UDC 517.53:512.9

MATHEMATICS

V. M. MIKLYUKOV

ON ORIENTED QUASICONFORMAL MAPPINGS IN SPACE

(Presented by Academician M. A. Lavrent’ev on 30 I 1968)

1. Let \(E^n\) be \(n\)-dimensional Euclidean space, \(|x|\) the length of the vector \(x=(x_1,\ldots,x_n)\in E^n\), and \(G\) a domain in \(E^n\). By \(C_0^\infty(G)\) we shall denote the class of infinitely differentiable finite functions with compact supports \(\operatorname{supp} u(x)\subset G\), and by \(W_n^1(G)\) the class of vector functions \(f(x)=[f_1(x),\ldots,f_n(x)]\) having first-order partial derivatives, generalized in the sense of S. L. Sobolev, locally summable to the power \(n\). Put

\[ \lambda(x,f)=\left[\sum_{i=1}^n\sum_{j=1}^n\left(\frac{\partial f_i}{\partial x_j}\right)^2\right]^{1/2}, \qquad I(x,f)=\frac{\partial(f_1,\ldots,f_n)}{\partial(x_1,\ldots,x_n)}. \]

A continuous vector function \(f(x)\) is said to realize a \(q\)-quasiconformal mapping of a domain \(G\subset E^n\) if \(f(x)\in W_n^1(G)\) and there exists a constant \(q\) such that almost everywhere in the domain the inequality

\[ \lambda^n(x,f)\leq n^{n/2}q^{\,n-1}|I(x,f)| \tag{1} \]

holds.

In the present note only oriented \(q\)-quasiconformal mappings are considered, i.e., mappings for which \(I(x,f)\geq 0\) almost everywhere in the domain.

1. Below we shall need the following assertion, which we give here without proof.

Lemma 1. If a continuous vector function \(f(x)\in W_n^1(G)\), \(u(x)\in C_0^\infty(G)\), then

\[ \left|\int_G u(x)I(x,f)\,dG\right| \leq C_1\int_G |\nabla u(x)|\,|f_1(x)|\,\lambda^{n-1}(x,f)\,dG, \tag{2} \]

where \(C_1\) is a constant depending only on \(n\).

The following theorem is a simple consequence of Lemma 1.

Theorem 1. Let a vector function \(f(x)\) realize an oriented \(q\)-quasiconformal mapping of a domain \(G\subset E^n\).

Then for any compact set \(F\Subset G\) the inequality

\[ \|\lambda(x,f)\|_{L^n(F)}\leq C_2\|f_1(x)\|_{L^n(G)} \tag{3} \]

holds, where \(C_2\) depends only on \(n\), \(q\), and \(\rho(F,\partial G)\).

This theorem makes it possible to sharpen somewhat the results of Yu. G. Reshetnyak \((^1)\) and E. D. Callender \((^2)\) concerning the equicontinuity of \(q\)-quasiconformal mappings.

Theorem 2. Under the conditions of Theorem 1, for any compact set \(F\Subset G\), for \(x',x''\in F\), one has

\[ |f(x')-f(x'')|\leq C_3|x'-x''|^\alpha, \tag{4} \]

where \(C_3\) is a constant depending only on \(n\), \(q\), \(\rho(F,\partial G)\), and \(\|f_1(x)\|_{L^n(G)}\), and \(0<\alpha\leq 1\) depends on \(q\) and \(n\).

This assertion follows directly from inequality (3) and Theorem 2 of paper \((^1)\).

Theorem 3. Let \(\{f(x)\}\) be a family of oriented \((1+\varepsilon)\)-quasiconformal mappings of the \(n\)-dimensional \((n>2)\) ball \(G: |x|<1\), normalized by the conditions: \(f(0)=0,\ f(1)=1\), and having uniformly bounded norms \(\|f_1(x)\|_{L^n(F)}\) on every compact set \(F \subset G\).

Then there exists a universal function \(\mu(\varepsilon,r)\ge 0\), defined for all \(0\le r<1\) and possessing the properties: \(\mu(\varepsilon,r)\to 0\) as \(\varepsilon\to 0\), for which

\[ |f(x)-x|\le \mu(\varepsilon,r). \tag{5} \]

The proof of this theorem is carried out analogously to \((^3)\), using the preceding theorem and Theorem 1 of paper \((^4)\).

For homeomorphic \(q\)-quasiconformal mappings this assertion was obtained by M. A. Lavrent’ev \((^5)\) (under some additional restrictions) and by Yu. G. Reshetnyak \((^3)\).

3. Let \(F\) be a compact set belonging to a domain \(G\subset E^n\). Define its conformal capacity by

\[ \operatorname{cap}_G F=\inf \int_G |\nabla u(x)|^n\,dG, \tag{6} \]

where the infimum is taken over all functions \(u(x)\in C_0^\infty(G)\) equal to \(1\) on \(F\).

In proving the following two theorems we use

Lemma 2. If the vector function \(f(x)\) realizes an oriented \(q\)-quasiconformal mapping of a domain \(G\subset E^n\), then for any compact set \(F\subset G\) one has

\[ \|\lambda(x,f)\|_{L^n(F)}\le C_4 \sup_G |f_1(x)|(\operatorname{cap}_G F)^{1/n}, \tag{7} \]

where \(C_4\) depends only on \(q\) and \(n\).

The following two theorems contain results that overlap with already known earlier (see \((^6)\)) assertions on the impossibility of mapping the whole space onto a part and on the removability of isolated singularities. Although our results are not generalizations of the known ones, they contain new information.

Theorem 4. If the vector function \(f(x)\) realizes an oriented \(q\)-quasiconformal mapping of the entire space \(E^n\) and if

\[ \max_{|x|\le r}|f_1(x)|=o\left(\ln^{(n-1)/n} r\right)\quad (r\to\infty), \tag{8} \]

then

\[ f(x)\equiv \mathrm{const}. \]

Theorem 5. Let the vector function \(f(x)\) realize an oriented \(q\)-quasiconformal mapping of the domain \(G: 0<|x|<1\), and suppose that

\[ \max_{r\le |x|\le 1/2}|f_1(x)|=o\left(\ln^{(n-1)/n} r\right)\quad (r\to 0), \tag{9} \]

\[ \lim_{x\to 0}|f(x)|\ne \infty. \tag{10} \]

Then \(f(x)\) is continuous at the origin.

Donetsk Computing Center
Academy of Sciences of the Ukrainian SSR

Donetsk State University

Received
25 I 1968

References

\(^1\) Yu. G. Reshetnyak, Sibirsk. matem. zhurn., 7, No. 5 (1966).
\(^2\) E. D. Kalleder, Pacific J. Math., 10, No. 2 (1960).
\(^3\) Yu. G. Reshetnyak, In: Some Problems of Mathematics and Mechanics, Novosibirsk, 1961.
\(^4\) Yu. G. Reshetnyak, Sibirsk. matem. zhurn., 8, 4 (1967).
\(^5\) M. A. Lavrent’ev, DAN, 95, No. 5 (1954).
\(^6\) Yu. G. Reshetnyak, B. V. Shabat, Proceedings of the IV Conference on Function Theory, 2, L., 1964.