UDC 517.53:512.9

MATHEMATICS

V. M. MIKLYUKOV

ON CERTAIN CLASSES OF MAPPINGS IN THE PLANE

(Presented by Academician M. A. Lavrent’ev on 1 IV 1968)

In the present note classes of mappings are introduced that are generalizations of the class (\widetilde{BL}_k), to the consideration of which the monographs of G. D. Suvorov ((^1)) and, in a somewhat less general situation, J. Lehto and K. I. Virtanen ((^2)) are devoted. Further, following the method developed by G. D. Suvorov for studying mappings of the class (\widetilde{BL}_k), two-sided estimates are established for the distortion of relative distance in closed domains, which makes it possible to transfer the greater part of the results from ((^1)) and partially from ((^2)) to mappings of the classes considered. Some results are also given that are new even for mappings of the class (\widetilde{BL}_k).

1. Let (w=f(z)=u(x,y)+iv(x,y)) be a continuous mapping of a domain (A) of the plane (z=x+iy), with the spherical metric ((r)) introduced in it, onto a domain (B) of the plane (w=u+iv) with metric ((R)). We shall say that (w=f(z)) belongs to the class (\widetilde{BL}_k^{\Phi}) in the domain (A) if (f(z)) has in (A) generalized partial derivatives (u_x,u_y,v_x,v_y) in the sense of S. L. Sobolev, for which

[
I_{\Phi}(f)=\int_A \Phi(|f'(z)|)\,dA \leq K<\infty,
\tag{1}
]

where (\Phi(t)) is a nonnegative, convex downward ((\Phi''>0)) function, defined for all (t\in[0,\infty)) and such that

[
\int_0^\infty \frac{\Phi(\tau)}{\tau^3}\,d\tau=\infty,
\tag{2}
]

and by (|f'(z)|) is denoted the expression

[
(u_x^2+u_y^2+v_x^2+v_y^2)^{1/2}/(1+|f(z)|^2/R^2).
]

For (\Phi(\tau)=\tau^2) this class of mappings coincides with the class (\widetilde{BL}_k) considered in ((^1)). At the same time, mappings of the classes (\widetilde{BL}_k^{\Phi}) encompass a significantly broader part of mappings, as may be judged, for example, from Remarks 1 and 2 to Theorem 1. It is not difficult to see that there exist such (\Phi_1) and (\Phi_2), satisfying condition (2), for which the inclusions

[
\widetilde{BL}_k^{\Phi_1}\supset \widetilde{BL}_k \supset \widetilde{BL}_k^{\Phi_2}
]

hold.

For simplicity of exposition we shall restrict ourselves here to the case when

[
\widetilde{BL}_k^{\Phi}\supset BL_k.
]

1. Let (z_1,z_2) be arbitrary points of a simply connected domain (A\subset E^2) containing the origin. Following G. D. Suvorov ((^1)), p. 50, we define the relative distance between the points (z_1,z_2) in the domain (A) by

[
\rho_A(z_1,z_2;r)=\min[\rho_1(z_1,z_2),\rho_2(z_1,z_2)],
\tag{3}
]

where (\rho_1(z_1,z_2)) is the greatest lower bound of the diameters, taken in the spherical metric ((r)), of curves contained in (A) and joining the points (z_1,z_2), and (\rho_2(z_1,z_2)) is the greatest lower bound of the spherical diameters of curves sepa-

ing points (z_1, z_2) from the origin. Complete the domain (A) with respect to the relative distance. The “boundary” elements adjoined in this process are identified with the prime ends of K. Carathéodory. The resulting space will be denoted by (\widetilde A).

The following theorem establishes the order of equicontinuity and equiopenness (see, for example, ((^1)), p. 94) of topological mappings of the class (\widetilde{BL}_k^{\Phi}) in closed domains.

Theorem 1. Let (w=f(z)) be a homeomorphic mapping of class (\widetilde{BL}_k^{\Phi}) of a domain (A \in E^2) onto a domain (B \in E^2), normalized by the conditions
(f(0)=0,\ f(z_0)=w_0).

Then (f(z)) extends to a continuous mapping of (\widetilde A) onto (\widetilde B), and for any two points (z', z'' \in \widetilde A) such that (\rho_{\widetilde A}(z', z''; r) \leqslant \varepsilon_1), the inequality
[
\rho_{\widetilde B}[f(z'), f(z''); R] \leqslant
\varphi_{\Phi}[\rho_{\widetilde A}(z', z''; r)]
\tag{4}
]
holds, where
[
\varphi_{\Phi}(t)=
\left[
C(k,\Phi)\int_1^{1/t} \frac{\Phi(\tau)}{\tau^3}\,d\tau
\right]^{-1/2}.
\tag{5}
]

If, moreover, the inverse mapping (z=f^{-1}(w)) belongs to the class (\widetilde{BL}k^{\Phi}) in the domain (B), then the lower estimate
[
\varphi(z', z''; r)]}^{-1}[\rho_{\widetilde A
\leqslant
\rho_{\widetilde B}[f(z'), f(z''); R],
\tag{6}
]
also holds, valid for all (z', z'' \in \widetilde A) for which (\rho_{\widetilde A}(z', z''; r) \leqslant \varepsilon_2). The constants (\varepsilon_1, \varepsilon_2) depend on (\Phi), (\rho_{\widetilde A}(0,\partial A; r)), (\rho_{\widetilde B}(0,\partial B; R)), and on the normalization conditions.

Remark 1. Estimates (4), (6) are sharp in the sense of order.

Remark 2. Since the rate of divergence of the integral (2) can, by a suitable choice of (\Phi(\tau)), be made arbitrarily small, it is evident that the classes (\widetilde{BL}_k^{\Phi}) may have arbitrarily low order of equicontinuity. This circumstance indicates the considerable breadth of the classes (\widetilde{BL}_k^{\Phi}) in comparison with the class (BL_k).

The existence of estimates (4), (6) makes it possible to carry over a large part of the results from ((^1)) to mappings of the classes (\widetilde{BL}_k^{\Phi}). As an example, one may mention the theorem on boundary correspondence by prime ends which follows directly from Theorem 1. The two-sided equicontinuous estimates of the distortion of spherical (Euclidean) distance inside domains, which follow easily from (4), (6), give an analogue of Carathéodory’s theorem on convergence to the kernel of a sequence of domains (see ((^1))). The two-sided equicontinuous estimates of the distortion of relative distance in closed domains make it possible to extend the theory of prime ends of a sequence of domains converging to a nondegenerate kernel ((^1)) to the case of mappings (f, f^{-1} \in \widetilde{BL}_k^{\Phi}), with all the consequences flowing from this; among them we mention here only the theorem on the stability of conformal mappings in closed domains ((^3)). In addition, on the basis of inequalities (4), (6) one can obtain various estimates of the distortion of boundary arcs and of the areas of boundary rings (see ((^4))).

1. The following assertions are new also for mappings of the class (\widetilde{BL}_k).

Theorem 2. Let (A) be a domain in (E^2) whose boundary contains a one-point component (z_0 \in \partial A). Let (f(z)) be an open mapping of class (\widetilde{BL}_k^{\Phi}) of the domain (A) into (E^2).

Then (f(z)) is continuous in the spherical metric at the point (z_0 \in \partial A).

This theorem indicates the fact that open mappings of the class (\widetilde{BL}_k^{\Phi}) may have isolated singularities only of the pole type. As a consequence of it we obtain an analogue of Liouville’s theorem, well known in the theory of analytic functions.

Theorem 3. There is no open mapping of the class (\widetilde{BL}_k^{\Phi}) of the entire plane (E^2) onto a proper part of it.

The following theorem gives a simple criterion for normality, in the sense of P. Montel ((^5)), of a family of mappings of the class (\widetilde{BL}_k^{\Phi}).

Theorem 4. A family of (F)-open mappings (f(z)) of the class (\widetilde{BL}_k^{\Phi}) in a domain (A \subset E^2), not assuming in this domain one finite value (w_0 \in E^2), is normal.

Donetsk Computing Center
Academy of Sciences of the USSR
Donetsk State University

Received
20 III 1968

CITED LITERATURE

1. G. D. Suvorov, Families of Plane Topological Mappings, Novosibirsk, 1965.
2. J. Lelong-Ferrand, Représentation conforme et transformations à intégrale de Dirichlet bornée, Paris, 1955.
3. G. D. Suvorov, Ukr. Mat. Zhurn., 20, No. 1 (1968).
4. G. D. Suvorov, DAN, 157, 802 (1954).
5. P. Montel, Normal Families of Analytic Functions, Moscow–Leningrad, 1936.