Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 24, 1999, 105–108

ON V. I. SMIRNOV DOMAINS

Peter W. Jones and Stanislav K. Smirnov

Yale University, Department of Mathematics New Haven, CT06520, U.S.A.; stas@math.yale.edu

Abstract. We show that complement of a non-V. I. Smirnov domain, coming from the Duren–Shapiro–Shields or Kahane construction must be a V. I. Smirnov domain. There is therefore a negative answer to the old question: need a complement of a non-V. I. Smirnov domain be a non-V. I. Smirnov domain itself?

The purpose of this note is to prove a general, elementary theorem that provides an answer to an old problem on V. I. Smirnov domains. Recall that if Γ is a rectifiable closed Jordan curve and Ω_{+} is the bounded component complementary to Γ, Ω_{+} is a V. I. Smirnov domain if F ′_{+} is an outer function. Here F_{+} is any choice of conformal map of D onto Ω_{+} . The class of such domains was introduced by V. I. Smirnov in [S] in connection with some questions of approximation theory. Consult the expository paper [D] of P. Duren for their properties and further references.

Several authors (starting with P. Duren, H. S. Shapiro, and A. L. Shields, see [DSS]) in the 1960’s found examples of Γ and Ω_{+} , with F ′_{+} a singular inner function,

(1)  F ′_{+} = exp{−(μ + iμ̃)}

where μ is a positive measure on the circle that is singular with respect to Lebesgue measure, dμ ⊥ dθ. When Condition (1) holds, Ω_{+} is not of V. I. Smirnov type (existence of non-Smirnov domains was established earlier by M. V. Keldysh and M. A. Lavrentiev in [KM]). If we let Ω_{−} denote the unbounded domain complementary to Γ, we have a similar definition of V. I. Smirnov (and non-V. I. Smirnov) domains.

Theorem. Let Γ, Ω_{+} , Ω_{−}, be as above. Then if F_{+} satisfies Condition (1), F_{−} (the conformal map of {|z| > 1} to Ω_{−}) satisfies

(2)  |F ′_{−}(z)| ≥ c > 0,  |z| > 1.

Corollary. If Ω_{+} is a non-V. I. Smirnov domain satisfying Condition (1), Ω_{−} is a V. I. Smirnov domain.

1991 Mathematics Subject Classification: Primary 30C20; Secondary 30C45, 30C85, 31A15.

The first author is supported by N.S.F. Grant No. DMS-9423746.

There is therefore a negative answer to the old question (see [T1], [T2] for dicsussion and references): Does Ω_{+} V. I. Smirnov imply Ω_{−} V. I. Smirnov?

Proof of Theorem. Let c_{1} := |F ′_{+}(0)| and let ω_{+} denote harmonic measure for Ω_{+} (on ∂Ω_{+} ) with respect to F_{+}(0). Then by Condition (1),

dω_{+} = \frac{1}{2πc_{1}} ds,

where ds = dℋ ^{1} is one dimensional Hausdorff measure on Γ. An immediate consequence of this is the inequality

(3)  ω_{+}(D(x, r)) ≥ \frac{r}{πc_{1}} ,  x ∈ Γ,

where D(x, r) is the disk centered at x with radius r < diam (Γ) (we use the notation ω_{+}(E) := ω_{+}(E ∩ Γ)).

Now we invoke the result of C. Bishop, L. Carleson, J. Garnett, P. Jones (see [BCGJ])

(4)  ω_{+}(D(x, r))ω_{−}(D(x, r)) ≤ c_{2}r^{2},

where ω_{−} is harmonic measure for Ω_{−} with respect (say) to ∞. Here the constant c_{2} depends on F_{+}(0), but not on x ∈ Γ, r < diam (Γ). This result is valid for harmonic measures on any two disjoint, simply connected domains Ω_{+} , Ω_{−}, and its proof is quite elementary. Now from (3) and (4) we obtain

(5)  ω_{−}(D(x, r)) ≤ c_{3}r,  x ∈ Γ, r < diam (Γ).

It is an easy exercise that Condition (5) implies Condition (2) (one can look at the harmonic measure of D(x, 2dist (x, Γ))), and that F ′_{−} is an outer function.

Remark. In certain cases, one can draw a stronger conclusion, than (2). For example, let μ come from Kahane’s construction in [K]. Let K_{+} ⊂ S^{1} be the closed support of μ. Then K_{−} := F −1_{−}(F_{+}(K_{+})) ⊂ S^{1} satisfies

(6)  Box Dimension (K_{−}) < 1.

On the other hand, general results (the sharp version is due to N. Makarov, [M]) show that if a set K ⊂ S^{1} has zero measure for the Hausdorff gauge function t\sqrt{log (1/t) log log log (1/t)}, then there is no Riemann mapping defined on D with the properties F′ ∈ H^{1} (the Hardy space),

F′ = G exp{−(μ + iμ̃)},

where G is outer, dμ ⊥ dθ, and μ is supported on K.

To verify (6) one can argue as in the following sketch. One first proves that for all e^{iθ} ∈ K_{+} and \frac{1}{2} ≤ r < 1, there is R ∈ (r, \frac{1}{2} + \frac{1}{2}r) such that

|SF_{+}(z)| ≥ ε(1 − |z|)^{−2}, z := Re^{iθ,}

where S denotes the Schwarzian derivative. Here ε > 0 is independent of Γ and θ. An easy normal families argument then yields

(7)  β_{Γ}(x, r) ≥ δ,  x ∈ F_{+}(K_{+}), r < diam (Γ).

Here β is the usual measure of “deviation from flatness” for the set Γ ∩ D(x, r) (see e.g. [J]). Then estimates on harmonic measure (similar to those proving Condition (4), but we get a stronger result because of the “twisting”, provided by Condition (7)) yield

ω_{+}(D(x, r))ω_{−}(D(x, r)) ≤ cr^{2+α}, x ∈ F_{+}(K_{+}).

Fortunately the argument for this inequality is given in the paper [R] of S. Rohde. S. Rohde proves his result for K_{+} = S^{1} , but one sees easily, that his argument will work in our case.

Combining the last inequality with (3) one obtains

(8)  ω_{−}(D(x, r)) ≤ cr^{1+α},  x ∈ F_{+}(K_{+}).

Now cover F_{+}(K_{+}) by disks of harmonic measure comparable to 2^{−n}. Applying the Besikovitch covering lemma and pulling back by F_{−} yields

Box Dimension (F −1_{−}(F_{+}(K_{+}))) < \frac{1}{1 + α}.

Here one must use that Γ has finite length. We remark that applications of estimates like (8) are the main point of S. Rohde’s paper.

References

[BCGJ] Bishop, C., L. Carleson, J. Garnett, and P. Jones: Harmonic measures supported on curves. - Pacific J. Math. 138, 1989, 233–236.

[D] Duren, P.L.: Smirnov domains. - Zap. Nauchn. Sem. LOMI 170, 1989, 95–101; Issled. Linein. Oper. Teorii Funktsii. 17; see also J. Soviet Math. 67, 1993, 167–170.

[DSS] Duren, P., H.S. Shapiro, and A.L. Shields: Singular measures and domains not of Smirnov type. - Duke Math. J. 33, 1966, 247–254.

[J] Jones, P.W.: Rectifiable sets and the traveling salesman problem. - Invent. Math. 102, 1990, 1–15.

[K] Kahane, J.P.: Trois notes sur les ensembles parfait linéares. - Enseign. Math. 15, 1969, 185–192.

108 Peter W. Jones and Stanislav K. Smirnov

[KL] Keldysh, M.V., and M.A. Lavrentiev: Sur la représentation conforme des domaines limités par des courbes rectifiables. - Ann. Sci. École Norm. Sup. 54, 1937, 1–38.

[M] Makarov, N.G.: Size of the set of singular points on the boundary of a non-Smirnov domain. - Zap. Nauchn. Sem. LOMI 170, 1989, 176–183 (Russian); Issled. Linein. Oper. Teorii Funktsii. 17; English transl.: J. Soviet Math. 67, 1993, 212–216.

[R] Rohde, S.: On conformal welding and quasicircles. - Michigan Math. J. 38, 1991, 111–116.

[S] Smirnov, V.I.: Sur la théorie des polynômes orthogonaux à la variable complexe. - Zh. Leningr. Fiz.-Mat. Ob-va 2, 1928, 155–179.

[T1] Tumarkin, G.Ts.: Boundary conditions for conformal mappings of certain classes of domains. - In: Some Problems in Modern Function Theory, Proc. Conf. Modern Problems of Geometric Theory of Functions, Inst. Mat., Akad. Sci. USSR, Novosibirsk, 1976, 149–160 (Russian); Akad. Nauk SSSR Sibirsk. Otdel. Inst. Mat., Novosibirsk, 1976.

[T2] Tumarkin, G.Ts.: Some problems concerning classes of domains determined by properties of Cauchy type integrals. - In: Linear and Complex Analysis Problem Book 3, Lecture Notes in Math. 1573, Springer-Verlag, Berlin, 1994, 411–413.

Received 29 April 1997