UDC 517.54

MATHEMATICS

I. S. OVCHINNIKOV

ON THE NONEXISTENCE OF MAPPINGS OF CLASS \(BL^{n/2}\) OF A BALL ONTO A DOMAIN

(Presented by Academician M. A. Lavrent’ev, 17 IV 1967)

1. A continuous vector function \(y=f(x)\), defined in a domain \(D\) of \(n\)-dimensional Euclidean space \(E^n\) and with values in \(E^m\), belongs to the class of functions \(BL_K^{n/2}\) if

\[ I(f,D)=\int_D \left[\sum_{i=1}^m \sum_{j=1}^n \left(\frac{\partial f_i}{\partial x_j}\right)^2\right]^{n/2} dx \leqslant K<\infty, \]

where \(dx\) is the volume element in \(E^n\), and the derivatives are understood in the sense of S. L. Sobolev.

If \(f\in BL_K^{n/2}\) for some \(K\), then \(f\in BL^{n/2}\). If the function \(y=f(x)\) realizes a homeomorphism of the ball \(D: |x|<R\) onto a domain \(\Delta\), with \(f, f^{-1}\in BL_K^{n/2}\), the domain \(\Delta\) contains the ball \(D_1: |y|<\delta\), and \(f(0)=0\), then the function \(f\) belongs to the class of functions \([BL]_K^{n/2}\). If \(f\in [BL]_K^{n/2}\) for some \(K\), then \(f\in [BL]^{n/2}\).

For \(n=m=1\), the integral \(I(f,D)\) gives the variation of the function \(f\) on the interval \(D\). For \(n=m=2\), \(I(f,D)\) is the usual Dirichlet integral, equal for conformal mappings to twice the measure of the domain \(f(D)\). For quasiconformal mappings with bounded measure of the domain \(f(D)\), the integral \(I(f,D)\) is bounded.

Of fundamental importance in studying the properties of mappings of class \(BL^{n/2}\) is the inequality of Theorem 2 of the work \((^1)\). If, instead of \(I(f,D)\), one substitutes into this inequality the integral

\[ V_f(D)=\int_D \lambda_f^n(x)\,dx, \]

where

\[ \lambda_f(x)=\varlimsup_{\Delta x\to 0}\frac{|f(x+\Delta x)-f(x)|}{|\Delta x|}, \]

then it is preserved up to a constant factor, since

\[ \lambda_f(x)\leqslant \left[\sum_{i=1}^m \sum_{j=1}^n \left(\frac{\partial f_i}{\partial x_j}\right)^2\right]^{1/2} \leqslant \sqrt{n}\lambda_f(x). \]

The quantity \(V_f(D)\) is naturally called the variation of the function over the domain \(D\). The class of functions \(BL_K^{n/2}\) is thus a generalization of the class of functions of bounded variation (more precisely, of the class of functions of bounded variation, absolutely continuous inside the interval \(D\)). The classes of functions \(BL_K^{n/2}\) and \([BL]_K^{n/2}\) have many properties in common with the class of conformal mappings in the plane. The most essential difference is the absence of an analogue of the Riemann theorem for \(n\geqslant 3\). The results given below show that, in the general case for \(n\geqslant 3\), the question of the existence of mappings of a ball onto a domain within the class under consideration is of exceptional difficulty. For \(n=m=2\), mappings of this class were studied by J. Lelong-Ferrand \((^2)\) and G. D. Suvorov \((^3)\). The study of the spatial case for \(n=m=3\) was initiated in the works \((^{4-7})\), and was subsequently continued for \(n\geqslant 2\) and arbitrary \(m\) in \((^1,{}^8,{}^9)\).

Below we shall use the following notation: \(\rho(M_1,M_2)\) is the distance between sets in \(E^m\); \(|x'-x''|\) is the distance between points-

in \(E^n\); \(\overline M\) is the closure of the set \(M\) in \(E^n\); \(\partial \Delta\) is the boundary of the domain \(\Delta\); \(d(M)\) is the diameter of the set \(M\) in \(E^n\); \(C_D(f,a,l)\) is the cluster set of the function \(f\) at the point \(a\) of the boundary of the domain \(D\) relative to the curve \(l \subset D\), i.e., it is the set of limit points of sequences \(\{f(a_m)\}\), \(a_m \in l\) \((m=1,2,\ldots)\), \(\lim_{m\to\infty} a_m=a\).

1. For bounded domains \(\Delta \subset E^n\), compact with respect to the relative distance \((^5)\) and belonging to the class of domains \(A_1(S)\) (for any sphere \(S\) in \(E^n\), every component of the set \(S\cap\Delta\) divides the domain \(\Delta\) into two subdomains), the compactification of the domain by prime ends, analogous to the prime ends of Carathéodory \((^{10})\), is a compactum \((^8)\). The prime ends of this class of domains contain 5 types of prime ends. To the 4 types in Carathéodory’s classification there is added a prime end of the 5th type, whose body has a disconnected closed set of principal points.* As the example of domains given below shows, for domains in \(E^n\) \((n\ge 3)\) all 5 types of prime ends are realized. On the other hand, by the corollary of Theorem 1, by means of a homeomorphism of class \(BL^{n/2}\) a ball cannot be mapped onto a domain having prime ends of the 5th type.

Theorem 1. If the function \(y=f(x)\) realizes a homeomorphic mapping of class \(BL^{n/2}\) of the ball \(D:\ |x|<1\) onto a bounded domain \(\Delta\), then for every non-tangential curve \(l\subset D\) tending to a point \(a\in\partial D\), the set \(C_D(f,a,l)\) coincides with the set \(|e|_1\) of principal points of the prime end \(e=f(a)\).

This theorem is an analogue of the well-known Lindelöf theorem for conformal mappings. For mappings of class \(BL\) it was established by J. Lelong-Ferrand \((^2)\). The proof of this theorem is based on the inequality from \((^1)\) and the results from \((^8)\).

Corollary. There do not exist homeomorphic mappings of class \(BL^{n/2}\) of the ball \(D:\ |x|<1\) onto a bounded domain \(\Delta\) having prime ends with a disconnected set of principal points.

We give an example of domains in \(E^3\), homeomorphic to the ball and compact with respect to the relative distance, which contain prime ends of the 5th type.

Example 1. Consider in \(E^3\) the cube \(R=\{x\in E^3:\ 0<x_i<1\ (i=1,2,3)\}\) and the sets \(T_{mp}=\{x\in R:\ x_1=1/2^m3^p\}\) \((m,p=1,2,\ldots)\). Let \(F\) be an arbitrary closed set on the face \(T_0=\{x\in \overline R:\ x_1=0\}\) of the cube \(R\). We construct a domain \(\Delta\) having a prime end \(e\) such that \(|e|=T_0,\ |e|_1=F\). We shall assume that \(F\) is an infinite set. For finite \(F\), the constructions are carried out analogously.

Let \(\{a_m\}\) \((m=1,2,\ldots)\), \(\{a_m\}\subset F\), be some countable set of points of the face \(T_0\), everywhere dense in \(F\), with the points of \(\{a_m\}\) not lying on the edges of the cube \(R\). Put \(Q_{mp}=\{x\in T_{mp}:\ |b_{mp}-x|<r_{mp}\}\), where \(b_{mp}=a_m+c_{mp}\), \(c_{mp}=(1/2^m3^p,0,0)\), \(r_{mp}=\min[1/2^m3^p,\rho(a_m,\partial T_0)]\), and \(\partial T_0\) is the boundary of \(T_0\), considered as a plane set (in the plane \(x_1=0\)). Then the set

\[ \Delta=R\setminus\left[\bigcup_{m,p=1}^{\infty}(T_{mp}\setminus Q_{mp})\right] \]

is a bounded domain that is homeomorphic to the ball and compact with respect to the relative distance. Consider a sequence of sections \(\{q_n\}\) \((n=1,2,\ldots)\), whose set coincides with the set of circles \(\{Q_{mp}\}\) \((m,p=1,2,\ldots)\) and which are numbered in the order of approach to the face \(T_0\). This sequence of sections defines a prime end \(e\) of the domain \(\Delta\) such that \(|e|=T_0,\ |e|_1=F\). Thus, if

* Let a prime end \(e\) of a domain \(\Delta\) be defined by means of a chain of subdomains \(\{g_m\}\). The body of the prime end \(e\) is the set \(|e|=\bigcap_{m=1}^{\infty}\overline g_m\). A point \(a\in |e|\) is called a principal point of the prime end \(e\) if there exists a chain of sections \(\{q_m\}\) defining the prime end \(e\) and contracting to the point \(a\).

If \(F\) is a disconnected set, then the simple end \(e\) of the constructed domain \(\Delta\) will be of type 5. For \(n=2\) the set of principal points of a simple end is always connected \({}^{(10)}\).

1. Let us give a lower estimate for the quantity \(K\) for a mapping \(y=f(x)\) of class \([BL]^{n/2}_K\) of the ball \(D\) onto the domain \(\Delta\). This estimate will involve the function \(\psi(a,\Delta)\) of the variable \(a\), which is defined as follows. Let \(G_a^0(\Delta)\) be the component of the set
   \[ G_a(\Delta)=\{x\in\Delta:\rho(x,\partial\Delta)>a\}, \]
   containing the point \(0\). Consider a point \(a\in\Delta\setminus G_a^0(\Delta)\) and define the function \(h(a,a)\) of the variable \(a\): \(h(a,a)=\inf d(K)\), where the infimum is taken over all continua \(K\subset\Delta\) separating the points \(a\) and \(0\) in \(\Delta\) and such that the sets \(K\cap G_a^0(\Delta)\) and \(K\cap\partial\Delta\) are nonempty. Then \(\psi(a,\Delta)=\sup h(a,a)\), where the supremum is taken over all points \(a\in\Delta\setminus G_a^0(\Delta)\).

Theorem 2. Let \(f(x)\in[BL]^{n/2}_K\). Then the following lower estimate for the quantity \(K\) holds:
\[ K\geq \sup_{0<a<\delta}\min\left\{ \frac{\psi^n(a,\Delta)}{M_n n+\psi^n(a,\Delta)} \left[\ln\frac{R^n}{M_n}+\ln\ln\frac{\delta}{a}\right], \frac{R^n}{M_n}\ln\frac{\delta}{a} \right\}, \tag{1} \]
where \(M_n\) is an absolute constant \((1)\).

The proof is based on the inequality of Theorem 2 from \((1)\).

Corollary. If
\[ \overline{\lim_{a\to0}}\ \psi(a,\Delta)\sqrt[n]{\ln\ln\frac{\delta}{a}}=\infty, \tag{2} \]
then there is no mapping \(y=f(x)\) of class \([BL]^{n/2}\) of the ball \(D\) onto the domain \(\Delta\).

Remark. Let \(y=f(x)\) be a \(Q\)-quasiconformal mapping of the ball \(D: |x|<R\) onto the domain \(\Delta\) of finite measure \(m\Delta\), and let \(f(0)=0\). Then \(f(x)\in[BL]^{n/2}_K\), where \(K\leq n^{n/2}Q^{\,n-1}\max(mD,m\Delta)\), and hence
\[ Q\geq [n^{n/2}\max(mD,m\Delta)]^{1/(n-1)}K^{1/(n-1)}. \tag{3} \]
Inequalities (1) and (3) give a lower estimate for the quasiconformality coefficient \(Q\) in terms of the structure of the domain \(\Delta\) and its boundary, which is expressed by means of the function \(\psi(a,\Delta)\).

The corollary of Theorem 2 makes it possible to construct various domains \(\Delta\subset E^n\) \((n\geq3)\), homeomorphic to a ball, onto which the ball cannot be mapped by means of functions of class \([BL]^{n/2}\), nor by quasiconformal mappings. It also makes it possible to construct sequences of domains \(\{\Delta_m\}\) such that there exist mappings of the ball \(\{f_m\}\), \(f_m(D)=\Delta_m\), \(f_m\in[BL]^{n/2}\) \((m=1,2,\ldots)\), but the relation \(f_m\in[BL]^K\) is not satisfied for any \(K\).

Example 2. Consider in \(E^3\) the cube
\[ R=\{x\in E^3: |x_i|<1\ (i=1,2,3)\}. \]
Let \(\{a_m\}\) be a sequence of positive numbers \(0<a_m<1/3\) \((m=1,2,\ldots)\), monotonically tending to \(0\). Put
\[ B_m=\{x\in E^3: |x_1|<b_m,\ a_{2m-1}<x_2<a_{2m},\ 1\leq x_3<b_m\}, \]
where
\[ b_m=\{\ln[-\ln(a_{2m}-a_{2m-1})]\}^{-1/6}. \]
The set \(\bar\Delta\),
\[ \Delta=R\cup\left(\bigcup_{m=1}^{\infty}B_m\right), \]
is a domain homeomorphic to the closed ball; moreover, for this domain relation (2) holds, and hence there is no mapping of class \([BL]^{3/2}\) of the ball onto this domain.

Example 3. Consider the cube
\[ R=\{x\in E^3: |x_i|<1\ (i=1,2,3)\}. \]
Let \(\{a_m\}\) be a sequence of positive numbers \(2/3\leq a_m<1\) \((m=1,2,\ldots)\), monotonically tending to \(1\). Put
\[ C_m=\{x\in R: x_2=a_m\},\qquad q_m=\{x\in C_m: |x_1|<a_{m+1}-a_m,\ |x_3|<b_m\}, \]
where
\[ b_m=\frac12\exp[-\exp(a_{m+1}-a_m)^{-6}]. \]
Then the set
\[ \Delta=R\setminus\left[\bigcup_{m=1}^{\infty}(C_m\setminus q_m)\right] \]
is a domain, homeomorphic to a ball and compact in the relative distance (it is constructed according to the type of the domains of Example 1). The domain \(\Delta\) has

a simple end of the second type, determined by the sequence of sections \(\{q_m\}\), and for it relation (2) is satisfied.

Example 4. Consider a domain having the form of a wedge with zero angle
\(\Delta=\{x\in E^3:\ |x_1|<1,\ |x_2|<1,\ |x_3|<\varphi(x_2)\}\), where \(\varphi(x_2)=\exp[-\exp(x_2+1)^{-6}]\). It is easy to see that for this domain relation (2) is satisfied. A mapping of class \([BL]^{n/2}\) onto a domain of this form will be impossible also in the case when, as the function \(\varphi(x_2)\), one takes one having a higher order of contact with the axis \(Ox_2\) in comparison with the function chosen in this example.

We note that, for quasiconformal mappings, as F. W. Gehring and J. Väisälä established [11], a mapping onto a domain having a ridge directed outward, or a peak directed inward, is impossible. One can give examples of mappings showing that, for mappings of class \([BL]^{n/2}\), such mappings are possible if the ridge and the peak are not too “sharp.”

Example 5. Put \(\Delta=\{x\in E^3:\ |x_i|<1\ (i=1,2,3)\}\), \(R_n=\{x\in\Delta:\ x_2=1-1/2n,\ -1<x_3\leq 0\}\) \((n=1,2,\ldots)\), and consider the sequence of domains \(\{\Delta_n\}\), \(\Delta_n=\Delta\setminus R_n\). Then there exists no number \(K\) and no sequence of mappings \(\{f_n(x)\}\) of the ball \(D\) onto the domains \(\Delta_n\), \(f_n(D)=\Delta_n\), such that \(f_n(x)\in[BL]^{3/2}_K\) for every \(n\).

Example 6. Put \(\Delta=\{x\in E^3:\ |x_1|<1,\ |x_2|<1,\ -1<x_3<2\}\), \(R_n'=\{x\in\Delta:\ x_2\in[-1/2n,1/2n],\ x_3=1\}\), \(\Delta_n=\Delta\setminus R_n\). Then there exists no sequence of mappings \(\{f_n(x)\}\) of the ball \(D\) onto the domains \(\Delta_n\), \(f_n(D)=\Delta_n\), and no number \(K\) such that \(f_n\in[BL]^{3/2}_K\) for all \(n\).

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

Received
3 IV 1967

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