UDC 517.53:512.9

MATHEMATICS

V. M. MIKLYUKOV

ON \(\varepsilon\)-QUASICONFORMAL MAPPINGS OF A BALL ONTO A BALL

(Presented by Academician M. A. Lavrent’ev, 18 XII 1968)

It is known that for \(q\)-quasiconformal mappings in space with \(q=1+\varepsilon\), smallness of \(\varepsilon\) guarantees the closeness of the mapping to a conformal one; and since the class of conformal mappings in space is exhausted only by Möbius transformations, under a corresponding normalization \(\varepsilon\)-quasiconformal mappings are close to the identity (see \(\left({}^{1-3}\right)\)). However, all these results are mainly qualitative in character.

In the plane case, for \(\varepsilon\)-quasiconformal mappings of a disk onto a disk, P. P. Belinskii \(\left({}^{4}\right)\) obtained an exact estimate for the deviation of the mapping from a conformal one. He also \(\left({}^{5}\right)\) gave an estimate for the change of measure under mappings of this class.

In the present note analogous results are established for the case of \(\varepsilon\)-quasiconformal mappings of the three-dimensional ball onto itself. In addition, an estimate is given for the deviation of the mapping from a conformal one in the metric of the space \(W_2^1\). However, the estimates obtained are not sharp.

1. Let the vector-function
   \[ y=f(x)\equiv (f_1(x), f_2(x), f_3(x)), \]
   where \(x=(x_1,x_2,x_3)\), realize an \(\varepsilon\)-quasiconformal mapping of the three-dimensional ball \(B: |x|<1\) onto itself, leaving fixed three fixed points on the boundary.

Theorem 1. If the vector-function \(y=f(x)\) realizes an \(\varepsilon\)-quasiconformal mapping of the three-dimensional ball \(B\) onto itself, normalized in the manner indicated above, then for small \(\varepsilon\)
\[ \|f(x)-x\|_{W_2^1}<\mathrm{const}\cdot \varepsilon^{1/4}+o(\varepsilon), \tag{1} \]
where the constant depends only on the normalization conditions, and by \(\|f\|_{W_2^1}\) is denoted
\[ \left(\sum_{i=1}^{3}\|f_i\|_{L^2(B)}^2+\sum_{i=1}^{3}\|\nabla f_i\|_{L^2(B)}^2\right)^{1/2}. \]

The proof of this theorem is carried out according to the following scheme. By virtue of a theorem of F. Gehring \(\left({}^{6}\right)\), the given mapping induces an \(\varepsilon\)-quasiconformal homeomorphism of the sphere \(S: |x|=1\) onto itself.

Hence, using the results of S. L. Krushkal’ \(\left({}^{7}\right)\), it is not difficult to show that everywhere on the sphere \(S\) the inequality
\[ |f(x)-x|<\mathrm{const}\cdot \varepsilon+o(\varepsilon), \tag{2} \]
holds, where the constant depends only on the normalization conditions.

Consider \(u_i(x)\) \((i=1,2,3)\)—harmonic functions coinciding on the boundary with the components of the vector-function \(f(x)\), i.e. \(u_i|_S=f_i|_S\) \((i=1,2,3)\). It can be shown that for \(\varepsilon\)-quasiconformal mappings
\[ \|\nabla(f-x)\|_{L^2}<\mathrm{const}\cdot \left(\varepsilon+\|\nabla(u-x)\|_{L^2}\right)^{1/2}+o(\varepsilon). \tag{3} \]

To estimate \(\|\nabla(u-x)\|_{L^2}\) we use Green’s formula

\[ \|\nabla(u-x)\|_{L^2}^{2} \equiv \sum_{i=1}^{3}\int_B |\nabla(u_i-x_i)|^2\,dB = \sum_{i=1}^{3}\int_S (u_i-x_i)\,\frac{\partial(u_i-x_i)}{\partial n}\,dS, \]

where \(\partial/\partial n\) denotes differentiation along the normal. Hence, by virtue of inequality (2), we obtain

\[ \|\nabla(u-x)\|_{L^2}^{2} < \mathrm{const}\cdot\varepsilon \sum_{i=1}^{3}\int_S \left|\frac{\partial(u_i-x_i)}{\partial n}\right|\,dS . \tag{4} \]

The uniform boundedness of the contour integrals in the right-hand side of (4) is established on the basis of M. I. Vishik’s inequality \({}^{8}\) for the normal derivatives of harmonic functions in a ball and the fact that \(f\) is quasiconformal on the sphere \(S\).

Thus, from (3) and (4) we have

\[ \|\nabla(f-x)\|_{L^2} < \mathrm{const}\cdot\varepsilon^{1/4}+o(\varepsilon). \tag{5} \]

Inequality (5), together with relation (2), gives an analogous estimate for \(\|f-x\|_{L^2}\), whence the validity of the theorem follows.

1. From Theorem 1, using the uniform continuity of \(\varepsilon\)-quasiconformal mappings of the ball onto itself (see, for example, \({}^{9}\)), we obtain:

Theorem 2. For \(\varepsilon\)-quasiconformal mappings of the three-dimensional ball onto itself, normalized in the same way as above, the inequality

\[ \max_B |f(x)-x| < \mathrm{const}\cdot\varepsilon^{1/16}+o(\varepsilon), \tag{6} \]

holds, where \(\mathrm{const}\) depends only on the normalization conditions.

The following theorem gives a quantitative estimate of the change of measure under \(\varepsilon\)-quasiconformal mappings.

Theorem 3. Let the vector function \(y=f(x)\) realize an \(\varepsilon\)-quasiconformal homeomorphism of the three-dimensional ball \(B\) onto itself, normalized in the manner indicated above. Then, for every measurable \(E\subset B\), the inequality

\[ |\operatorname{mes} f(E)-\operatorname{mes} E| < \mathrm{const}\cdot\varepsilon^{1/2}+o(\varepsilon), \tag{7} \]

is valid, where the constant depends only on the normalization conditions.

The proof of this theorem is carried out essentially analogously to \({}^{5}\), using relation (2).

Donetsk Computing Center
Academy of Sciences of the Ukrainian SSR

Donetsk State University

Received
2 XII 1968

REFERENCES

\({}^{1}\) M. A. Lavrent’ev, DAN, 95, No. 5 (1954).
\({}^{2}\) Yu. G. Reshetnyak, in the collection Some Problems of Mathematics and Mechanics, Novosibirsk, 1961.
\({}^{3}\) P. P. Belinskii, DAN, 147, No. 5 (1962).
\({}^{4}\) P. P. Belinskii, DAN, 91, No. 5 (1953).
\({}^{5}\) P. P. Belinskii, DAN, 121, No. 1 (1958).
\({}^{6}\) F. W. Gehring, Trans. Am. Math. Soc., 103, No. 3 (1962).
\({}^{7}\) S. L. Krushkal’, Siberian Math. J., 8, No. 4 (1967).
\({}^{8}\) M. I. Vishik, UMN, 6, No. 2 (1951).
\({}^{9}\) A. V. Sychev, DAN, 166, No. 2 (1966).