Annales Academiæ Scientiarum Fennicæ Series A. I. Mathematica Volumen 20, 1995, 373–378

ON THE BEHAVIOR OF MEROMORPHIC FUNCTIONS AROUND SOME NONISOLATED SINGULARITIES II

Pentti Järvi

University of Helsinki, Department of Mathematics P.O. Box 4, FIN-00014 Helsinki, Finland

Abstract. Let G ⊂ C be a domain, let E be a closed subset of G, and let f be meromorphic in G \ E with at least one essential singularity in E. We show that lim sup f^{∗}(z) d(z, E)^{β} = ∞ as d(z, E) → 0 for every β < min(1, 2 − α) provided that E is locally a null set for uniform domains and dim_{H}(E) ≤ α. A similar result is obtained if E is a closed, totally disconnected subset of G lying on a quasicircle.

1. This paper is a supplement to [6]. The terminology and notation is as in [6]. Let G ⊂ C be a domain, let E ⊂ G be a relatively closed set, and let f be meromorphic in G \ E with at least one essential singularity in E. In [6] we were concerned with the growth rate of the spherical derivative f^{∗}(z) as z tends to E. It was shown that

(1)  lim sup_{d(z,E)→0} f^{∗}(z) d(z, E)^{β} = ∞  for all β < 1 − \frac{1}{2}α

provided that E is of class N_{D} and the α-dimensional Minkowski content of E is finite [6, Theorem 3]. The result is sharp in the sense that 1 − \frac{1}{2}α cannot be replaced by a larger constant without additional restrictions. In this note we show that the above estimate can be substantially improved provided that the distribution of the points of E is regular in a certain sense. For instance, this is the case if E lies on a quasicircle (in particular, if E is a linear set) or E is a Cantor set with constant ratio. The outline of proof is as follows. We first show that if (1) does not hold for some β, then any point of E possesses a neighborhood, in which f satisfies the so-called local Lipschitz condition of order 1 − β. Secondly, owing to the geometry of E the local Lipschitz condition implies the global one. This in turn makes it possible to employ known results about removable singularities of holomorphic Lipschitz functions so as to arrive at a contradiction.

1991 Mathematics Subject Classification: Primary 30D30; Secondary 30D40.

2. Let σ(·, ·) denote the spherical metric in Ĉ, i.e., σ is the metric induced by the density 1/(1 + |z|^{2}). If f is meromorphic in a domain G ⊂ C and α ∈ (0, 1], we say that f is in Lip_{α}(G) provided that

(2)  σ(f(z_{1}), f(z_{2})) ≤ m|z_{1} − z_{2}|^{α}

for some m ≥ 0 and for all z_{1}, z_{2} ∈ G. Furthermore, we say that f ∈ loc Lip_{α}(G) if (2) holds whenever z_{1} and z_{2} belong to an open disk contained in G (cf. [4]). Given α ∈ (0, 1] and m > 0, we set r_{0} = (π/4m)^{1/α} and define φ(·; α, m): G → R as follows

φ(z; α, m) = \{ \begin{matrix}\frac{4m}{π} d(z, ∂G)^{α−1}, & when d(z, ∂G) ≤ r_{0}, \\ (\frac{4m}{π})^{1/α}, & when d(z, ∂G) ≥ r_{0}. \end{matrix}

In case G = C it is understood that d(z, ∂G) ≥ r_{0} for all z ∈ G. We begin by proving the spherical version of the Hardy–Littlewood theorem (see [1, Theorem 5.1]).

Lemma 1. Let G ⊂ C be a domain, let f be meromorphic in G, and let α ∈ (0, 1]. Then f ∈ loc Lip_{α}(G) if and only if

f^{∗}(z) ≤ φ(z; α, m) in G for some m > 0.

Proof. Suppose that f ∈ loc Lip_{α}(G). Then there exists m ≥ 0 such that

(3)  σ(f(z_{1}), f(z_{2})) ≤ m|z_{1} − z_{2}|^{α}

whenever z_{1} and z_{2} belong to an open disk in G. Fix z_{0} ∈ G such that

d(z_{0}, ∂G) ≥ r_{0} = (\frac{π}{4m})^{1/α}.

Performing a rotation of the Riemann sphere, we may assume that f(z_{0}) = 0. By (3) we have

σ(f(z), f(z_{0})) = σ(f(z), 0) ≤ m|z − z_{0}|^{α} ≤ mr_{0}^{α} = \frac{1}{4}π

for all z ∈ B(z_{0}, r_{0}). This implies that |f(z)| ≤ 1 for all z ∈ B(z_{0}, r_{0}). By uniform continuity, f can be taken to be continuous in \overline{B}(z_{0}, r_{0}). Hence we may apply the Cauchy integral formula to obtain

\begin{aligned}f^{∗}(z_{0}) & = |f′(z_{0})| = \frac{1}{2π} |∫_{|z−z_{0}|=r_{0}} \frac{f(z) − f(z_{0})}{(z − z_{0})^{2}} dz| \\ & ≤ \frac{1}{2π} ∫_{|z−z_{0}|=r_{0}} \frac{1}{|z − z_{0}|^{2}} |dz| = \frac{1}{r_{0}} . \end{aligned}

In case d(z_{0}, ∂G) = r < r_{0} , we have similarly

f^{∗}(z_{0}) = |f′(z_{0})| = \frac{1}{2π} |∫_{|z−z_{0}|=r} \frac{f(z) − f(z_{0})}{(z − z_{0})^{2}} dz| ≤ \frac{1}{2π} ∫_{|z−z_{0}|=r} \frac{|f(z)|}{|z − z_{0}|^{2}} |dz|

≤ \frac{1}{r} max\{|f(z)| | |z − z_{0}| = r\} ≤ \frac{4}{πr} max\{σ(f(z), 0) | |z − z_{0}| = r\}

≤ \frac{4m}{πr}r^{α} = \frac{4m}{π}r^{α−1} = \frac{4m}{π}d(z_{0}, ∂G)^{α−1}.

Hence f^{∗}(z) ≤ φ(z; α, m) for all z ∈ G.

The proof of the converse implication proceeds along the same lines as in the case of holomorphic functions (cf. [1, p. 75]). Hence we omit the details.

A domain G ⊂ C is called uniform if there exist constants a, b ∈ [1, ∞) such that each pair of points z_{1} , z_{2} in G can be joined by a rectifiable arc γ ⊂ G such that

l(γ) ≤ a|z_{1} − z_{2}|

and

min(l(γ_{1}), l(γ_{2})) ≤ bd(z, ∂G) for each z ∈ γ.

Here γ_{1} and γ_{2} are the components of γ \ {z}. This concept was introduced by Martio and Sarvas [9]. They also observed that quasidisks are always uniform domains [9, Corollary 2.33]. Recall that a domain in C is a quasidisk if it is the image of an euclidean disk under a quasiconformal self-mapping of Ĉ. The boundaries of quasidisks are called quasicircles.

Arguing as in the proof of [3, Theorem 2.1], we deduce from Lemma 1 another useful result.

Lemma 2. Let G ⊂ C be a uniform domain, let f be meromorphic in G, and let α ∈ (0, 1]. Then f ∈ loc Lip_{α}(G) if and only if f ∈ Lip_{α}(G).

3. A set E in the euclidean n-space R^{n} is porous in R^{n} if there exists a constant c ∈ (0, 1] such that each closed ball \overline{B}(x, r) in R^{n} contains a point z such that the open ball B(z, cr) does not meet E; see e.g. [10, p. 525]. For example, Cantor sets with constant ratio in R^{n} are porous in R^{n}.

We follow [11, p. 118] and say that a closed set E ⊂ C is a null set for uniform domains, or an NUD set, if E is nowhere dense and C \ E is a uniform domain. It is easily seen that a closed set E ⊂ R is porous in R if and only if E is NUD. More generally, invoking Väisälä’s compactness criterion [11, Corollary 3.8] one realizes that a closed subset of C of the form E = E_{1} ×E_{2} with E_{i} ⊂ R, i = 1, 2, is NUD if and only if both E_{1} and E_{2} are porous in R. It is known that compact parts of an NUD set always belong to N_{D} (cf. [11, Remark 5.3.4]). In particular, each NUD set is totally disconnected. Furthermore, we say that a closed, totally disconnected subset E of a domain G ⊂ C is locally NUD in G if each point of E has a neighborhood U such that E ∩ U is NUD. For instance, the set of integers is locally NUD in C but not NUD.

In what follows, H^{α} refers to the α-dimensional Hausdorff measure.

Theorem 1. Let G ⊂ C be a domain and let E be locally NUD in G with dim_{H}(E) ≤ α for some α ∈ [0, 2). Let f be meromorphic in G \ E with at least one essential singularity in E. Then

lim sup_{d(z,E)→0} f^{∗}(z) d(z, E)^{β} = ∞ for all β < min(1, 2 − α).

Proof. Suppose, on the contrary, that there are positive constants C and r_{0} such that f^{∗}(z) d(z, E)^{β} ≤ C for all z ∈ G \ E with d(z, E) ≤ r_{0} and for some β < min(1, 2 − α). Fix z_{0} ∈ E and let U be a simply connected neighborhood of z_{0} with smooth boundary such that U ⊂ B(z_{0}, r_{0}) and E ∩ U is NUD. It follows from Lemma 1 that f | U \ E belongs to loc Lip_{1−β}(U \ E). Since U \ E is a uniform domain [11, Theorem 5.4], we deduce from Lemma 2 that f | U \ E ∈ Lip_{1−β}(U \ E). This implies that f admits a continuous extension f̂ to the whole of U. By an auxiliary rotation of the Riemann sphere, we may assume that f̂ is bounded in some disk B(z_{0}, r) ⊂ U. This means that f̂ satisfies in B(z_{0}, r) an euclidean Lipschitz condition of order 1−β. Now α < 2−β implies H^{2−β}(E) = 0. Hence we may apply [3, Corollary III.4.5] to conclude that f̂ is holomorphic in B(z_{0}, r). This completes the proof.

Remark 1. We show that the above result is sharp in the sense that the constant min(1, 2−α) cannot be replaced by a larger one without further restrictions. Assume first that α ∈ (1, 2). There is a compact NUD set E, say a Cantor set with constant ratio, such that 0 < H^{α}(E) < ∞. The existence of a meromorphic function f in C \ E, without meromorphic extension to C, with

lim sup_{d(z,E)→0} f^{∗}(z) d(z, E)^{2−α} < ∞

now follows from [3, Corollary III.4.5] in view of Lemma 1. On the other hand, by [8, Theorem 1] there is a meromorphic function f in C \ {0} with an essential singularity at 0 such that

lim sup_{z→0} |z|f^{∗}(z) = \frac{1}{2}.

This settles the case α ∈ [0, 1].

There is another instance, in which one can exploit Lemma 2. We have in mind the case that the exceptional set E lies on the common boundary of two adjacent uniform domains, i.e., E is a subset of a quasicircle.

Theorem 2. Let G ⊂ C be a domain, let C ⊂ Ĉ be a quasicircle, and let E ⊂ C be a closed, totally disconnected subset of G with dim_{H}(E) ≤ α for some α ∈ [0, 2). Let f be meromorphic in G \ E with at least one essential singularity in E. Then

lim sup_{d(z,E)→0} f^{∗}(z) d(z, E)^{β} = ∞ for all β < min(1, 2 − α).

Proof. Suppose there are positive constants c and r_{0} such that f^{∗}(z) d(z, E)^{β} ≤ c for all z ∈ G\E with d(z, E) ≤ r_{0} and for some β < min(1, 2−α). Fix z_{0} ∈ E and let U ⊂ B(z_{0}, r_{0}) be a quasidisk with z_{0} ∈ U such that U \ C consists of two quasidisks U_{1} and U_{2} with C ∩ \overline{U} = ∂U_{1} ∩ ∂U_{2} . Now f | U_{j} ∈ loc Lip_{1−β}(U_{j}), j = 1, 2. By [9, Corollary 2.33], U_{1} and U_{2} are uniform domains. Hence by Lemma 2, f | U_{j} ∈ Lip_{1−β}(U_{j}), j = 1, 2. This implies that both of them have a continuous extension to the common boundary C ∩ \overline{U}. Since the boundary values coincide in (C ∩ \overline{U}) \ E and this set is dense in C ∩ \overline{U}, the boundary values are equal throughout C ∩\overline{U}. In other words, f has a continuous extension to U. It is now a simple matter to verify that f | U \ E belongs to Lip_{1−β}(U \ E). Arguing as in the proof of Theorem 1, we then realize that f is even meromorphic in U. The proof is complete.

Remark 2. Again the constant min(1, 2 − α) is the largest possible. If α ∈ (1, 2), pick a self-similar Cantor set E with 0 < H^{α}(E) < ∞. Thanks to the construction of Gehring and Väisälä [5], E can be realized as a subset of a quasicircle. The assertion now follows as in Remark 1. In case α ∈ [0, 1], we may again refer to [8, Theorem 1].

Corollary. Let G ⊂ C be a domain and let E be a linear, closed, totally disconnected subset of G. Suppose that f is meromorphic in G \ E with at least one essential singularity in E. Then

lim sup_{d(z,E)→0} f^{∗}(z) d(z, E)^{1−ε} = ∞ for all ε > 0.

4. It is clear that the statements corresponding to Theorems 1 and 2 are valid also in the setting of holomorphic functions. Defining the global and local Lipschitz classes for holomorphic functions as in Section 2 but using the euclidean metric instead of the spherical one, we obtain the following result (cf. [7, Theorem C]).

Theorem 3. Let G ⊂ C be a domain, let E be locally NUD in G, and let f ∈ loc Lip_{α}(G \ E) be holomorphic in G \ E. If H^{1+α}(E) = 0, then f admits a holomorphic extension to G. The same conclusion holds if E is a closed, totally disconnected subset of G lying on a quasicircle.

378 Pentti Järvi

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Received 24 March 1994