MATHEMATICS

A. M. RODNYANSKII

ON MAPPINGS OF THE PRODUCT OF A TOPOLOGICAL SPACE BY A EUCLIDEAN SPACE INTO A EUCLIDEAN SPACE

(Presented by Academician P. S. Aleksandrov, 28 II 1957)

In this note \(X\) is a topological space; \(R_y^q, R_u^q\) are \(q\)-dimensional Euclidean oriented spaces; \(Z=[X, R_y^q]\) is the topological product of \(X\) by \(R_y^q\); \(x, y, z, u\) (possibly with an index) are points respectively of the spaces \(X, R_y^q, Z, R_u^q\); \(O^x=O^x(x_0)\), \(O^y=O^y(y_0)\), \(O^z=O^z(z_0)=O^z(x_0,y_0)\), \(O^u=O^u(u_0)\) are absolute neighborhoods of the points \(x_0, y_0, z_0=(x_0,y_0), u_0\) in the spaces \(X, R_y^q, Z, R_u^q\), respectively; the same notations are adopted for absolute neighborhoods of subsets of the spaces \(X, R_y^q, Z, R_u^q\) relative to these spaces; \(\pi_x, \pi_y\) are the projections of the space \(Z\) respectively onto the spaces \(X, R_y^q\); \(G\) is a nonempty open subset of the space \(Z\); \(f\) is a continuous mapping of \(G\) into \(R_u^q\); \((x_0,y_0)\) is a point of \(G\); \(u_0=f(x_0,y_0)\); \(F^u\) is a closed subset of the space \(R_u^q\); \(\Phi^z\) is a compact subset \(\subset G\); \(\Lambda\) is the empty set; \(M\) is a subset of the space \(Z\).

In addition, the following notation and definitions are used:

1) \(M(x)=\{y:(x,y)\in M\}\).

2) \(\widetilde M=\{(x,y):(x,y)\in \overline M,\ x\in\pi_xM\}=\{(x,y):x\in\pi_xM,\ y\in\overline{M(x)}\}=\overline M\cap\pi_x^{-1}\pi_xM\).

3) \(M\) is locally bounded with respect to \(y\) if, for every \(x_1\in\pi_xM\), there is an \(O^x=O^x(x_1)\) such that \(\pi_y(M\cap\pi_x^{-1}O^x)\) is bounded in \(R_y^q\).

4) \(M\) is connected with respect to \(y\) if \(M(x)\) is connected for every \(x\in\pi_xM\).

5) \(f_x\) is the mapping of \(G(x)\) into \(R_u^q\) given by the formula \(f_x(y)=f(x,y)\) \((y\in G(x))\).

6) \(\hat f\) is the mapping of \(G\) into \([X,R_u^q]\) given by the formula \(\hat f(x,y)=(x,f(x,y))\).

7) \(E_g=(E)_g\) is the boundary of the set \(E\) relative to that one of the spaces \(X, R_y^q, Z, R_u^q\) which contains it.

8) \(m(E)\) is equal to the cardinality of the set \(E\) if \(E\) is finite or \(\Lambda\), and is equal to \(+\infty\) if \(E\) is infinite.

9) \(k(\Phi^z(x),f_x)=\sup m(\Phi^z(x)\cap f_x^{-1}u)\).

10) \(k(\Phi^z,f)=\sup_{x\in X}(\Phi^z(x),f_x)\).

11) If \(f_{x_0}\) is differentiable at the point \(y_0\), then \(J(f_{x_0},y_0)\) denotes the Jacobian of the mapping \(f_{x_0}\) at the point \(y_0\).

12) If, for every \(x\in\pi_xG\), the mapping \(f_x\) is differentiable on \(G(x)\), then we put
\[ G^+=\{(x,y):(x,y)\in G,\ J(f_x,y)>0\},\quad G^-=\{(x,y):(x,y)\in G,\ J(f_x,y)<0\},\quad G^0=\{(x,y),(x,y)\in G,\ J(f_x,y)=0\}. \]

13) \(y_0\) is an isolated point of the mapping \(f_{x_0}\) if, for all sufficiently...

for sufficiently small \(h\ne 0\) we have \(f_{x_0}(y_0+h)\ne f_{x_0}(y_0)\). In this case the local degree \(\gamma(f_{x_0},y_0)\) is defined, equal to the degree \(\gamma(O^y,f_{x_0},u_0)\), where \(O^y=O^y(y_0)\) is sufficiently small. If \(f_{x_0}\) is differentiable at \(y_0\), and \(J(f_{x_0},y_0)\ne 0\), then, as is known, \(y_0\) is an isolated point of the mapping \(f_{x_0}\), and we have \(\gamma(f_{x_0},y_0)=\operatorname{sign} J(f_{x_0},y_0)\).

§ 1. In this section \(G\) is locally, with respect to \(y\); \(f\) is continuous in \(\widetilde G\).

Theorem 1. Let \(\overline G(x_0)\cap f_{x_0}^{-1}F^u\subseteq G(x_0)\). Then there exists \(O^x=O^x(x_0)\subseteq \pi_xG\) such that for all \(x\in O^x\) we have:

1.1) \(\overline G(x)\cap f_x^{-1}F^u\subseteq G(x)\);

1.2) for every \(u\in F^u\) the degree \(\gamma(G(x),f_x,u)=\gamma(G(x_0),f_{x_0},u)\) is defined.

Corollary 1. Let \(\overline G(x_0)\cap f_{x_0}^{-1}F^u\subseteq G(x_0)\), and suppose \(\gamma(G(x_0),f_{x_0},u)\ne 0\) for all \(u\in F^u\). Then there exists \(O^x=O^x(x_0)\) such that \(F^u\subseteq f_xG(x)\) for all \(x\in O^x\).

Corollary 2. Let \(C^x\) be connected, \(C^x\subseteq \pi_xG\), and suppose \(\overline G(x)\cap f_x^{-1}u_0\subseteq G(x)\) for all \(x\in C^x\). Then \(\gamma(G(x),f_x,u_0)=\mathrm{const}\) for all \(x\in C^x\).

Corollary 3. If \(F^u\cap f_{x_0}\overline G(x_0)=\Lambda\), then there exists \(O^x=O^x(x_0)\subseteq \pi_xG\) such that for all \(x\in O^x\), \(u\in F^u\), the degree \(\gamma(G(x),f_x,u)=0\) is defined.

Theorem 2. Let \(\overline G(x_0)\cap f_{x_0}^{-1}u_0\subseteq G(x_0)\), \(\gamma(G(x_0),f_{x_0},u_0)\ne 0\). Then for every sufficiently small \(O^u=O^u(u_0)\) there is an \(O^z=O^z([x_0,\overline G(x_0)\cap f_{x_0}^{-1}u_0])\subseteq G\) such that for all \(x\in \pi_xO^z\) we have:

2.1) \(f_xO^z(x)=O^u\);

2.2) \(\overline G(x)\cap f_x^{-1}O^u=O^z\);

2.3) for every \(u\in O^u\) the degree \(\gamma(O^z(x),f_x,u)=\gamma(G(x_0),f_{x_0},u_0)\) is defined.

Remark 1. All the results of the present section are, as far as I know, essentially new even in the case \(X=R^p\) with \(p>0\). For the case \(p=0\) (\(f\) is a mapping of \(G\), open in \(R_y^q\), into \(R_u^q\)), analogous results were obtained by me in \((^1,^2)\). I note that no restrictions are imposed on the space \(X\), except the fulfillment of the four Kuratowski axioms; it may even fail to be a \(T_0\)-space and need not satisfy the first axiom of countability. Also of interest is the case when \(X\) is an arbitrary subset of the space \(R^p\), and \(G\) is a set open in \([X,R_y^q]\).

§ 2. In this section \(y_0\) is an isolated point of the mapping \(f_{x_0}\), and \(\gamma(f_{x_0},y_0)\ne 0\).

Theorem 3. Let \(O^z=O^z(x_0,y_0)\) be given. Then for every sufficiently small (connected) \(O^u=O^u(u_0)\) there is a (connected with respect to \(y\)) \(O_1^z=O_1^z(x_0,y_0)\), contained in \(O^z\) and such that:

3.1) \(\widetilde O_1^z\subseteq G\);

3.2) \(\overline O_1^z\) is locally bounded with respect to \(y\);

3.3) \(\overline O_1^z(x_0)\cap f_{x_0}^{-1}u_0=\{y_0\}\);

3.4) \(f_xO_1^z(x)=O^u\) for every \(x\in \pi_xO_1^z\);

3.5) \(\overline O_1^z(x)\cap f_x^{-1}O^u=O_1^z(x)\bigl(\overline O_1^z(x)\cap f_x^{-1}O^u=O_1^z(x)\bigr)\) for every \(x\in \pi_xO_1^z\);

3.6) for any \(x\in \pi_xO_1^z\), \(u\in O^u\) the degree \(\gamma(O_1^z(x),f_x,u)=\gamma(f_{x_0},y_0)\) is defined;

3.7) \(\hat f\,\widetilde O_1^z=[\pi_xO_1^z,O^u]\);

3.8) \(\hat f\,(\widetilde O_1^z\setminus O_1^z)=[\pi_xO_1^z,\overline O^u\setminus O^u]\) \(\bigl(\hat f\,\widetilde O_1^z=[\pi_xO_1^z,\overline O^u]\bigr)\);

3.9) if \(x\in \pi_xO_1^z\), and \(f_x\) is differentiable on \(O_1^z(x)\), then

\[ \operatorname{mes}\{y:y\in O_1^z(x),\ \operatorname{sign}J(f_x,y)=\operatorname{sign}\gamma(f_{x_0},y_0)\}>0, \]

\[ \operatorname{mes} f_x\{y:y\in O_1^z(x),\ \operatorname{sign}J(f_x,y)=\operatorname{sign}\gamma(f_{x_0},y_0)\}=\operatorname{mes}O^u. \]

Corollary 1. The mapping \(\hat f\) is open at the point \((x_0,y_0)\).

Remark. All the results of the present paragraph are, as far as I know, essentially new even in the case \(X=R^p\) for \(p>0\). For the case \(p=0\) (\(f\) is a mapping of an open subset \(G\) in \(R_y^q\) into \(R_u^q\)) analogous results were obtained by me in \((^{3,4})\) under the additional assumption that \(f\) is differentiable in \(G\).

§ 3. In this paragraph the mapping \(f_x\) is differentiable in \(G(x)\) for every \(x\in \pi_x G\).

Theorem 4. Each of the sets \(\pi_x G^+\), \(\pi_x G^-\) is open in the space \(X\).

Theorem 5. If \(G^0=\Lambda\), then each of the sets \(G^+\), \(G^-\) is open in the space \(Z\).

Corollary 1. Let \(G^0=\Lambda\), and let \(C\) be a connected subset of the set \(G\). Then \(\operatorname{sign} J(f_x,y)=\mathrm{const}\) for all \((x,y)\in C\).

Corollary 2. If \(G^0=\Lambda\), and \(G\) is a domain, then \(\operatorname{sign} J(f_x,y)=\mathrm{const}\) for all \((x,y)\in G\).

Theorem 6. Let \(O^z=O^z(x_0,y_0)\subseteq G\setminus G^0\). Then for every sufficiently small connected \(O^u=O^u(u_0)\) there exists a locally bounded in \(y\) and connected in \(y\) neighborhood \(O_1^z=O_1^z(x_0,y_0)\), contained in \(O^z\), such that the mapping \(\varphi\), defined on the set \([\pi_x O_1^z,O^u]\) by the formula

\[ \varphi(x,u)=O_1^z(x)\cap f_x^{-1}u \qquad ((x,u)\in[\pi_x O_1^z,O^u]), \]

is a single-valued continuous open mapping of the set \([\pi_x O_1^z,O^u]\) into the space \(R_y^q\), and we have:

6,1) for every \(x\in\pi_x O_1^z\) the mapping \(f_x\) is a differentiable topological mapping of \(O_1^z(x)\) onto \(O^u\);

6,2) for every fixed \(x\in\pi_x O_1^z\) the mapping \(\varphi_x\) \((\varphi_x(u)=\varphi(x,u))\) is a differentiable topological mapping of \(O^u\) onto \(O_1^z(x)\), inverse to the mapping \(f_x\), considered on \(O_1^z(x)\);

6,3) for every fixed \(u\in O^u\) the mapping \(\varphi_u\) \((\varphi_u(x)=\varphi(x,u))\) is a continuous mapping of \(\pi_x O_1^z\) into \(R_y^q\), and we have:

\[ f(x,\varphi_u(x))=u \qquad (x\in\pi_x O_1^z), \]

i.e. \(\varphi_u\) is the unique continuous implicit mapping determined from the equation

\[ f(x,y)=u \qquad ((x,y)\in O_1^z); \]

6,4) \(\hat f\) is a topological mapping of \(O_1^z\) onto \([\pi_x O_1^z,O^u]\);

6,5) \(\varphi(x,u)=\pi_y(\hat f^{-1}(x,u))\) for all \((x,u)\in[\pi_x O_1^z,O^u]\).

Theorem 7. Let \(q=1\), and let \(O^z=O^z(x_0,y_0)\) be such that for every \(x\in\pi_x O_1^z\) the set \((O^z\cap G^0)(x)\) contains not more than one point. Suppose, further, that \(k\) is a nonnegative integer such that the function \(f_x\) is differentiable \(2k\) times in some neighborhood of the point \(y_0\) and has a finite derivative of order \(2k+1\) at the point \(y_0\), and moreover

\[ \frac{d^l f_{x_0}}{dy^l}(y_0)=0 \qquad (1\leq l\leq 2k), \qquad \frac{d^{2k+1}f_{x_0}}{dy^{2k+1}}(y_0)\neq 0. \]

Then for every sufficiently small interval \(O^u=O^u(u_0)\) there exists a locally bounded in \(y\) and connected in \(y\) neighborhood \(O_1^z=O_1^z(x_0,y_0)\),

contained in \(O^z\) and such that the function \(\varphi\), defined on the set \([\pi_x O_1^z, O^u]\) by the formula

\[ \varphi(x,u)=O_1^z(x)\cap f_x^{-1}u \qquad ((x,u)\in[\pi_x O_1^z,O^u]), \]

effects a one-to-one continuous open mapping of the set \([\pi_x O_1^z,O^u]\) onto the line \(R_y^1\); moreover assertions 6, 1)—6, 5) hold.

Theorem 8. Let \(G^0=\Lambda\); let \(X\) be a Hausdorff space satisfying the first axiom of countability. Then \(k(\Phi^z,f)<+\infty\).

Theorem 9. Let \(G^0=\Lambda\); let \(X\) be a metric space with a countable base, and suppose that an outer Carathéodory measure \(\mu_x\) is given in \(X\) such that every point \(x\in X\) has a neighborhood of finite \(\mu_x\)-measure. By \(\mu_y,\mu_u\) denote the \(q\)-dimensional Lebesgue measures, respectively, in the spaces \(R_y^q,R_u^q\). Further, let \(\varphi(u)\) be a complex function, defined on \(f\Phi^z\) (almost everywhere on \(f\Phi^z\)) and summable on \(f\Phi^z\). Finally, suppose that

\[ \inf_{(x,y)\in\Phi^z}\operatorname{vrai}|J(f_x,y)|=m>0. \tag{1} \]

Then the function \(\varphi(f(x,y))\) is defined on \(\Phi^z\) (almost everywhere on \(\Phi^z\)), is summable on \(\Phi^z\), and

\[ \int_{\Phi^z}|\varphi(f(x,y))|\,d(\mu_x\times\mu_y) \leq \frac{2}{m}k(\Phi^z,f)\,\mu_x(\pi_x,\Phi^z) \int_{f\Phi^z}|\varphi(u)|\,d\mu_u<+\infty. \]

If, however, \(\operatorname{sign}J(f_x,y)=\mathrm{const}\) for all \((x,y)\in\Phi^z\), then the constant 2 on the right-hand side of the last inequality is replaced by the constant 1.

Remark. All the results of the present paragraph are essentially new even in the case \(X=R^p\) with \(p>0\). For \(p=0\) (\(f\) is a mapping, open in \(R_y^q\), of the set \(G\) into \(R_u^q\)) results analogous to Corollary 2 of Theorem 5 and Theorem 6 were obtained by me in \((^3)\). Particular cases of Corollary 2 of Theorem 6 were obtained by L. D. Kudryavtsev in \((^{5,6})\).

Theorems 4, 5, and 7 are essentially new also in the case when \(X=R^p\), and \(f\) is differentiable jointly in all the variables \(x_1,\ldots,x_p; y_1,\ldots,y_q\).

Theorems 8 and 9 are essentially new in the case \(X=R^p\), \(\mu_x\) is \(p\)-dimensional Lebesgue measure, and under any assumptions concerning the smoothness of the mapping \(f\). Moreover, if the partial derivatives of the mapping functions \(\partial f_i/\partial y_j\) \((i,j=1,\ldots,q)\) are continuous jointly in the variables \(x_1,\ldots,x_p; y_1,\ldots,y\), then the conditions \(G^0=\Lambda\) and (1) in Theorems 8 and 9 are unnecessary. It is enough to require that \(\Phi^z\subseteq G\setminus G^0\).

Moscow Institute of Physics and Technology

Received
27 II 1957

REFERENCES

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