UDC 517.53:517.947.42

MATHEMATICS

V. M. MIKLYUKOV

BOUNDARY PROPERTIES OF (n)-DIMENSIONAL QUASICONFORMAL MAPPINGS

(Presented by Academician M. A. Lavrent'ev on 19 I 1970)

1. Let (R^n) be (n)-dimensional Euclidean space; (\widetilde R^n = R^n \cup {\infty}) the corresponding Möbius space; (x=(x_1,\ldots,x_n)) a point (or vector) in (R^n); (|x|=\sqrt{x_1^2+\cdots+x_n^2}) the length of a vector. For an arbitrary domain (\Omega \subseteq R^n): (L_n(\Omega)) is the class of measurable functions summable over (\Omega) with exponent (n); (W_n^1(\Omega)) is the set of functions (\varphi(x)\in L_n(\Omega)) having generalized derivatives (\partial\varphi/\partial x_i\in L_n(\Omega)) ((i=1,\ldots,n)); (L_1^n(R^n)) is the set of functions (\varphi(x)) representable in the form of a Bessel potential (see ((^{1,2})))

[
\varphi(x)=(G_1u)(x)\equiv \int_{R^n} G_1(|x-y|)u(y)\,dy,\qquad u\in L_n(R^n),
\tag{1}
]

where (G_1(t)) is the Bessel kernel of order 1. A vector function (f(x)=(f_1(x),\ldots,f_n(x))) belongs to one of the listed classes if and only if each of its components (f_i(x)) ((i=1,\ldots,n)) has this property.

Let (A) be an arbitrary set in (R^n). Consider all possible nonnegative functions (u(x)\in L_n(R^n)) such that for all (x\in A)

[
(G_1u)(x)\ge 1.
]

The exact lower bound of the quantities (\left(|u(x)|_{L_n(R^n)}\right)), taken over the set of all such functions (u(x)), is called the ((1,n))-capacity (or simply the capacity) of the set (A).

This definition of ((1,n))-capacity is a special case of the definition given in ((^3)). In the same work, relations between ((1,n))-capacity and other capacities are also indicated.

1. Let (B) be an open ball in (R^n), and let (S) be its boundary. For an arbitrary vector function (f:B\to R^n), set: (E(f)) is the set of points on (S) at which the mapping (f) has angular boundary values; (E'(f)) is the set of points on (S) such that for any point (x_0\in E'(f)) there exists a path (\gamma_{x_0}\subset B), tending to (x_0), along which the mapping (f) has a limit; (E''(f)) is the set of points (x_0\in S) for each of which there exists a point (y_0\in \overline{R^n}) such that

[
\lim_{h\to 0}\frac{1}{h^n}\int_{B_h(x_0)} |f(x)-y_0|\,dx=0,
\tag{2}
]

where (B_h(x_0)={x\in B: |x-x_0|<h}).

Lemma 1. Let (f:B\to R^n) be a monotone mapping of the class (W_n^1(B)), and let (\gamma_{x_0}\subset B) be an arbitrary path tending to a point (x_0\in S). If, for some sequence of points ({x_n}) ((n=1,2,\ldots)) with the properties

[
x_n\in \gamma_{x_0},\qquad \lim_{n\to\infty}x_n=x_0,\qquad \lim_{n\to\infty}\frac{\operatorname{diam} x_n\widetilde{x}_{n+1}}{\rho(x_n,S)}<1,
]

where (\rho(x_n,S)) is the distance from the point (x_n) to (S), and if the limit (\lim\limits_{n\to\infty} f(x_n)=y_0) exists, then the mapping has the same limit along the path (\gamma_{x_0}).

The proof is carried out analogously to the proof of Lemma 3 of paper ((^4)).

Lemma 2 (see ((^4))). Let (f:B\to R^n) be a monotone mapping of class (W_n^1(B)). Then (E(f)=E'(f)).

Proceeding from condition (2) and applying the preceding lemmas successively, it is not difficult to verify the validity of the following assertion.

Lemma 3. Let (f:B\to R^n) be a monotone mapping of class (W_n^1(B)). Then (E''(f)\subseteq E(f)).

The following Lemma 4 is a simple consequence of Theorem 5.7 of paper ((^3)) (see also ((^2))).

Lemma 4. Let (f:\dot R^n\to R^n) be a vector-function of class (L_1^n(R^n)), and let (f_1=f|_B) be its restriction to (B). Then the set (S\setminus E'''(f_1)) is a set of zero capacity.

The following theorem is a generalization of the corresponding results from ((^{4-6})).

Theorem 1. Let (f:B\to R^n) be a monotone mapping of class (W_n^1(B)). Then the set (S\setminus E(f)) is a set of capacity zero.

We give the idea of the proof. By the extension theorem (see, for example, ((^1)), p. 144) there exists a vector-function (f^:R^n\to R^n) of class (W_n^1(R^n)) such that (f^|_B=f). Further, by Theorem 5.1 of paper ((^3)) (see also ((^2))) there is a vector-function (f^{*}:R^n\to R^n) of class (L_1^n(R^n)), coinciding almost everywhere in (R^n) with (f^).

Consider (f_1=f^{**}|_B). For almost all (x\in B) we have (f_1(x)=f(x)), and, consequently, (E''(f_1)=E''(f)). By Lemma 4 the set (S\setminus E''(f_1)=S\setminus E''(f)) is a set of capacity zero. Applying Lemma 3, we verify the theorem.

Remark. This theorem can be extended (in the corresponding terms) also to monotone mappings of class (W_p^1(B)), where (p>n-1) (see ((^7))).

1. Let (A\subseteq\Omega) be a set, closed relative to the domain (\Omega\subset R^n), having no interior points, and let (f:(\Omega\setminus A)\to R^n) be a quasiconformal mapping, distinct from the identical constant one. To each point (x_0\in A) we assign the following sets of values (see ((^8))).

The cluster set (C_{\Omega\setminus A}(f,x_0)). A value (y\in C_{\Omega\setminus A}(f,x_0)) if there exists a sequence of points ({x_n}) with the properties

[
x_n\in \Omega\setminus A,\qquad \lim_{n\to\infty}x_n=x_0,\qquad \lim_{n\to\infty}f(x_n)=y.
]

The set of repeated values (R_{\Omega\setminus A}(f,x_0)). A value (y\in R_{\Omega\setminus A}(f,x_0)) if there exists a sequence of points ({x_n}) with the properties

[
x_n\in \Omega\setminus A,\qquad \lim_{n\to\infty}x_n=x_0,\qquad f(x_n)=y.
]

In paper ((^{10})) we gave certain criteria for the removability of special sets.* The results formulated below concern the structure of the sets (C_{\Omega\setminus A}(f,x_0)), (R_{\Omega\setminus A}(f,x_0)) in the case when (A) is a set of essentially singular points.

Theorem 2. Let (A\subseteq\Omega) be an arbitrary compact set relative to the domain (\Omega\subset R^n) of capacity zero. Suppose that (f:(\Omega\setminus A)\to R^n) is a quasiconformal mapping (nonconstant) having an essential singularity at each point (x_0\in A). Then the cluster set

[
C_{\Omega\setminus A}(f,x_0)=\dot R^n.
]

* Taking the opportunity, we note that Theorem 1 formulated in ((^{10})) is meaningful only for (\alpha=n). For (\alpha>n) the class of sets of zero (\alpha)-capacity is empty.

For two-dimensional quasiconformal mappings this assertion was proved in (⁹) (see also (⁸)).

Theorem 3. Let the hypotheses of Theorem 2 be satisfied. Then in any neighborhood of a point (x_0 \in A) the mapping (f) assumes infinitely often every value (y \in \overline{R}^{\,n}), with the possible exception of a set of capacity zero; i.e., the set (\overline{R}^{\,n}\setminus R_{\lambda\setminus A}(f,x_0)) is a set of capacity zero.

In the case where (x_0) is an isolated essentially singular point, this theorem is due to Yu. G. Reshetnyak (communication at the Donetsk Colloquium on quasiconformal mappings, 1968).

Donetsk Computing Center
of the Academy of Sciences of the Ukrainian SSR

Received
7 I 1970

CITED LITERATURE

¹ S. M. Nikol’skii, Approximation of Functions of Several Variables and Imbedding Theorems, “Nauka,” 1969. ² N. Aronszajn, Mulla Fuad, P. Szeptycki, Ann. de l’Inst. Fourier, 13, 211 (1963). ³ Yu. G. Reshetnyak, Siberian Math. J., 10, No. 5, 1109 (1969). ⁴ V. M. Miklyukov, Questions of the Geometric Theory of Functions, vol. 4, Tr. Tomsk State Univ., 189, 80 (1966). ⁵ A. Beurling, Acta Math., 72, No. 1–2, 1 (1940). ⁶ V. A. Zorich, DAN, 177, No. 4, 771 (1967). ⁷ V. A. Zhukov, V. M. Miklyukov, Questions of the Geometric Theory of Functions, vol. 5, Tr. Tomsk State Univ., 200, 88 (1968). ⁸ K. Nosiro, Cluster Sets, IL, 1963. ⁹ M. Ohtsuka, Nagoya Math. J., 11, 131 (1957). ¹⁰ V. M. Miklyukov, DAN, 188, No. 3, 525 (1969).