Homogeneous Subsets of a Lipschitz Graph and the Corona Theorem

For a domain bounded by homogeneous subsets of a Lipschitz graph, we show the Corona Theorem is affirmative.


§1 Introduction
Let A : R → R be an M -Lipschitz continuous function. Thus A has a derivative almost everywhere such that A L ∞ = M . Let Γ be the Lipschitz graph parametrically defined by z(x) = x + iA(x) in the extended complex plane, and let E 0 be a closed set contained in Γ with Set Ω = C * \E 0 , and let H ∞ (Ω) denote the space of bounded analytic functions on Ω. In this paper, we prove: Theorem 1.1 (The Corona Theorem). Given f 1 , . . . , f n ∈ H ∞ (Ω) and µ > 0 with the property that µ ≤ max{|f j (z)| : 1 ≤ j ≤ n} ≤ 1 for every z ∈ Ω, there exist g 1 , . . . , g n ∈ H ∞ (Ω) such that f 1 g 1 + · · · + f n g n ≡ 1 on Ω.
We will refer to the functions {f j } n j=1 and {g j } n j=1 as the corona data and corona solutions respectively, and we will refer to µ and n as the corona constants.
There is an alternative way of viewing the theorem in the language of uniform algebras. Let us denote by M = M(H ∞ (Ω)) the maximum ideal space of H ∞ (Ω). When H ∞ (Ω) separates the points of Ω, we can identify elements of Ω with pointwise evaluation functionals in M. Under this identification, the theorem becomes equivalent to determining whether Ω is dense in M in the Gelfand topology. It is in this context where the theorem gets its name; whereby, in the special case where Ω is the unit disk, D, we can think of D as being the sun, and M\D as being the sun's corona.
Lennart Carleson (1962) proved the first corona theorem for the case of the disk [4]. His proof was subsequently simplified (using a ∂ equation) by Hörmander [15], and later by a clever proof by Wolff ( [8], [10]). The theorem was swiftly adapted to the case of finitely connected domains (Alling [1], [2]; Stout [24], [25], [26]; and others [6], [7], [23]). Each proof gave new insight into the structure of H ∞ . The finitely connected domain proofs were fundamentally based upon admixing localized corona solutions for overlapping simply connected components. One major drawback to this method was that the bounds of the corona solutions, g j ∞ , were dependent on the number of boundary components. This was an unfortunate hindrance as any planar domain can be exhausted by a sequence of finitely connected domains. Without a uniform bound on the corona solutions for the approximating domains, any method of taking normal limits was futile. In that direction, Gamelin [9] observed that the corona theorem for all planar domains would be true if and only if there existed a uniform bound for finitely connected domains which is independent of the number of boundary components. A proof to the corona problem for all planar domains remained a mystery.
Further investigations into the corona problem revealed a connection between interpolating sequences, boundary thickness, and the Cauchy transform. Along those lines, Carleson made another breakthrough by proving the corona theorem for domains with homogeneous boundary contained in the real line (homogeneous Denjoy domains). The significance of his result was that these domains are infinitely connected. Carleson lifted the corona data to the universal covering surface (where the corona solutions exist) and then mapped the solutions back to the original domain by an explicit projection operator invented by Forelli [7]. This concept was later simplified by Jones and Marshall [19]. They determined that if the critical points of the Green's function for a domain form an interpolating sequence, then there exists a projection operator and the corona theorem is affirmative. Moreover, they gave conditions necessary for determining when the critical points are indeed an interpolating sequence; one such condition can easily be proved when the boundary is homogeneous. Following these results, the corona problem for all planar domains bounded by a homogeneous subset of a graph seemed promising.
Peter Jones was the first to propose the idea of the corona problem for domains whose boundary lies in a Lipschitz graph [18]. He was motivated by the Denjoy conjecture, a consequence of Calderón's theorem on Cauchy integrals, which suggested that the space of bounded analytic functions was significantly abundant for these domains. Thereby, one might be able to construct "by hand" the corona solutions. As mentioned by Jones, the difficulty in the Lipschitz case was the lack of symmetry. At that time, the deepest results for the corona theorem were in the Denjoy domains (Ω = Ω) as in Carleson [5], Jones and Marshall [19], and Garnett and Jones [11]. These proofs made explicit use of the symmetry of the domains, either by confining the critical points to real intervals or by creating analytic functions by means of Schwarz reflection. Nonetheless, Jones (unpublished) proved the corona theorem for domains bounded by a homogeneous subset of a Lipschitz graph. He constructed by hand a projection operator akin to Forelli's.
For our proof, we work directly on the underlying space Ω without localizing the critical points of the Green's function, which can be cumbersome. We divide Ω into two overlapping simply connected regions, Ω + and Ω − . On each region, we use Carleson's simply connected result to obtain regional corona solutions, {g + j } n j=1 and {g − j } n j=1 . Starting in Ω + , we constructively solve a particular ∂ equation to modify {g + j } n j=1 so that max j |g + j (z) − g − j (z)| is reduced in the overlap of the regions. After the modification, we do a similar procedure in Ω − to reduce the differences even more, then iterate the procedure. The result of the iteration gets us two uniformly bounded sequences of solutions on each region. The ∂ equation was constructed specifically so that the normal limits of the sequences agree on the overlap of the regions.
In proving the theorem, we assume that E 0 consists of a finite union of closed intervals in Γ, two of which are unbounded. This assumption is easily removed by a normal families argument provided that the number of intervals does not control the bounds of the corona solutions. To be clear, when we use the phrase, "J is an interval in Γ" we mean p(J) is an interval in R for the projection p : Γ → R defined by p(z(x)) = x. It will also be convenient for us to consider the Lipschitz angle α = tan −1 (M ) for most of our calculations, instead of the slope M .
We mention here that there are two conditions equivalent to (1.1): When Γ is an M -Lipschitz Graph and E 0 ⊂ Γ, the following three conditions are equivalent: i) E 0 is homogeneous with a Carleson lower density 0 .
iii) There exists an 2 > 0 such that if we denote by J z,r = J L ∪ J R the interval in Γ containing z; J L is the subinterval having z as a right endpoint, and J R is the subinterval having z as a left endpoint with Λ(J L ) = Λ(J R ) = r, then Λ(J z,r ∩ E 0 ) > 2 r, for all z ∈ E 0 and all r > 0.
In addition, if either i), ii), or iii) hold, then iv) There exists an 3 > 0 such that cap (B(z, r) ∩ E 0 ) > 3 r for all z ∈ E 0 , and all r > 0.
The third item, iii), has the advantage that it applies to more general curves, while iv) is even more general: it says E 0 is uniformly perfect (see Pommerenke [22]). The crux of the proof for Lemma 1.2. is based upon the relationship of the projected length: Λ(J) ≥ |p(J)| ≥ cos(α)Λ(J) for an interval J ⊂ Γ. 1 The homogeneous condition combined with the fact that E0 is closed implies that 0 ≤ 1/2. The reason being that a complementary open interval F ⊂ Γ\E0 is not empty. As such, the double of F has a density less than 1/2.
Proof of Lemma 1.2: Let us first assume that i) holds. Fix z = x + iy ∈ Γ and r > 0. Since the projected mass of B(z, r) ∩ E 0 lies inside of (x − r, x + r) ∩ p(E 0 ) and condition ii) holds with 1 = cos(α) 0 . Now assume that ii) holds, and fix z ∈ Γ and r > 0. By simple geometric considerations, we see that B(z, r cos(α)) ∩ Γ ⊂ J z,r . This implies The last inequality is from ii). This implies condition iii) with 2 = cos(α) 1 .
Showing that iii) implies i) is simple as we can make the interval J z,r inside the ball B(z, r). Then condition iii) implies Λ(B(z, r) ∩ E 0 ) > 2 r. Thus E 0 is homogeneous with a Carleson lower density 2 .
Lastly, from the proof of Theorem III.11 in Tsuji [27], we have the relationship for This tells us that i) implies iv) with 3 = 0 cos(α) 2e .
For the proof of Theorem 1.1, we make the additional assumption that the tangent to Γ at a point ζ ∈ Γ is constant whenever ζ ∈ Γ\E 0 . This comes without any loss of generality. Specifically, if we write Γ\E 0 = ∪ k F k , then we define (see Figure 1.) Let us now fix some notation that will be used throughout the whole paper. We define a tent region over an interval J = (z 1 , z 2 ) with acute angle γ by Notice that Γ divides the plane into two simply connected components, Ω + and Ω − , where Ω + lies above Γ and Ω − lies below Γ. With these components, we fix two conformal maps and their inverses: We ask that Φ + (∞) = ∞ and Φ − (∞) = ∞. From Carathéodory's theorem we can extend our maps to homeomorphisms so that Φ + : H + → Ω + and Φ − : H − → Ω − respectively (see [20], Theorem I.1.). 2 We will be using two facts about about Φ + and Φ − : where * denotes reflection across F j .
2 By placing suitable minus signs, we may assume that Re{Ψ ± (z1)} < Re{Ψ ± (z2)} whenever z1, z2 ∈ Γ and Re{z1} < Re{z2}. Lemma 1.3. tells us that homogeneity is preserved by the maps (although the Carleson lower densities may be different), while Lemma 1.4. tells us that obtuse tents are mapped into obtuse tents. It should be mentioned that π/12 is not crucial for Lemma 1.4. We made this choice since the acute angle gets tripled in the lemma and, throughout this paper, we will only consider tents that have an acute angle less than π/4.

Proof of Lemma 1.3:
We use a result of Kenig [20]: if ν is the measure on R whose density is . Without loss of generality, let us assume that Λ(K R ) ≤ Λ(K L ). The A 2 relationship gives us a lower bound for Λ(K R ), Let J z be the interval inside K, as defined as in Lemma 1.2, with J z = J L ∪ J R , where J R = K R and J L is the interval with right endpoint z and length equal to Λ(K R ). From the proof of Lemma 1.2, we know that Λ(E 0 ∩J z ) > 0 cos(α)Λ(K R ), and when we combine this inequality with (1.2) we have By a result of Muckenhoupt [21], ν ∈ A 2 on R implies ν ∈ A ∞ on R. Hence, there exist constants c 1 > 0 and c 2 > 0 independent of E + and r such that, Combining the above relationship with (1.3), we see that E + is homogeneous with a Carleson lower density depending only upon 0 and α.
Proof of Lemma 1.4: The appearance of the * and the conjugation bar for the statement in the lower half plane arise since the tents have an orientation to be above the intervals. It is not difficult to see that the two statements remain alike upon modifying the arguments in the definition of the tents, and we will only prove the result for the upper half plane.
Fix a tent domain T (I + j ,γ) over I + j = Ψ + (F j ) ∈ R and write log [(Φ + ) ] (z) = f 1 (z) + if 2 (z) (take a principle determination). Again from Kenig [20], we have a bounded argument for the derivative, that is |f 2 (z)| ≤ α for all z ∈ H. As such, we can represent f 2 (z) with a Poisson integral of the values coming from its non-tangential limits on the real line: The final equality holds since f 2 = c j over I + j . Taking absolute values of the above equality Additionally, if z ∈ T (I + j ,γ) and γ ≤ π/12, then |f 2 (z) − c j | ≤ 4αγ/π by taking simple estimates for harmonic measure. This lets us conclude that the values of the derivative lie in the cone domain: We conclude that Φ + (z) lies in T (F j ,3γ) . §2 Four Crosscuts Merging the two tents together for all j, we make the diamonds D j = D + j ∪ D − j . The parameters for α M were chosen so that α M < π/4 and D j ∩ D k = ∅ for j = k (see Figure 2).
In this section, we construct four families of crosscuts that encompass the open intervals F j and lie inside D j . To do so, we will first need some elementary harmonic measure estimates.
In the upper half plane H + , This is shown by decomposing the Poisson kernel for the upper half plane into a sum of box kernels centered around x ∈ E + (P z (t) = k a k X A k (t)). With this representation, If we write the complement of E + in the real line as j I + j = R\E + = j Ψ + (F j ), then ω(z, ∪I + j , H + ) < 1 − when Re{z} ∈ E + . This tells us that we have the bounds ω(z, ∪I + j , H + ) < 1 − on the sides of the vertical half strip defined with base I + j extending vertically in the upper half plane. We can apply these bounds to get harmonic measure estimates on the boundaries of the diamonds: Proof of Lemma 2.1: Fix z ∈ ∂T (I + k ,γ) for some k and normalize I + k into (−π/2, π/2), and let us denote the half strip over I + k by S + = {x + iy : −π/2 < x < π/2 : y > 0}. From the preceding remarks, By symmetry, this tells us that sin(z) Hence, We remark that Lemma 2.1. can easily be proved without conformal maps but with a weaker bound on harmonic measure. This comes from the observation that if we denote by d = dist(z, E + ), then |E + ∩ B(z, 2d)| > d. This implies for each z ∈ j ∂T (I + j ,γ) there exists a subset of E + with linear measure proportionate to the distance of z and the real axis. The upper bounds for harmonic measure now follow from estimating the Poisson kernel over these sets.
With the estimates following from Lemma 2.1, we can now define our desired crosscuts.
Recall, a Carleson contour in the upper half plane is a countable union C of rectifiable arcs in H + such that for every interval I ⊂ R, This implies arc length on C is a Carleson measure with constant C(C).
and by taking a derivative, Separating the real and imaginary parts gives us the ratio It is also clear that ω y (z) > 0 for all z in the tents, so that by comparing the like terms in the sums of (2.1) with the ratio in (2.2), As the curve δ + 1 is a level set, the gradient of ω at any point is perpendicular to the tangent of the curve. With the above ratio, we conclude that the tangent to the level curves is bounded in argument by 2α M 3 . This means that δ + 1 is a Carleson curve with a constant of sec 2α M 3 . §3 The Regions D + and D − Now that we have the cross cuts {γ ± j } j=1,2 , we may define the following extended domains: let Ω + be the simply connected domain containing Ω + that is bounded by the closed intervals of E 0 and the bottom crosscuts γ − 1 . As Ψ + (z) has a constant argument on each F j , and the crosscuts of γ − 1 lie in disjoint diamonds, Ψ + (z) can be extended (by reflecting across each F j ) to a map Ψ + : Ω + → H + , where H + is the domain containing H + that is bounded by E + and Ψ + (γ − 1 ). 4 4 We are not identifying the crosscuts Ψ + (γ − 1 ) with the crosscuts δ − 1 .
For a nice discussion on interpolating sequences and a proof of Carleson's interpolation theorem see [10], §VII.
Since the sequence lies in a Carleson contour (by Lemma 2.2) and it is hyperbolically separated, we know that δ H + ({z m }) = C(A, , α M ) > 0 (see [10], §VII). Carleson's interpolation theorem then implies that {z m } ∞ m=1 is an interpolating sequence for H + . It is also the case that {z m } ∞ m=1 is an interpolating sequence for the extended domain H + . This follows from Garnett and Jones [11] (Theorem IV.1), and applies in our case since H + is a subset of a Denjoy domain. Alternatively, {z m } ∞ m=1 can be shown to be an interpolating sequence for H + by a result of González and Nicolau [13]. From their result, it suffices to have δ H + > 0 for the image of {Φ + (z m )} under the canonical quasi-conformal map that takes the domain (∪ j D j ) ∪ Ω + to the upper half plane. Since the quasi-conformal map is explicit, it is easy to verify; we omit the details. In any case, there exist interpolating functions for the domain H + with a constant of interpolation N = N (A, α M , ).
Working again in the upper half plane, let B = min A, 1 6N 2 and denote the region We chose A so that by Harnack's inequality D + lies above δ + 2 , that is ω(z, ∪I + j , H + ) < β 2 , for all z ∈ D + (see Figure 3.).

Now fix a sequence {z
for all z ∈ D + .

(3.2)
The existence of such a sequence follows by taking a maximal sequence satisfying (3.1). We now follow a standard argument as originated in Garnett and Jones [11] (Lemma II.2) and as used in Handy [14] (Lemma III.2) to obtain a specific set of interpolation functions:

4)
and Proof of Lemma 3.1: By a stopping time argument used to group {z + m } into generations (see [10], pg. 416), we may split {z + m } into a finite union of disjoint subsequences for all k.
Since the points of S k are hyperbolically separated by A, our earlier discussion implies that each S k has a constant of interpolation less than N .
Let us restrict our attention to a fixed subsequence S k . If we assume that S k = {z 1 , z 2 , . . . , z n 0 } is finite, then there exists f j ∈ H ∞ ( H + ) such that f j H ∞ ≤ N and f j (z m ) = ω mj , where ω = e 2πi/n 0 . Moreover, if we define Therefore, by exhausting each S k and taking the normal limits, we have The technique used above of averaging interpolating functions is due to Varopoulos [28]. We made our choice of N 2 B ≤ 1/6 specifically so that if we write D + as the disjoint union of sets D + n ⊂ {z : |z − z + n | ≤ 3By + n }, then with (3.4) and Schwarz Lemma (3.6) |h + n (z)| > 1/2 whenever z ∈ D + n . Since throughout this chapter we could change all plus signs to minus signs, we could likewise define our friends:

Blending of Corona Solutions
We now begin the process of "sewing" together corona solutions from the simply connected domains Ω + and Ω − . Let {g 0 j } n j=1 be an arbitrary corona solution set for Ω + and let {g 1 j } n j=1 be an arbitrary corona solution set for Ω − . These solution sets exist from Carleson's simply connected corona theorem; furthermore, there is a uniform bound for the sets: The First Stitchings.
The first sewing of the corona solutions will be across the region D + in H + . Let us denote by ω(z) = ω(z, ∪I + j , H + ) as the harmonic measure for the open intervals j I + j in the upper half plane. Although ω(z) is not defined on the extended domain, we use the convention ω(z) = 2 − ω(z, ∪I + j , H + ) to extend ω to H + . (When working in H − , we will also be denoting with ω(z) and it should be clear from context.) We can think of both {f j (z)} n j=1 and {g 0 j (z)} n j=1 as being defined in H + under the map Φ + (z), and we will not change our notation. On the other hand, when we write {g 1 j (z)} n j=1 we must remember that these functions are only defined in H + between the curves { Ψ + (γ − 1 )} and {δ + 1 } (see Figure 4). Because of a calculation advantage, we have chosen not to sew on the Ω + side where the corona data and solutions are originally defined, but instead work in the extended half planes where we have defined D + , {z + m }, and {h + m (z)}. Let {ϕ + (z), 1−ϕ + (z)} be a smooth partition of unity for H + across D + , with ϕ + (z) = 1 when ω(z) ≤ β 1 . By standard arguments, |y + n ||∇ϕ + (z)| ≤ CB −1 for all z ∈ D + n , where C is a positive constant. Using our partition, let us piece together the two families of corona solutions: . . , n. These smooth functions are a well defined solution set to the corona equation for the region H + , but they are not necessarily analytic. Therefore, we consider a technique of Hörmander [15] (and as used in [11] and [14]). We seek to find functions Indeed, such functions provide the necessary cancelation to make the collection a solution set while simultaneously solving the equation ∂g 2 j = 0 in the sense of distributions. Then upon modifying each a 2 j,k on a set of measure zero, Weyl's lemma will allow us to conclude that the collection {g 2 j } n j=1 is a bona fide corona solution set in H + .
For our construction, we require not only that the functions {a 2 j,k } are bounded, but also have an additional convergence factor. Fix 1 > b 1 > 0 (to be determined later) and denote byω(z) as the harmonic conjugate for ω(z). Consider the equation Formally ∂a 2 j,k = G 2 j ∂G 2 k , so we need to check the convergence of the sum Using (3.6) and recalling ω(ζ) < β 2 when z ∈ D + , Before we show the above is a convergent sum, we would like to identify some key numbers that will appear in the iterative process. Using the notation · D + and · D − for the supremum of the modulus in the region D + and D − respectively, let us label k D − when m is even, when m is odd, when m is even.
So that in our context, and by using (3.5) we reduce the inequality to We conclude that {a 2 j,k } ⊂ L ∞ ( H + ) as it is easy to verify x 1 and y 1 are bounded with N . Moreover, if we apply these bounds to the relationship (4.1), then we get the bounded equation where K is an absolute constant which depends only upon the corona constants (µ, δ, and n), the geometric considerations ( and α M ), and our choice of A. Lastly, under the map Φ + (z) we can regard the newly constructed {g 2 j } n j=1 and {G 2 j } n j=1 as being functions defined on Ω + .

The Subsequent Stitchings.
In the same fashion that we used to construct the relationship (4.1), we could construct the third generation of solutions, {g 3 j } n j=1 and {G 3 j } n j=1 , by stitching the newly formed {g 2 j } n j=1 to {g 1 j } n j=1 across the region D − in H − . As soon as the third generation of solutions are constructed, we repeat the process, just as we did in the first stitchings, to obtain the fourth generation of solutions, {g 4 j } n j=1 and {G 4 j } n j=1 , by stitching {g 3 j } n j=1 to {g 2 j } n j=1 across D + in H + . Iterating this procedure with the sequences {b m } ∞ m=1 , {x m } ∞ m=1 , and {y m } ∞ m=1 , we deduce the analogues of (4.1) and (4.2): Since we will be referring to (4.4) many times from here, we consider some variations. Each variation is customized to the location of the variable z. Recall, when m is odd, z ∈ H + , and z lies below δ + 2 , g m−1 j (z) when m is odd, z ∈ H + , and z lies above δ + 1 , when m is even, z ∈ H − , and z lies above δ − 2 , g m−1 j (z) when m is even, z ∈ H − , and z lies below δ + 1 .
Next, let us deduce a recursive relationship for {y m } ∞ m=1 . It is easy to verify y 1 ≤ N . Now suppose m is odd, then y m+2 = max k G m+3 Let us take a look at the bounds for the functions in the above equality. Recall, ω(z) > β 1 when z ∈ D + so that (4.4) reduces to In addition, (4.4d) and the previous inequality imply Since the bound for |g m+2 k (z)| is greater than the bound for |g m+1 k (z)|, we deduce that We could repeat verbatim the case where m is even, and for the case of y 2 with the bound g 1 j ∞ ≤ N , so that we obtain the second recursive relationship: The last two terms in this expression arise from the error in stitching over two generations. It will be our goal to show that these errors are summable. §5 Convergence of x m In this chapter we select the factors {b m } ∞ m=1 so that {x m } ∞ m=1 ∈ 1 and {y m } ∞ m=1 ∈ ∞ simultaneously: such that for any pair of positive sequences {x m } ∞ m=1 and {y m } ∞ m=1 that satisfies the difference equations with initial data x 1 ≤ 2N, y 1 ≤ N , and y 2 ≤ N + Kb Proof of Lemma 5.1: Since the pair of sequences ({x m } ∞ m=1 and {y m } ∞ m=1 ) that have equality holding in the initial data and equality holding in (R1) and (R2) dominate all admissible pairs, it suffices to solve for {b m } ∞ m=1 for this particular pair. In addition, without loss of generality, we may assume that y 1 , y 2 ≥ 1. Now, fix 1 > r > 0 (to be determined later) and take b m so that Next, we substitute the right hand side of the above equation to reduce (R2), When m ≥ 2, we can solve for b m in terms of r and y m by looking at successive generations of (5.1). Specifically, the right hand side of (5.1) at the m th generation is x m y m = r m , and by substituting the left hand side of (5.1) for x m makes If we raise both sides of the above equality to the power β 1 −1 For the case where m = 1 we can repeat the preceding, but with If we substitute these relations for b m and b 1 into (5.2), then we have the ordinary difference equation: To get bounds for y m , we look at the difference The last inequality holds since 1 ≤ · · · ≤ y m ≤ y m+2 . By telescoping the differences, starting with y 4 and y 5 for m even and m odd respectively, We recall, y 1 = N , y 2 = N + r; and by using (5.4a) and (5.4b), y 3 = O(r −1 ), y 4 = O(1), and y 5 = O(r −1 ). So that for r sufficiently small, Fix such an r small enough so that the above inequalities holds and make sure r > r 0 > 0 so that, where C is some absolute constant depending only upon 0 , α, and A. We begin by looking at g m j H ∞ ( H + ) in the extended upper half plane. Fix z ∈ H + and m odd. From (4.4) and variation (4.4a) for the regions {ω(z) ≤ β 1 }, {β 1 < ω(z) ≤ β 2 }, and {β 2 < ω(z)} respectively, we have Let us treat each region as its separate own special case.
From the first relationship (6.1), if we telescope the differences over the even generations of {g m j }, then since sup k {y k } < C 2 , k x k < C 1 , and inf k {b k } ≥ r 0 KC 2 2 1−β 1 from Lemma 5.1.
For this case and the next we need some estimates similar to the ones we obtained when we derived the second recursive relation, (R2). Recall the relationship we have x k y k ≤ C < ∞.
Moreover, (6.5) implies that g + k (z) = g − k (z) for all z ∈ Γ\E 0 . Therefore, we can merge the two solutions together across Γ\E 0 , and obtain corona solutions on all of Ω.
For our proof, the homogeneous condition was critical. Without it, we would not have been able to bound the crosscuts γ + 1 and γ − 1 into the disjoint diamonds, leaving the extended domains as multiply connected. A proof for the non-homogeneous case still eludes the author. One might hope to avoid this obstacle by directly applying the results of the non-homogeneous cases, (e.g., the Denjoy domains).
The present work is part of the author's Ph.D. dissertation. Most of all, the author would like to express his genuine gratitude to his thesis advisor, John Garnett, for countless hours of insightful conversations and guidance throughout the past couple years. The author is truly indebted for his support.