Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 23, 1998, 283–290

Quasiconformal mappings preserving interpolating sequences

María J. González and Artur Nicolau

Universitat Autònoma de Barcelona, Departament de Matemàtiques E-08193 Bellaterra, Barcelona, Spain

Abstract. We consider quasiconformal mappings from the upper half plane onto itself and we show that a necessary and sufficient condition for such a mapping to preserve interpolating sequences is that its restriction to the boundary is a strongly quasisymmetric function, that is absolutely continuous with derivative an A_{∞} weight.

We will characterize the quasiconformal mappings from the upper half plane, R2^{+} , onto itself that preserve interpolating sequences.

Before stating the theorem, let us recall a few definitions and results.

A sequence of points {z_{n}} in R2^{+} is an interpolating sequence for H^{∞}(R2^{+}) if for any sequence (a_{n}) ∈ l^{∞}, there exists a function f ∈ H^{∞}(R2^{+}) such that f(z_{n}) = a_{n}, for all n.

Related to interpolating sequences is the notion of Carleson measure. A positive measure μ on R2^{+} is called a Carleson measure if there is a constant N(μ) such that

μ(Q) ≤ N(μ)l(Q)

for all squares

Q = {x_{0} < x < x_{0} + l(Q) : 0 < y < l(Q)}.

The smallest such constant N(μ) is the Carleson norm of μ.

Carleson proved that a sequence {z_{n}} in R2^{+} is an interpolating sequence if and only if the following two conditions hold:

(i) The points {z_{n}} are separated, that is, there exists a constant a such that

ϱ(z_{n}, z_{k}) = |\frac{z_{n} − z_{k}}{z_{n} − z̄_{k}} | ≥ a > 0, n ≠ k.

(ii) ∑ y_{n}δ_{z_{n}} is a Carleson measure [G].

1991 Mathematics Subject Classification: Primary 30C62.

Supported by DGICYT Grant PB95-0956 and by CIRIT grant 1996SRG00026.

A homeomorphism F defined in a domain Ω is quasiconformal if it has locally integrable distributional derivatives and Fz̄ = μF_{z}, where μ ∈ L^{∞}(C), ∥μ∥_{∞} < 1. We use the notation

\begin{aligned}Fz̄ & = \overline{∂}F = \frac{1}{2}(\frac{∂}{∂x} + i\frac{∂}{∂y} )F, \\ F_{z} & = ∂F = \frac{1}{2}(\frac{∂}{∂x} − i\frac{∂}{∂y} )F. \end{aligned}

We will say that a quasiconformal map F from R2^{+} onto itself preserves interpolating sequences if for any interpolating sequence {z_{n}} in R2^{+} , its image {F (z_{n})} is also interpolating.

If F is a quasiconformal map from R2^{+} onto itself, its restriction to R, h, gives a homeomorphism on R that satisfies a doubling condition:

\frac{1}{M} ≤ |\frac{h(x + t) − h(x)}{h(x) − h(x − t)}| ≤ M for all x, t.

Such a homeomorphism is called quasisymmetric, and in general it is not even absolutely continuous. On the other hand, any quasisymmetric function h: R → R, can be extended to a quasiconformal map on R2^{+} whose restriction to R is h [A].

Let w(x) > 0 be locally integrable on R. Let w(E) = ∫_{E} w(x) dx and let |E| denote Lebesgue measure of E. We say that w is an A_{∞} weight if there exists a pair of constants α, β with 0 < α, β < 1 so that, for all intervals I and all subsets E ⊆ I,

|E| ≤ α|I| ⟹ |w(E)| ≤ β|w(I)|.

See [S] for equivalent definitions.

Suppose that h is an increasing homeomorphism from R onto R. We say that h is strongly quasisymmetric if it is locally absolutely continuous and h′ ∈ A_{∞}. Note that these conditions imply that h is quasisymmetric.

We are ready to state the main result:

Theorem. Let F be a quasiconformal map from R2^{+} onto R2^{+} . Then F preserves interpolating sequences if and only if F |_{R} is strongly quasisymmetric.

This is not a surprising result if we consider Jones’ result on homeomorphisms on R that preserve BMO(R) [J]. He proved that if h is an increasing homeomorphism on R, then for any function f ∈ BMO(R), f ∘ h is also in BMO(R) if and only if h′ ∈ A_{∞}. Now, as is very well known, BMO(R) and Carleson measures in R2^{+} are closely related.

On the other hand, the fact that if F |_{R} is strongly quasisymmetric then it preserves interpolating is already proved in [AZ]. In any case, we will give a different proof of this result in a more geometric way.

We would like to extend this result to any Carleson measure. We will say that F: R2^{+} → R2^{+} preserves Carleson measures if given any Carleson measure μ in R2^{+} , the measure ν defined in R2^{+} as

ν(E) = ∫_{F^{−1}(E)} a_{F} (z) dμ(z)

is a Carleson measure. The function a_{F} (z) is somehow the quasiconformal substitute of |F′| in the Koebe distortion theorem. To be more precise, set

a_{F} (z) = \frac{1}{|B_{z}|} ∫∫_{B_{z}} J_{F} (ξ)^{1/2} dξ d\overline{ξ}

where B_{z} is the disk of center z and radius \frac{1}{2}y and J_{F} is the Jacobian of F. Then, Im F (z) ≅ a_{F} (z) · y [AG].

Corollary 1. Let F be a quasiconformal map from R2^{+} onto R2^{+} . Then F preserves Carleson measures if and only if F |_{R} is strongly quasisymmetric.

Proof. [G, p. 360] shows that given any Carleson measure μ, there is a sequence of Carleson measures {μ_{n}} such that μ_{n} → μ in the weak sense and

μ_{n} = \frac{1}{N} ∑_{1}^{4N} μ̃_{k}, N = N(n),

with μ̃_{k} = ∑ y_{jk}δ_{z_{j,k}} and {z_{jk} : j = 1, 2, …} an interpolating sequence having uniformly bounded constants. Therefore it is enough to prove the result for such μ′_{n} s. So, given any Carleson cube Q:

∫_{F^{−1}(Q)} a_{F} (z) dμ_{n}(z) = ∑_{k=1}^{4N} ∫_{F^{−1}(Q)} a_{F} (z)\frac{1}{N} dμ_{k}

= ∑_{k=1}^{4N} \frac{1}{N} ∑_{z_{j,k}∈F^{−1}(Q)} a_{F} (z_{j,k})y_{j,k}

≅ ∑_{k=1}^{4N} \frac{1}{N} ∑_{F (z_{j,k})∈Q} Im(F (z_{j,k})) ≤ c\frac{4N}{N} |l(Q)|,

with c depending on N(μ̃_{k}) and F.

Let Γ be a quasicircle passing through ∞, that is, Γ is the image of a line under a quasiconformal mapping of the plane. Let Ω_{+} , Ω_{−} be the two domains bounded by Γ and Φ_{+} , Φ_{−} the corresponding conformal mappings from R_{2}^{±} onto Ω^{±}. Then both conformal mappings extend to be quasiconformal on the whole plane [A]. The map Φ^{−1} ∘ Φ_{+}: R → R is quasisymmetric and it is called the conformal welding of Γ.

Also, any quasicircle Γ admits a quasiconformal reflection, that is, there is a sense reversing quasiconformal map which interchanges Ω^{+} , Ω^{−} and keeps Γ pointwise fixed [A].

As before we say that a sequence {z_{n}} ⊂ Ω is an interpolating sequence in Ω if it is interpolating for H^{∞}(Ω) and that a quasiconformal reflection r preserves interpolating sequences if given any interpolating sequence {z_{n}} in Ω^{+} , the reflected sequence {r(z_{n})} is interpolating in Ω^{−}.

Corollary 2. Let Γ be a quasicircle bounding the domains Ω^{+} , Ω^{−}. Then any quasiconformal reflection through Γ preserves interpolating sequences if and only if the conformal welding of Γ is strongly quasisymmetric.

Proof. Any quasiconformal reflection r through Γ is of the form r = f ∘ j ∘ f^{−1} where f is a quasiconformal map from R_{2} onto R_{2} and j is the reflection through R.

Let

{w_{n}} = {Φ−1_{+}(z_{n})}  and  {w_{n}^{∗}} = {j ∘ Φ−1_{−} ∘ r(z_{n})}.

Then w_{n}^{∗} = j ∘ Φ−1_{−} ∘ (f ∘ j ∘ f^{−1}) ∘ Φ^{+}(w_{n}). So, {w_{n}^{∗}} is the image of {w_{n}} under a quasiconformal map of the plane that sends R onto itself and whose restriction to R is the conformal welding of Γ. Applying the theorem to this function we get the corollary.

The rest of the paper will be devoted to the proof of the theorem.

Proof of the necessity. First we will show that if F preserves interpolating sequences then F |_{R} is absolutely continuous. This will be an easy consequence of the following lemma.

Lemma 1. Let E ⊂ R be a compact set with |E| = 0. Then there exists an interpolating sequence {z_{n}} whose set of non-tangential accumulation points is E, that is

E = {x ∈ R : #(Γ(x) ∩ {z_{n}}) = ∞}

where Γ(x) = \{z ∈ R2^{+} : |z − x| ≤ \sqrt{2} Im z\}.

Proof. We consider coverings of E by a set of disjoint open intervals E ⊂ ⋃_{j=1}^{N(k)} I_{j}^{(k)} , k = 1, 2, …, satisfying

(1) ⋃_{j} I_{j}^{(k+1)} ⊆ ⋃_{l} I_{l}^{(k)} , k = 1, 2, …;

(2) E = ⋂_{k} ⋃_{j} I_{j}^{(k)} ; (3) ∑_{I_{j}^{(k+1)}⊆I_{l}^{(k)}} |I_{j}^{(k+1)}| ≤ \frac{1}{2}|I_{l}^{(k)}|.

Given an interval I ⊂ R, we associate to I the point z_{I} = x_{I} +i\frac{1}{2}|I| where x_{I} is the center of I. Let {z_{n}} be the sequence associated to the intervals {I_{j}^{(k)}}_{j,k}. Then {z_{n}} is interpolating because of (3). Also, if I is an interval in R, the point z_{I} ∈ Γ(x) if and only if x ∈ I. Therefore z_{n} ∈ Γ(x) for infinitely many n if and only if x ∈ I_{j}^{(k)} for infinitely many k, that is, x ∈ E.

Let h be the restriction of F to R. Let us show that h is absolutely continuous.

Suppose it is not. Then there is a compact set E ⊂ R such that |E| = 0 and |h(E)| > 0. Consider the interpolating sequence given by the lemma. Because of the circular distortion theorem [L, p. 17], the non-tangential accumulation set of {F (z_{n})} contains h(E). Actually,

h(E) ⊆ {x ∈ R : |x − F (z_{n})| ≤ c Im F (z_{n}), n = 1, 2, …},

where c is a constant only depending on the quasiconformal constant of F. Hence, for each k the intervals centered at Re F (z_{I_{j}^{(k)}}) with length 2c Im F (z_{I_{j}^{(k)}}), j =

1, …, N(k), cover h(E). Thus

∑_{j=1}^{N(k)} Im F (z_{I_{j}^{(k)}}) ≥ \frac{1}{2c}|h(E)| > 0

and ∑_{k=1}^{∞} ∑_{j=1}^{N(k)} Im F (z_{I_{j}^{(k)}}) = ∞.

So {F (z_{n})} is not interpolating and we get a contradiction.

Let us now finish the proof of the sufficiency by showing that h′ ∈ A_{∞}. Assume h′ ∉ A_{∞}, then for each n = 1, 2, …, there is an interval I_{n} and a

set E_{n} ⊂ I_{n} such that

|E_{n}| < 2^{−n}|I_{n}|  and  |h(E_{n})| > (1 − 2^{−n})|h(I_{n})|.

We can also assume that |I_{n+1}| < \frac{1}{10}|E_{n}| (to see this, split I_{n} into two intervals of the same length In^{+} and In^{−} and consider E_{n}^{±} = E ∩ I_{n}^{±}), and also that E_{n} is union of dyadic intervals of I_{n}.

We will build an interpolating sequence {z_{n}} such that {F (z_{n})} is not interpolating. To do so we need another lemma.

Lemma 2. Let I be an interval in R and let E ⊂ I be an open set with |E| < \frac{1}{100}|I|. Then there is an open set Ẽ = ⋃_{k} J_{k} such that

(i) E ⊂ Ẽ ⊂ I; (ii) |E ∩ J_{k}| ≤ \frac{1}{8}|J_{k}|, k = 1, 2, …; (iii) |Ẽ| ≤ 16|E|.

Proof. Let us apply a stopping time argument. Choose {J_{k}} so that they are the maximal dyadic intervals of I with

|E ∩ J_{k}| ≥ \frac{1}{16}|J_{k}|.

Set Ẽ = ⋃_{k} J_{k}. Then (i)–(iii) obviously hold.

For each n, n ≥ 10, apply Lemma 2 several times to obtain a nested sequence of subsets of I_{n}, {E_{n,k}}_{k=0}^{K(n)} such that E_{n} = E_{n,0} and E_{n,k−1} ⊂ E_{n,k}. Note that K(n) → ∞ if |E_{n}|/|I_{n}| → 0.

To each dyadic interval J ⊂ E_{n,k}, 0 < k < K(n), n ≥ 10, associate the corresponding complex number z_{J} as before. We claim that such a sequence {z_{j}} is interpolating but its image {F (z_{j})} is not.

Since the points {z_{j}} lie in different top halves of dyadic squares, the sequence is separated. On the other hand, it will be enough to show the Carleson measure condition for squares Q that contain some element of the sequence on its top half. Therefore we can assume that the base of Q is one of the dyadic intervals in E_{n,k} for some k, n, let us denote it by I_{Q}. Then

\begin{aligned}∑_{z_{j}∈Q} y_{j} & ≅ ∑_{{J∈E_{n,j}∩I_{Q} \atop 0≤j≤k}} |J| + ∑_{{J∈E_{m,j}∩I_{Q} \atop 0≤j≤K(m), m≥n}} |J| \\ & ≤ ∑_{j=1}^{k} \frac{1}{8^{j}} |I_{Q}| + ∑_{{I_{m}∩I_{Q}≠∅ \atop m≥n}} l(I_{m}) ≤ c|I_{Q}|. \end{aligned}

To show that {F (z_{j})} is not interpolating consider the squares {Q_{n}} with the base {I_{n}}. By the circular distortion theorem F (Q_{n}) is “almost” a square, that is, it contains a square of comparable diameter. Also if J is an interval and z_{J} its associated complex number, Im F (z_{J}) ≅ |h(J)|. So,

∑_{F (z_{j})∈F (Q_{n})} Im F (z_{j}) ≥ c ∑_{k=0}^{K(n)} |h(E_{n,k})|
≥ K(n)|h(E_{n})| ≥ K(n)(1 − 2^{−n})|h(I_{n})|
≃ K(n)diam(F (Q_{n})).

Since K(n) → ∞, the first part of the theorem is proved.

Proof of the sufficiency. Let {z_{n}} be an interpolating sequence in R2^{+} and F be a quasiconformal map such that F |_{R} = h is strongly quasisymmetric.

The fact that {F (z_{n})} is separated follows immediately from the circular distortion theorem.

To show the Carleson measure condition for {F (z_{n})} we first split the sequence {z_{n}}. Given 0 < δ < 1, split {z_{n}} into N = N(δ) disjoint subsequences S_{1}, …, S_{N} such that

inf_{z_{k}∈S_{j}} ∏_{{z_{l}∈S_{j} \atop z_{l}≠z_{k}}} |\frac{z_{l} − z_{k}}{z_{l} − z̄_{k}} | ≥ δ, j = 1, …, N,

(see [G, Ch. VII]). The number δ will be fixed later; it depends only on F. So it is enough to show that

∑_{z_{k}∈S_{j}} Im F (z_{k})δ_{F (z_{k})}

is a Carleson measure with norm independent of j.

Fix j and consider S_{j} = {w_{k}}. Let Q_{F (w_{k})} be the Carleson square

Q_{F (w_{k})} = \{x + iy : |x − Re F (w_{k})| ≤ \frac{1}{2} Im F (w_{k}), 0 < y ≤ Im F (w_{k})\}.

Note that if F (w_{l}) ∈ Q_{F (w_{k})} , then because of the circular distortion theorem, w_{l} ∈ MQ(w_{k}) where M is a constant only depending on F. Also Im F (w_{l}) ≅ |h(I_{l})| where I_{l} is the interval associated to w_{l}. Therefore

∑_{F (w_{l})∈Q_{F (w_{k})}} Im F (w_{l}) ≅ ∑_{w_{l}∈MQ(w_{k})} |h(I_{l})|.

Since

inf_{k} ∏_{l≠k}|\frac{w_{l} − w_{k}}{w_{l} − \overline{w}_{k}} | ≥ δ,

using the estimate log x^{−1} ≤ 1 − x^{2} for 2^{−1} < δ < x < 1, it follows

∑_{l≠k} \frac{Im w_{l} Im w_{k}}{|w_{l} − \overline{w}_{k}|^{2}} ≤ \frac{1}{2} log δ^{−1}.

Hence,

∑_{w_{l}∈MQ(w_{k})} Im w_{l} ≤ (2M^{2} log δ^{−1}) Im w_{k}.

If δ is close enough to 1, then 2M^{2} log δ^{−1} < 1. Since h′ ∈ A_{∞}, there exists β < 1 such that

∑_{w_{l}∈MQ_{w_{k}}} |h(I_{l})| ≤ β|h(I_{k})|.

This shows that ∑ Im F (w_{l})δ_{F (w_{l})} is a Carleson measure with norm depending only on δ and F.

290 María J. González and Artur Nicolau

References

[A] Ahlfors, L.V.: Lectures on Quasiconformal Mappings. - Van Nostrand, New York, 1966.

[AG] Astala, K., and F.W. Gehring: Quasiconformal analogues of theorems of Koebe and Hardy–Littlewood. - Mich. Math. J. 32, 1985, 99–107.

[AZ] Astala, K., and M. Zinsmeister: Teichmüller spaces and BMOA. - Math. Ann. 289, 1991, 613–625.

[G] Garnett, J.: Bounded Analytic Functions. - Academic Press, London–New York, 1981.

[J] Jones, P.: Homeomorphism of the line that preserve BMO. -Ark. Mat. 21, 1983, 229–231.

[L] Lehto, O.: Univalent Functions and Teichmüller Spaces. - Springer-Verlag, New York, 1987.

[S] Stein, E.M.: Harmonic Analysis. - Princeton University Press, 1993.

Received 1 August 1996