UDC 517:514

MATHEMATICS

B. P. KUFAREV

ABSOLUTE CONTINUITY OF FUNCTIONS OF THE CLASS $\widetilde W_p^1$ ON LEVEL SETS OF A FUNCTION OF THE CLASS $\widetilde W_q^1$ AND SOME BOUNDARY PROPERTIES OF MAPPINGS WITH GENERALIZED DERIVATIVES IN A PLANE DOMAIN

(Presented by Academician M. A. Lavrent’ev on 30 X 1967)

Let $G$ be an arbitrary domain of the plane $R^2$. Denote by $W_p^1(G)$ (respectively $W_\infty^1(G)$) the class of functions $r(z)$ continuous in $G$, $z=(x,y)$, having there first-order generalized derivatives in the sense of S. L. Sobolev (see $(^{1,2})$), summable to the power $p \geqslant 1$ (respectively essentially bounded) on every compact set $K \subset G$. The totality of functions of the class $W_p^1(G)$ differentiable almost everywhere in $G$ will be denoted by $\widetilde W_p^1(G)$.

Let $T(z)$ be a mapping of $G$ into $R^2$, $T=u+iv$. We shall say that $T \in \widetilde W_p^1(G)$ if $u,v \in \widetilde W_p^1(G)$.

As is known, a function of the class $W_1^1(G)$ is absolutely continuous inside almost all $x$-sections and $y$-sections of the domain $G$. In the present work a more general fact is established, with the aid of which some boundary properties of mappings $T \in \widetilde W_p^1(G)$ are proved, in particular, of $\widetilde{BL}$-homeomorphisms, to which the monograph of G. D. Suvorov $(^3)$ is devoted.

If the domain $G$ is finitely connected, then the definitions and theorems given below in Sec. 3 can be modified for sets of simple ends of $G$, for example as was done in the paper $(^4)$.

1. If a function $r \in \widetilde W_1^1(G)$, then the orthogonal vectors $\nabla r(z)$ and $s(z)=(-\partial r/\partial y,\ \partial r/\partial x)$ are defined almost everywhere in $G$.

For a function $f \in \widetilde W_1^1(G)$ and a vector field $s$ we put in $G$ almost everywhere

$$ D_s f \overset{\mathrm{def}}{=} \begin{cases} \left(-\dfrac{\partial f}{\partial x}\dfrac{\partial r}{\partial y} +\dfrac{\partial f}{\partial y}\dfrac{\partial r}{\partial x}\right)\cdot |\nabla r|^{-1}, & \text{if } \nabla r \ne 0,\\[1.2em] 0, & \text{if } \nabla r=0. \end{cases} $$

If a mapping $T \in \widetilde W_1^1(G)$, then, by definition, $D_sT=D_su+iD_sv$.

A plane curve is a continuous mapping $z:I\to R^2$, where $I$ is a closed interval of the numerical axis. If $\Gamma=z(I)$, and the mapping $T:\Gamma\to R^2$ is continuous, then by the symbol $l[T(\Gamma)]$ we denote the length of the curve $T\circ z$.

Further, if $r$ is a function defined in $G$, and $A\subset G$, then

$$ {}_t A \overset{\mathrm{def}}{=} \{z\in A\mid r(z)=t\}. $$

Let $t_1=\inf r(z)$ and $t_2=\sup r(z)$, $z\in G$.

Throughout the sequel $H$ denotes linear Hausdorff measure; $\overline M$ is the closure of the set $M\subset R^2$; $\operatorname{Fr} M$ is the boundary of $M$.

From the results of A. S. Kronrod $(^5)$, taking into account the paper $(^6)$, it follows that for $r\in \widetilde W_1^1(G)$ the following assertions are valid for almost all $t\in(t_1,t_2)$:

a) if $\Gamma\subset {}_t G$ is a simple arc, then $\Gamma$ is rectifiable,

b) the components of the set \(G_t\) are either points or homeomorphic images of a circle or of an open interval (in a countable number), and if \(G_t^0 \subset G_t\) is the collection of point-components, then \(H(G_t^0)=0\).

Definition. Let \(B\) be a domain with compact closure \(\overline B \subset G\). For any continuous mapping \(T:G\to R^2\) set

\[ l[T(B_t)]=\sum_k l[T(\beta_t^k)], \]

where \(\beta_t^k\) is the closure of a nonpoint component of the set \(B_t\) \((k=1,2,\ldots)\). By \(l[T(G_t)]\) we denote \(\lim_n l[T(B_t^n)]\), where \((B^n)_{n=1,2,\ldots}\) is some increasing sequence of domains with compact closure \( \overline{B^n}\subset G\), exhausting the domain \(G\).

This definition has meaning for almost all \(t\), and \(l[T(G_t)]\) does not depend on the choice of the sequence \((B^n)\).

Theorem 1. If the mapping \(T\) belongs to the class \(\widetilde W_p^1(G)\), \(p\ge 1\), and the function \(r(z)\in \widetilde W_q^1(G)\), \(p^{-1}+q^{-1}=1\), then for almost all \(t\in(t_1,t_2)\) the components \(u,v\) of the mapping \(T\) are absolutely continuous inside \(G_t\), i.e., on every simple arc \(\Gamma\subset G_t\) the restrictions \(u|_\Gamma\) and \(v|_\Gamma\) are absolutely continuous as functions of the arc length \(H\). Moreover,

\[ l[T(G_t)]=\int_{G_t} |D_sT|\,dH. \]

Corollary. Suppose that, under the assumptions of Theorem 1, the function \(|D_sT|\cdot |\nabla r|\in L_1(G)\). Then for almost all \(t\in(t_1,t_2)\) the following assertion is true: if a nonpoint component \(\Gamma\subset G_t\) is homeomorphic to the open interval \(I=(0,1)\): \(\Gamma=z(I)\), and \(\alpha\in I\), then the finite limits

\[ \lim_{\alpha\to 0} T(z(\alpha)) \quad \text{and} \quad \lim_{\alpha\to 1} T(z(\alpha)) \]

exist.

This assertion is broader than certain existing analogues of the well-known theorem of P. Fatou (see \((^7)\), p. 66) on boundary values of analytic functions bounded inside a disk. For example, the results of \((^8,^9)\) easily follow from it when \(r(z)=\arg z\).

p. 2. Let \(\lambda(z)\) and \(\nu(z)\) be nonnegative measurable functions on \(G\). It can be proved that if \(T\) and \(r\) are a mapping and a function of the class \(\widetilde W_1^1(G)\), then almost everywhere on \((t_1,t_2)\) the measurable functions

\[ l_\lambda(G_t)=\int_{G_t}\lambda\,dH \quad \text{and} \quad l_\nu[T(G_t)]=\int_{G_t}\nu |D_sT|\,dH \]

are defined.

Below, by \(\varphi(h)\) we denote a function defined for \(h\ge 0\) and such that \(\varphi(h)\ge 0\), \(\varphi(0)=0\), \(\varphi''(h)>0\).

Theorem 2. Let \(T\) and \(r\) be a mapping and a function of the class \(\widetilde W_1^1(G)\), and let \(g\subset(t_1,t_2)\) be a Borel set. If almost everywhere on \(g\)

\[ 0<l_\lambda(G_t)<\infty, \tag{*} \]

then

\[ \int_g l_\lambda(G_t)\cdot \varphi\bigl(l_\nu[T(G_t)]/l_\lambda(G_t)\bigr)\,dt \le \iint_{r^{-1}(g)} \lambda\cdot \varphi\left(\frac{\nu}{\lambda}|D_sT|\right)\cdot |\nabla r|\,dx\,dy. \]

This is a generalization of the well-known inequality underlying the investigations \((^3,^ {10})\).

p. 3. In what follows, let \(\lambda(z)\) and \(\mu(w)\) be nonnegative \(\mathfrak B\)-functions given on \(\overline G\) and \(\overline{T(G)}\), respectively.

For any \(H\)-measurable set \(B \subset R^2\) and \(\mathfrak B\)-function \(\rho(z) \geqslant 0\), defined in \(R^2\), we shall call the quantity

\[ S_\rho(B) \stackrel{\mathrm{def}}{=} \int_B \rho(z)\,dH \]

the \(\rho\)-length of the set \(B\). (We note that the function \(\rho\) is \(H\)-measurable \((^{11,12})\).)

A homeomorphic image \(\pi \subset G\) of the half-open interval \(I=[0,1)\) will be called a path (lying in \(G\)). Let \(\pi=z(I)\), and \(\pi_\alpha=z(I_\alpha)\), where \(z\) is a homeomorphism, and \(I_\alpha=(\alpha,1)\), \(\alpha\in(0,1)\). The set

\[ |\pi|=\bigcap_\alpha \pi_\alpha \]

will be called the cluster set of \(\pi\). Let

\[ |\Pi|\stackrel{\mathrm{def}}{=}\bigcup_{\pi\in\Pi}|\pi|. \]

Let, further, \(K\) be a nondegenerate continuum lying inside \(G\).

Definition. We shall say that a family \(\Pi\) of paths covers a set \(A\subset G\), if:

1) \(A\cap K=\varnothing\);

2) for every \(\pi\in\Pi\) there exists an \(\alpha\in(0,1)\) such that \(\pi_\alpha\subset G\setminus G_K^A\),

where \(G_K^A\supset K\) is the component of connectivity of the set \(G\setminus A\).

Definition. A set \(\Pi\) of paths is called a \(0^\rho\)-set (a null-\(\rho\)-set) if for every \(\varepsilon>0\) there exists a Borel set \(B\) covering \(\Pi\) with \(\rho\)-length \(S_\rho(B)<\varepsilon\).

Let now \(W\subset \widetilde W_q^{\,1}(G)\) be some family, and let \(\Pi\) be a certain family of paths (lying in \(G\)).

Definition. \(\Pi\) is called a \(0_\varphi^\lambda(W)\)-set if there exist a function \(r\in W\) and a Borel set \(g\subset(t_1,t_2)\) such that, for each \(t\in g\), the set \(G_t=\{z\in G\mid r(z)=t\}\) covers \(\Pi\) and condition \((*)\) is satisfied, and for any constant \(c>0\)

\[ \int_g l_\lambda(G_t)\cdot \varphi\bigl(c/l_\lambda(G_t)\bigr)\,dt=\infty. \]

By the symbol \(BL\varphi(\lambda,\mu,W)\) we shall denote the class of mappings \(T\in \widetilde W_p^{\,1}(G)\) for which the integral

\[ I(T,G)=\iint_G \lambda\cdot \varphi\left(\frac{\mu(T)}{\lambda}\,|D_sT|\right)\cdot|\nabla r|\,dx\,dy<\infty \]

for every function \(r\in W\), \(p^{-1}+q^{-1}=1\). Obviously,

\[ BL\varphi(\lambda,\mu,W)\subset BL\varphi(\lambda,\mu,W_1), \]

if \(W_1\subset W\).

If \(p=1\), \(\varphi(h)=h^2\), \(\lambda(z)=1\), \(\mu(w)=1/(1+|w|^2)\), and \(W=\{|z-\zeta|\mid \zeta\in R^2\}\), then the class \(BL\varphi(\lambda,\mu,W)\) includes the class \(\widetilde{BL}\), studied in \((^3)\).

Theorem 3. A homeomorphism \(T:G\to\Delta\) of the class \(BL\varphi(\lambda,\mu,W)\) carries every \(0_\varphi^\lambda(W)\)-set of paths \(\Pi\) (lying in \(G\)) into a \(0^\mu\)-set

\[ T(\Pi)=T(\pi)_{\pi\in\Pi}. \]

If the domain \(G\) is finitely connected, \(\mu(w)=1/(1+|w|^2)\) and \(S_\mu(\operatorname{Fr}\Delta)<\infty\), then \(S_\mu(|T(\Pi)|)=0\); in particular, for \(\pi\in\Pi\) the cluster set of each path \(T(\pi)\) is a point, i.e. there exists (finite or not)

\[ \lim_{z\in\pi}T(z). \]

Here and below \(\lim\) is to be understood conveniently in the sense of Moore—Smith; see \((^{13})\).

For the formulation of one of the corollaries of this theorem we give the following

Definition. We shall call a point \(w\) of the extended plane \(R^2\) a boundary value of the mapping \(T:G\to R^2\), if there exists such a path \(\pi\subset G\) that \(|\pi|\subset \operatorname{Fr}G\) and

\[ \lim_{z\in\pi}T(z)=w. \]

Denote by \(\Omega\) the totality of all functions of the form \(|w-\omega|\), where \(\omega\in R^2\) is an arbitrary fixed point.

Corollary. Let the function \(\varphi\) satisfy the condition

\[ \int_0^1 t\varphi(1/t)\,dt=\infty, \]

and $\mu(w)=1/(1+|w|^2)$. Then there does not exist a homeomorphism $T:G\to\Delta$, $T^{-1}\in BL\varphi(\mu,\lambda,\Omega)$, taking one and the same boundary value $w$ on a set of $\Pi$ paths (lying in $G'$) which is not an $O^\lambda$-set.

From this there follows, for example, S. Agmon’s result \((^{14})\) on the nonconstancy of the boundary function on a nonzero set of the boundary of a disk under a quasiconformal mapping of the latter.

Remark. It is easy to see that the results given above extend almost verbatim to mappings of a plane domain into Euclidean $n$-space.

Tomsk State University
named after V. V. Kuibyshev

Received
24 X 1967

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