Reports of the Academy of Sciences of the USSR
1967. Volume 175, No. 1

UDC 517.54

MATHEMATICS

A. V. SYCHEV

ON SPATIAL QUASICONFORMAL MAPPINGS SATISFYING A HÖLDER CONDITION AT BOUNDARY POINTS

(Presented by Academician M. A. Lavrent'ev on 10 IX 1966)

In papers \((^3,^4,^6,^7)\) it was proved that every \(Q\)-quasiconformal mapping \((^2)\) of a domain \(D\) of three-dimensional Euclidean space \(R^3\) satisfies a Hölder condition on any closed subdomain \(\overline{D'}\), \(\overline{D'} \subset D\). In the present note the fulfillment of this condition is established for \(Q\)-quasiconformal mappings of certain classes of domains also at boundary points.

For a given set \(E \subset R^3\), \(\partial E\) denotes its boundary, \(\overline{E}\) its closure, \(m(E)\) the three-dimensional Lebesgue measure, and for given sets \(E_1, E_2 \subset R^3\), \(r(E_1,E_2)\) the distance between them. Further, for a finite point \(P\) and \(t>0\), \(B^3(P,t)\) denotes the ball \(|x-P|<t\), and \(S^2(P,t)\) its boundary sphere; if \(P=0\), the following abbreviations are also used:

\[ B^3(t)=B^3(0,t),\quad B^3=B^3(1),\quad S^2(t)=S^2(0,t),\quad S^2=S^2. \]

A bounded domain \(D\) is called a ring domain if it is homeomorphic to the domain enclosed between concentric spheres.

Let \(D\) be a ring domain; let \(B_0\) and \(B_1\) be its inner and outer boundary components; and let \(F_0 \subset B_0\) and \(F_1 \subset B_1\) be certain distinguished simply connected boundary continua.

We shall say that a curve \(\gamma \subset \overline{D}\) connects \(F_0\) and \(F_1\) in \(D\) if \(\gamma \cap F_0 \ne \varnothing\), \(\gamma \cap F_1 \ne \varnothing\), where \(\varnothing\) is the empty set. We shall also say that a surface \(\sigma \subset \overline{D}\) separates \(F_0\) and \(F_1\) in \(D\) if every curve connecting \(F_0\) and \(F_1\) in \(D\) intersects it in at least one point.

Lemma. Let \(\Sigma\) be a certain family of surfaces contained in the ball \(B^3\); let \(\Sigma^*\) be the image of \(\Sigma\) under the transformation \(y=x/|x|^2\), and let \(\Sigma_1\) be the family of surfaces \(\sigma_1=\sigma \cup \sigma^*\), \(\sigma\in\Sigma\), \(\sigma^*=y(\sigma)\in\Sigma^*\).
Then

\[ M(\Sigma)=\sqrt{2}\,M(\Sigma_1). \tag{1} \]

We shall say that a domain \(D\) is \(a\)-locally simply connected at the boundary point \(P\) if one can indicate an \(a>0\) such that \(\partial D \cap B^3(P,t)\) is simply connected for all \(t\le a\). If \(D\) is \(a\)-locally simply connected at every boundary point, then we shall say that it is \(a\)-locally simply connected on the boundary.

Theorem 1. Let \(y=f(x)\), \(f(0)=0\), be a \(Q\)-quasiconformal mapping of a bounded domain \(D\), \(a\)-locally simply connected at the point \(P\in\partial D\) \((a<R_0/2)\), onto the ball \(B^3\). Then for every point \(Q\in \overline{D}\cap B^3(P,a)\)

\[ |f(P)-f(Q)|<C|P-Q|^{1/K(Q)}, \tag{2} \]

where \(C=C(R_0,Q)\), \(K(Q)=Q\sqrt{2}\), \(R_0=r(0,\partial D)\).

Proof. In view of the \(a\)-local simple connectedness of the domain \(D\) at the point \(P\), for every point \(Q\in\overline{D}\cap B^3(P,a)\) there exists a curve \(\gamma_{P,Q}\), connecting \(P\) and \(Q\), such that: a) \(\gamma_{P,Q}-Q\subset \partial D\cap B^3(P,|P-Q|)\), if \(Q\in\partial D\); b) \(\gamma_{P,Q}-P-Q\subset D\cap B^3(P,|P-Q|)\), if \(Q\in D\).

Let \(Q_0\) be a point for which \(|Q_0|<R_0/2\). Denote
\(r=|Q_0|\), \(r^*=|f(Q_0)|\), \(\tilde r=|P-Q|\), \(\tilde r^*=|f(P)-f(Q)|\);
\(F_0=[0,Q_0]\), \(F_0^*=f(F_0)\), \(F_1=\gamma_{P,Q}\), \(F_1^*=f(F_1)\). Then
\(D-(F_0\cup F_1)\) and \(B^3-(F_0^*\cup F_1^*)\) are ring domains with boundary components
\(B_0=F_0\), \(B_1=\partial D\cup F_1\) and \(B_0^*=F_0^*\), \(B_1^*=S^2\cup F_1^*\), respectively. Let \(\Sigma\) and \(\Sigma^*\) be the families of surfaces separating \(F_0,F_1\) in \(D-(F_0\cup F_1)\) and \(F_0^*,F_1^*\) in \(B^3-(F_0^*\cup F_1^*)\).

By the \(Q\)-quasiconformality of \(f(x)\),

\[ M(\Sigma)\leq QM(\Sigma^*). \tag{3} \]

Using the known properties of moduli \((^1)\), Theorem 4 \((^5)\), and \((^1)\), we obtain the estimates

\[ M^*(\Sigma)<\frac{2}{2\sqrt{\pi}}\ln\frac{R_0^2}{2r\tilde r},\qquad M(\Sigma^*)<\frac{V^2}{2\sqrt{\pi}}\ln\frac{2\lambda^2}{r^* \tilde r^*},\qquad 4\leq\lambda\leq 1,2,4,\ldots, \]

whose substitution in (3) gives (2).

Corollary 1. The family \(\{f(x)\}\) of normalized \(Q\)-quasiconformal mappings of a bounded domain \(D\) with an \(a\)-locally simply connected boundary onto the ball \(B^3\) is equicontinuous in \(\overline D\).

Let \(y=\varphi(x)\) be a homeomorphism of the domain \(D\subset R^3\); we shall call \(\varphi(x)\) a \(C\)-isometry, \(1\leq C<\infty\) (a Lipschitz mapping with constant \(C\)), if

\[ C^{-1}|P_1-P_2|\leq |f(P_1)-f(P_2)|\leq C|P_1-P_2| \]

for all \(P_1,P_2\in D\).

We shall say that a domain \(D\ni 0\), homeomorphic to a ball, satisfies at the point \(P\in\partial D\) the condition \((a,C)\), if one can indicate a sufficiently small \(a>0\) such that: 1) there exists a \(C\)-isometry \(\varphi_P\) mapping \(D\cap B^3(P,a)\) onto a half-ball \(H\), under which \(\partial D\cap B^3(P,a)\) goes into the flat part \(\partial H\), and the point \(P\) goes to the center of \(H\); 2) the area of any surface separating \(S^2(a)\) and \(D\cap S^2(P,a)\) in \(D-\bigl(\overline{B^3(a)}\cup\overline{B^3(P,a)}\bigr)\) is not less than the area of \(D\cap S^2(P,a)\).

Theorem 2. Let \(y=f(x)\), \(f(0)=0\), be a \(Q\)-quasiconformal mapping of a bounded domain \(D\), homeomorphic to a ball, onto a bounded domain \(D^*\), and suppose that the domains \(D\) and \(D^*\), at the boundary points \(P\) and \(P^*=f(P)\), satisfy the condition \((a,C)\). Then for every point \(Q\in\overline D\), sufficiently close to \(P\),

\[ C_1|P-Q|^{K(C,Q)}<|f(P)-f(Q)|<C_2|P-Q|^{1/K(C,Q)}, \]

where
\(C_1=C_1(a,C,R_0,R_0^*,V,Q)\), \(C_2=C_2(a,C,R_0,R_0^*,V^*,Q)\),
\(K(C,Q)=C^2Q\sqrt{2}\), \(R_0=r(0,\partial D)\), \(R_0^*=r(0,\partial D^*)\),
\(V=m(D)\), \(V^*=m(D^*)\).

The proof of Theorem 2 is analogous to the proof of Theorem 1.

Corollary 2. The family \(\{f(x)\}\) of normalized \(Q\)-quasiconformal mappings of a bounded domain \(D\), homeomorphic to a ball and satisfying the condition \((a,C)\) on the boundary, onto a domain \(D^*\) of the same kind is equicontinuous in \(\overline D\).

Institute of Mathematics
Siberian Branch
Academy of Sciences of the USSR

Received
1 IX 1966

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