+t)αN (t)dsdt.
s2 E
Z0 Z0
The concavity assumption on w implies that t → t1−1/γw′(t) is decreasing,
and thus, as in [7],
w2(t+s)−w2(t) ≤ 2γw(t+s)w′(t)t (1+s/t)1/γ −1 ,
logw(t+s)−logw(t) ≤ tw′(t)(1+s/(cid:0)t)1/γ log(1+s/t(cid:1)).
w(t+s)
8 EL–FALLAH,KELLAY,ANDRANSFORD
Combining these estimates, we obtain
π π (w2(t+s)−w2(t))(logw(t+s)−logw(t))
(s+t)αdsN (t)dt
s2 E
Z0 Z0
π π ds
≤ 2γw′(t)2t2+α (1+s/t)1/γ −1 (1+s/t)1/γ+αlog(1+s/t) N (t)dt
s2 E
Z0 Z0
π π(cid:0)/t (cid:1) dx
= 2γw′(t)2t1+α 2γ (1+x)1/γ −1 (1+x)1/γ+αlog(1+x) N (t)dt
x2 E
Z0 Z0
π (cid:0) (cid:0) (cid:1) (cid:1)
≤ C(α,γ) w′(t)2t1+αN (t)dt.
E
Z0
In the last inequality we used the fact that γ > 2/(1−α). (cid:3)
4. Generalized Cantor sets
The notion of the generalized Cantor set E associated to a sequence (a )
n
was defined in §1. In this section we briefly describe some pertinent prop-
erties of these sets. We shall write
a
n+1
λ := sup .
E
a
n≥0 n
Recall that, by hypothesis, λ < 1/2.
E
Our first result shows that generalized Cantor sets satisfy (5), and hence
that Theorem 3.2 is applicable to such sets.
Proposition 4.1. Let E be a generalized Cantor set and let α ∈ [0,1).
Then, for each arc I ⊂ T,
1
d(ζ,E)α|dζ| ≥ C(α,λ )|I|α.
E
|I|
ZI
Proof. Let I be an arc with |I| ≤ 2a , and choose n so that 2a < |I| ≤
0 n
2a . Recall that the n-th approximation to E consists of 2n arcs, each of
n−1
length a , and that the distance between these arcs is at least a −2a .
n n−1 n
If I meets at least two of these arcs, then I \E contains an arc J of length
a −2a , and if I meets at most one of these arcs, then I\E contains an
n−1 n
arc J of length (|I|−a )/2. Thus I \E always contains an arc J such that
n
|J|/|I| ≥ min{1/2−λ , 1/4}. Consequently
E
1 1 1 |J|α+1
d(ζ,E)α|dζ| ≥ d(ζ,E)α|dζ|≥ ≥ C(α,λ )|I|α.
E
|I| |I| |I| α+1
ZI ZJ
(cid:3)
In the next two results, we write E := {ζ ∈ T :d(ζ,E) ≤ t}.
t
Proposition 4.2. If E is a generalized Cantor set, then |E | = O(tµ) as
t
t → 0, where µ := 1−log2/log(1/λ ).
E
CANTOR SETS AND CYCLICITY IN WEIGHTED DIRICHLET SPACES 9
Proof. Givent ∈ (0,a ],choosensothata < t ≤ a . Thenclearly|E | ≤
0 n n−1 t
2n(a +2t) ≤ 3.2nt. Also2n−1 = (1/λn−1)log2/log(1/λE) ≤ (a /t)log2/log(1/λE).
n E 0
Hence |E |≤ C(a ,λ )t1−log2/log(1/λE). (cid:3)
t 0 E
In particular, every generalized Cantor set E is a Carleson set, that is,
π
(|E |/t)dt < ∞. Taylor and Williams [10] showed that Carleson sets are
0 t
zero sets of outer functions in A∞(D). This justifies a remark made in §1.
R
The final property that we need concerns the α-capacity, C , which was
α
defined in §1.
Theorem 4.3. Let E be a generalized Cantor set and let α ∈ [0,1). Then
π dt
C (E) = 0 ⇐⇒ = ∞.
α tα|E |
Z0 t
Proof. This follows easily from [5, §IV, Theorems 2 and 3]. (cid:3)
5. Regularization
We shall need the following regularization result. The proof is the same,
with minor modifications, as that of [7, Theorem 5.1].
Theorem 5.1. Let α ∈ [0,1), let σ ∈ (0,1) and let a > 0. Let φ : (0,a] →
R+ be a function such that
• φ(t)/t is decreasing,
• 0 < φ(t) ≤ tσ for all t ∈(0,a],
a dt
• = ∞.
tαφ(t)
Z0
Then, given ρ ∈ (0,σ), there exists a function ψ :(0,a] → R+ such that
• ψ(t)/tρ is increasing,
• φ(t) ≤ ψ(t) ≤tσ for all t ∈ (0,a],
a dt
• = ∞.
tαψ(t)
Z0
6. Completion of the proof of Theorem 1.2
For α = 0, thistheorem was provedin[7, §6]. Theproofforα ∈ (0,1) will
follow the same general lines, and once again we shall concentrate mainly
on the places where the proofs differ.
Let f be the function in the theorem. Our goal is to show that 1 ∈[f] .
Dα
Let E be as in the theorem, and let g be the outer function such that
|g∗(ζ)|= d(ζ,E)4 a.e. on T.
Then |g(z)| ≤ (π/2)4dist(z,E)4 (z ∈ D): indeed, for every ζ ∈ E, we have
0
1 1−|z|2
log|g(z)| ≤ 4log (π/2)|ζ −ζ | |dζ| =4log (π/2)|z −ζ | .
2π T |z−ζ|2 0 0
Z
(cid:0) (cid:1) (cid:0) (cid:1)
10 EL–FALLAH,KELLAY,ANDRANSFORD
By assumption, the zero set F := {ζ ∈ T :|f(ζ)| = 0} in contained in E, so
|g(z)| ≤ (π/2)4dist(z,F)4 (z ∈ D). Theorem 2.1 therefore applies, and we
can infer that g ∈ [f] . It thus suffices to prove that 1 ∈ [g] .
Dα Dα
We shall construct a family of functions w : [0,π] → R+ for 0 < δ < 1
δ
such that the associated outer functions f belong to [g] and satisfy:
wδ Dα
(i) |f∗ |→ 1 a.e. on T as δ → 0,
wδ
(ii) |f (0)| → 1 as δ → 0,
wδ
(iii) liminf kf k < ∞.
δ→0 wδ α
If such a family exists, then a subsequence of the f converges weakly to
wδ
1 in D , and since they all belong to [g] , it follows that 1 ∈ [g] , as
α Dα Dα
desired.
By Proposition 4.2, there exists µ > 0 such that |E | = O(tµ) as t → 0.
t
Fix ρ,σ satisfying
1−α 1−α+µ
<ρ < σ < min 1−α, .
2 2
n o
Define φ :(0,π] → R+ by
φ(t) := max min{|E |, tσ}, t1−α (t ∈ (0,π]).
t
n o
Clearly φ(t)/t is increasing and 0 ≤ φ(t) ≤ tσ for all t. We claim also that
π dt
= ∞. (8)
tαφ(t)
Z0
To see this, note that
π ds t π ds t1−α
≥ = C(α) ,
sα|E | |E | sα+1 |E |
Zt s t Zt t
whence
π dt π dt
≥
tαφ(t) max{t,tα|E |}
Zǫ Zǫ t
π ds
≥ C(α)
tα|E |( πds/sα|E |)
Zǫ t t s
π dt
≥ C(α)log R .
tα|E |
Zǫ t
Since E is a generalized Cantor set of α-capacity zero, Theorem 4.3 shows
that
π dt
= ∞.
tα|E |
Z0 t
Consequently (8) holds, as claimed.
Wehavenowshownthatφsatisfiesallthehypothesesoftheregularization
theorem, Theorem 5.1. Therefore there exists a function ψ : (0,π] → R+
satisfying the conclusions of that theorem, namely: ψ(t)/tρ is increasing,
t1−α ≤ φ(t) ≤ ψ(t) ≤ tσ for all t, and πdt/tαψ(t) = ∞.
0
R