UDC 517.53:517.947.42

MATHEMATICS

B. P. KUFAREV

ZERO-SETS AND A HOMEOMORPHISM WITH FINITE DIRICHLET INTEGRAL

(Presented by Academician M. A. Lavrent'ev, 15 VI 1966)

One says that a mapping \(T(z)=u(x,y)+iv(x,y)\) of a domain \(G\) of the plane \(R^2\) belongs to the class \(BL(G)\) if the functions \(u\) and \(v\) are continuous, have generalized first derivatives with respect to \(x\) and \(y\) in the domain \(G\), and have finite Dirichlet integral

\[ D(G)\overset{\mathrm{def}}{=}\iint_G \left(|\nabla u|^2+|\nabla v|^2\right)\,dx\,dy. \]

It is known that every \(KQC\)-mapping (\(K\)-quasiconformal mapping) of a domain \(G\) onto a domain \(\Delta\) of finite area belongs to \(BL(G)\).

Let \(G\) be a simply connected domain, \(\dot G=\overline{G}\setminus G\) the boundary of \(G\), and \(\widetilde G\) a compact extension of \(G\), whose elements are the prime ends in the sense of Carathéodory (if \(G\) is unbounded, the corresponding construction using the spherical metric on the plane is understood) (see \((^{1-3})\)).

In this note we give a condition for the invariance of zero-sets \(\subset \dot G\) with respect to homeomorphisms of class \(BL\), in particular, the conditional boundary \(N\)-property of \(BL\)-homeomorphisms of \(G\) onto a domain \(\Delta\) with rectifiable boundary.

Our condition (see Definition 2 below) resembles the metric criteria for the null boundary of Riemann surfaces:

\[ \int^\rho \frac{dt}{\nu(M_t)} \]

diverges (see \((^4)\), pp. 408–418). This analogy is not merely external.

I. A homeomorph \(\gamma\subset G\) of an open or half-open interval \(I\) (of the number line) is called a section of \(G\) if \(G\setminus \gamma\) consists of exactly two components.

Let \(G_a^\gamma\) be the component of the set \(G\setminus \gamma\) containing \(a\in G\), and let \(U\subset G\) be some closed disk with center at the point \(a\).

By definition, a prime end \(e\in\dot G\) is covered by the section \(\gamma\) if \(e\) belongs to \(G\setminus \overline{G_a^\gamma}\) and \(U\subset G_a^\gamma\); a set \(E\subset\dot G\) is covered by a system of sections \(\{\gamma\}\) if every element \(e\in E\) is covered by at least one \(\gamma\in\{\gamma\}\).

Definition 1 (cf. \((^5)\)). A set \(E\subset\dot G\) is called a \(0\)-set (a generalized null-set) if, for every \(\varepsilon>0\), there exists a countable system \(\{\gamma_i\}\) of sections of \(G\) covering \(E\), with

\[ \sum_i l(\gamma_i)<\varepsilon, \]

where \(l\) is the length of \(\gamma_i\).

For each \(e\in E\subset\dot G\), let \(Z^e\) be some nonempty set of principal points \(e\) (\((^3)\), p. 194). Put

\[ M\overset{\mathrm{def}}{=}M(E)=\bigcup_{e\in E} Z^e,\qquad L_t\overset{\mathrm{def}}{=}\{z\in R^2:\ r(z,M)=t\}\quad\text{and}\quad M_t\overset{\mathrm{def}}{=}L_t\cap G; \]

here \(r\) is Euclidean distance.

Definition 2. A set \(E\subset\dot G\) is called a \(0_1\)-set if there exists such a bounded set \(M(E)=M\) that \(r(M,U)=\)

\(=\rho>0\) and

\[ \int_0^\rho \frac{dt}{\nu(M_t)} \]

diverges (\(\nu\) is Hausdorff length, see \((^6)\), p. 92).

The condition \(\rho>0\) is, obviously, equivalent to the nonemptiness of \(\dot G \setminus \widetilde E\), where \(\widetilde E\) is the closure of \(E\) in \(G\).

It is proved that an \(\dot\theta_1\)-set is an \(\dot\theta\)-set.

It is known that a \(BL\)-homeomorphism \(T:G\to\Delta\) can be extended to a continuous mapping of \(\dot G\) onto \(\Delta\) (\((^3)\), p. 66).

Let \(l[T(M_t)]\) be the sum for the images of the component arcs of the set \(M_t\) (see \((^7)\)). With the aid of the inequality (see \((^8)\), p. 14)

\[ \int_0^\rho \frac{l^2[T(M_t)]}{\nu(M_t)}\,dt \leqslant D(G) \]

one proves

Theorem 1*. A homeomorphism \(T\in BL(G)\) of a simply connected domain \(G\) onto a domain \(\Delta\) takes every \(\dot\theta_1\)-set into a \(\dot\theta\)-set \(T(E)\subset\Delta\). If \(E\) is closed in \(\dot G\), and \(\nu(\Delta)<\infty\) (here \(\Delta\) is a continuous image of the circle; see \((^{10})\), p. 51, \(\overline u\) \((^{11})\), pp. 411–412), then \(\nu(|T(E)|)=0\), and therefore (see \((^{12})\)) \(|T(E)|\) is a \(0\)-set in the sense of Painlevé (\(|T(E)|\) is the sum of the bodies of all \(e\in T(E)\)).

If \(G\) is homeomorphic to a circle, then the sets \(E\) and \(M(E)\) can be identified (see \((^2)\), p. 408).

As is known \((^{13})\), there exists a \(KQC\)-mapping \(T(z)\) of the disk \(G':|z|<1\) onto the disk \(\Delta':|w|<1\) such that for some perfect set \(E\subset G'\), with \(\nu(E)=0\), one has \(\nu(|T(E)|)>0\). Hence we have

Corollary 1. There exists a perfect set of measure zero on the boundary of the disk (and hence also a \(0\)-set in the sense of Painlevé) which is not an \(\dot\theta_1\)-set. Moreover, there exists on the \(x\)-axis in the plane \(R^2\) a perfect set \(P\subset[0,2\pi]\) of measure zero for which

\[ \int_0^\rho \frac{dt}{\nu(P_t)} \]

converges for every \(\rho\in(0,\infty)\) and, consequently,

\[ \lim_{t\to 0}\nu(P_t)/t=\infty; \]

here

\[ R_t=\{z\in R^2:r(z,P)=t\}. \]

II. A homeomorph \(p\subset G\) of the half-open interval \(I=[0,1)\) is called a path. Let \(p=f(I)\), and \(p_\alpha=f(I_\alpha)\), where \(f\) is a homeomorphism, and \(I_\alpha=(\alpha,1)\), \(\alpha\in(0,1)\). One says that the path \(p\) goes to a prime end \((p\to e)\) if, for every open set \(g\) containing the prime end \(e\), there exists an \(\alpha\) such that \(f(I_\alpha)\subset g\) (see \((^{14})\), p. 5).

Let \(|e|\) be the body (the limiting set) of the prime end \(e\in\dot\Delta\),

\[ \Delta_w=\{e\in\dot\Delta:w\in|e|\} \]

and

\[ \Phi^w=\{e\in\Delta_w:w\text{ is the principal point of }e\}. \]

Definition 3. A homeomorphism \(T:G\to\Delta\) concentrates a set \(E\subset\dot G\) if there exists a point \(w\in\overline\Delta\) with the following property: for each \(e\in E\) there is a path \(p_e\to e\) such that \(T(p_e)\to\varphi\in\Phi^w\).

Theorem 2. If \(E\subset\dot G\) is not a \(\dot\theta\)-set, then there does not exist a homeomorphism \(T:G\to\Delta\), \(T^{-1}\in BL(\Delta)\), concentrating \(E\) (i.e., it concentrates only sufficiently “sparse” sets).

Corollary 2. Let \(G\) be a simple rectifiable closed arc. If \(E\subset G\) and \(\nu(E)>0\), then a homeomorphism \(T:G\to\Delta\), \(T^{-1}\in BL(\Delta)\), does not concentrate \(E\); in particular, there does not exist a homeomorphism \(T\), \(T^{-1}\in BL(\Delta)\), taking one and the same angular boundary value \(w\) on a set \(E\) (for conformal mappings the latter is, of course, a consequence of the well-known theorem of Luzin–Privalov, \((^9)\), p. 292).

* It is interesting to compare this assertion with Theorems 3 and 4 of \((^5)\), and also with the theorem of M. A. Lavrent'ev; see \((^9)\), p. 293.

We shall state one theorem concerning the topological structure of the set \(\dot{\Delta}_w\).

Theorem 3. The set \(\dot{\Delta}_w\) is closed in \(\dot{\Delta}\).

Remark 1. The assertions are easily extended to finitely connected domains.

Remark 2. In many respects analogous results can be obtained for mappings of spatial domains.

Tomsk State University
named after V. V. Kuibyshev

Received
1 VI 1966

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