L. D. IVANOV

ON SMOOTH MAPPINGS OF THE SPACE \(R_n\) INTO \(R_1\)

(Presented by Academician A. N. Kolmogorov, November 29, 1963)

Let \(R_n\) be \(n\)-dimensional real Euclidean space, \(\mathbf{x}_1,\ldots,\mathbf{x}_n\) its fixed orthonormal basis; \(x^1,\ldots,x^n\) the coefficients in the expansion of the vector \(\overrightarrow{(0,x)}\) with respect to this basis; \(\rho(E_1,E_2)\) the distance between the sets \(E_1\) and \(E_2\); \(dE\) the diameter of the set \(E\); \(S(a,r)=\{x:\rho(x,a)\le r\}\); \((E,x)\) the connected component of the set \(E\) containing the point \(x\); \(|E|\) the length of the set \(E\); \(\dim E\) the dimension of the set \(E\); \(d_k(E)\) \((k=1,\ldots,n-1)\) the maximum of the measures of the projections of the set \(E\) onto all possible \(k\)-dimensional coordinate planes; \(d_nE=\operatorname{mes}E\); \(E(f,t)=\{x:f(x)=t\}\); \(k(f,t)\) the number of connected components of the set \(E(f,t)\). Some quantities will be denoted by the letters \(a_0,a_1,\ldots\); in parentheses we shall indicate the numbers on which they depend. Let \(s\) be a nonnegative integer, \(0\le \alpha\le 1\), \(l=s+\alpha>0\), and let \(f(x)\) be a function mapping the space \(R_n\) into \(R_1\), equal to zero outside the ball \(S(0,R)\), and having all derivatives up to order \(s\) inclusive. Suppose that the derivatives of the last order \(f_i(x)\) satisfy the condition

\[ |f_i(x)-f_i(y)|\le M[\rho(x,y)]^\alpha . \tag{1} \]

Put \(N=\{x:|\operatorname{grad} f(x)|=0\}\); \(N_1=\{x:(E(f,f(x)),x)\cap N\ne 0\}\); \(G=CN_1\). The components \(K_i\) of the open set \(G\) will be called rings.

Lemma 1. The boundary of a ring \(\partial K_i\) consists of two connected components \(F_i,H_i\), one of which, say \(H_i\), separates the other from infinity. The function \(f(x)\) maps the ring \(K_i\) onto some interval \((a_i,b_i)\) in such a way that the set \(K_i\cap E(f,t)\) is connected for \(t\in(a_i,b_i)\).

For the proof see (1), p. 136.

The set \(F_i\) will be called the inner boundary of the ring \(K_i\), and \(H_i\) the outer boundary. Let \(h_i=|b_i-a_i|\) be the length of the image of the ring \(K_i\) under the mapping by the function \(f(x)\), \(m_i=h_i^{1/l}\), \(n_i=m_i^n\).

Theorem 1. Under the conditions on the function \(f(x)\) stated above, the following estimate holds:

\[ \sum_{K_i} n_i \le a_1(n,l) M^{n/l} R^n . \tag{2} \]

Theorem 1 was proved by A. G. Vitushkin in two cases: 1) \(n=1\) and 2) \(l=n\).

Theorem 2. Under the conditions of Theorem 1, if \(l\ge n\), then

\[ \int_{-\infty}^{\infty} [k(f,t)]^{l/n}\,dt \le [a_1(n,l)]^{l/n}MR^l . \tag{3} \]

There is an example of a function \(f_l(x)\) satisfying the conditions of Theorem 1 and such that, for every \(\varepsilon>0\), the series \(\sum_{K_i} n_i^{1-\varepsilon}\) and the integral \(\int_{-\infty}^{\infty} [k(f,t)]^{l/n+\varepsilon}\,dt\) diverge ...

…This example shows that Theorems 1 and 2 cannot be improved. We now prove Theorem 1.

Lemma 2 (Oleinik). If \(f(x)\) is a polynomial of degree \(s\), then for any \(t\)

\[ k(f,t)\leqslant a_0(n,s). \tag{4} \]

Lemma \(2'\). Let \(f(x)\) be a polynomial of degree \(s\), constant on the boundary of each of \(N\) domains \(G_i\) such that \(G_i\cap G_j=0\) for \(i\ne j\). Then either \(f(x)\equiv \mathrm{const}\) or \(N\leqslant a'_0(n,s)\).

Lemma 3. Let \(f(x)\) be a polynomial of degree \(s\), \(E=(E(f,0)\cap S(a,r),x)\). Let \(y\in E\), \(\varepsilon>0\). There exist no more than \(a_2(n,s)\) points \(x_i\) and polynomials \(f_i(x)\), the degree of each of which does not exceed \(a_3(n,s)\), such that: a) \(f_i(x_i)=0\); \(x_i\in S(a,r)\); b) \(\dim E_i=1\), where \(E_i=(E(f_i,0)\cap S(a,r),x_i)\); c) \(x\in\bigcup_i E_i\); \(y\in\bigcup_i E_i\); d) the set \(\bigcup_i E_i\) is connected; e) if \(z\in E_i\), then \(\rho(z,E)\leqslant \varepsilon\).

Lemma 4. Let \(f(x)\) be a polynomial of degree \(s\), \(E_t=\{x: |f(x)|\leqslant t\}\). If \(x\in E_t\), \(y\in E_t\), \((E_t\cap S(a,r),x)=(E_t\cap S(a,r),y)\), then the points \(x\) and \(y\) can be joined by a curve \(\gamma\) such that \(\gamma\in(E_t\cap S(a,r),x)\); \(|\gamma|\leqslant a_4(n,s)r\).

Proof of Theorem 1. If \(l\leqslant 1\), then the theorem is obvious, since in each ring \(K_i\) there is a point \(A_i\) such that

\[ S\left(A_i,\frac12\left(\frac{h_i}{2M}\right)^{1/l}\right)\subset K_i. \]

Let \(l>1\), and let \(x_i\) be a point of the inner boundary of the ring \(K_i\) at which \(|\operatorname{grad} f(x)|=0\). Put

\[ \widetilde G_{\varepsilon,t}^{\,i} = S(x_i,2^t m_i)\cap \left\{x:|\operatorname{grad} f(x)|\leqslant \frac{\varepsilon t}{n}\,m_i^{\,l-1}\right\}; \qquad G_{\varepsilon,t}^{\,i}=(\widetilde G_{\varepsilon,t}^{\,i},x_i). \tag{5} \]

We divide the collection of rings \(K_i\) into \(n+1\) classes \(\sigma_k\) in the following way: \(K_i\in\sigma_1\) if \(dG_{\varepsilon,1}^{\,i}\leqslant 0.1\,n^{-2}m_i\). For \(2\leqslant k\leqslant n\), \(K_i\in\sigma_k\) if \(d_{k-1}G_{\varepsilon,k-1}^{\,i}>(0.1\,n^{-2}m_i)^{k-1}\), but \(d_kG_{\varepsilon,k}^{\,i}\leqslant (0.1\,n^{-2}m_i)^k\), and \(K_i\in\sigma_{n+1}\) if \(d_nG_{\varepsilon,n}^{\,i}>(0.1\,n^{-2}m_i)^n\). Let the ring \(K_i\in\sigma_1\). Put \(S_i=S(y_i,m_i)\), where \(y_i\) is some point satisfying the condition \(\rho(x_i,y_i)=2m_i\).

Let the ring \(K_i\in\sigma_k\), \(2\leqslant k\leqslant n\). Take numbers \(j_1,\ldots,j_{k-1}\) such that the measure of the projection of the set \(G_{\varepsilon,k-1}^{\,i}\) onto the \((k-1)\)-dimensional plane \(\Pi_i\), spanned by the vectors \(x_{j_1},\ldots,x_{j_{k-1}}\), exceeds \((0.1\,n^{-2}m_i)^{k-1}\). One can find a set \(E^i\) of points of the plane \(\Pi_i\) such that \(\operatorname{mes}E^i\geqslant \frac12(0.1\,n^{-2}m_i)^{k-1}\), and the \((n-k+1)\)-dimensional plane \(\Pi_{i,x}\), orthogonal to the plane \(\Pi_i\) and such that \(\Pi_i\cap\Pi_{i,x}=x\in E^i\), contains a point \(y_i(x)\in G_{\varepsilon,k-1}^{\,i}\) and \(d(G_{\varepsilon,k}^{\,i}\cap\Pi_{i,x},y_i(x))\leqslant \frac15 m_i\). In this case the points \(z_i(x)\) can be chosen so that: 1) \(z_i(x)\in\Pi_{i,x}\), 2) \(\rho(y_i(x),z_i(x))=2m_i\), and 3) the set

\[ S_i=\bigcup_{x\in E^i} S(z_i(x),m_i)\cap\Pi_{i,x} \]

is measurable. If \(K_i\in\sigma_{n+1}\), let

\[ S_i=\{x: |x^k-x_i^k|\leqslant m_i,\ k=1,\ldots,n\}. \]

It is clear that, in order to prove the theorem, it is enough to estimate \(\sum_i \operatorname{mes} S_i\).

In doing so one may assume that the numbers \(\varepsilon\) and \(M\) are sufficiently small, say

\[ \varepsilon<2^{-4l-n-1}[a_4(n,2s)]^{-1}, \qquad M=12^{-2l-1}n^{-2}\varepsilon . \]

Lemma 5. Let the rings \(K_i\) and \(K_j\) be such that \(h_j\leqslant h_i\leqslant 2^{2l}h_j\). Then \(G_{\varepsilon,n}^{\,i}\cap G_{\varepsilon,n}^{\,j}=0\).

Indeed, otherwise, using Lemma 4, we would join the points \(x_i\) and \(x_j\) by a curve \(\gamma\) such that: 1) \(|\gamma|\leqslant a_4(n,2s)(2^{2l+n}+2^n)m_i\), and 2) for \(x\in\gamma\),

\[ |\operatorname{grad} f(x)|\leqslant \varepsilon\,2^{2l}m_i^{\,l-1}. \]

This contradicts the inequality

\[ \max_{x\in\gamma} f(x)-\min_{x\in\gamma} f(x)\geqslant h_j. \]

Lemma 6. Let the rings \(K_{i_r}\in\sigma_k\) for one and the same \(k\), \(1\leqslant k\leqslant n\). Suppose moreover that, if \(k\geqslant 2\), the planes \(\Pi_i\) coincide.

There is no point common to more than \(a_5(n,l)\) of the sets \(S_{i_r}\).

Proof. Consider the case \(k=1\). Suppose that, contrary to the assertion of Lemma 6, the point \(t \in S_{i_r}\) for \(r=1,\ldots,a_5(n,l)+1\).

From Lemma 5 and the construction of the sets \(S_{i_r}\) it follows that, for \(r_1 \ne r_2\),
\(G_{\varepsilon,1}^{i_{r_1}} \cap G_{\varepsilon,1}^{i_{r_2}}=0\). Therefore one can find either \(a_0(n,2s)+1\) sets \(G_{\varepsilon,1}^{i_r}\) contained one in another, or \(a'_0(n,2s)+1\) sets \(G_{\varepsilon,1}^{i_r}\) lying outside one another. Let \(P(k,t,x)\) be a segment of the Taylor series of the function \(\dfrac{\partial}{\partial x^k} f(x)\) with center at the point \(t\), and
\[ P(t,x)=\sum_{k=1}^{n} P^2(k,t,x). \]
In the first of the resulting cases we have \(h_j \ge 2^{-2l}h_i\), where \(h_j\) is the minimum and \(h_i\) the maximum of the numbers \(h_{i_r}\) in the system under consideration. Using the estimate for the number \(M\), we obtain:
\[ k\left(P(t,x),\left[\frac12\,\frac{\varepsilon}{n}\,2^{-2l}m_i^{\,l-1}\right]^2\right) \ge a_0(n,2s)+1, \]
which contradicts Lemma 2. In the second case the polynomial \(P(t,x)\) remains constant on the boundary of \(a'_0(n,2s)+1\) domains \(H_{i_r}\) such that \(H_{i_r}\subset G_{\varepsilon,1}^{i_r}\). This contradicts Lemma \(2'\). Lemma 6 is proved in the case \(k=1\). The case \(2\le k\le n\) is treated analogously.

Lemma 7. If \(K_i \in \sigma_{n+1}\), then \(S_i \in G_{\varepsilon,a_6(n,l)}^{\,i}\). If \(\rho(x,x_i)\ge m_i\), then
\[ |\operatorname{grad} f(x)| \le M a_7(n,l)\,[\rho(x,x_i)]^{l-1}. \]

Let the number \(\varepsilon\) also satisfy the two inequalities
\[ 12^{-2l} n^{-2} a_7(n,l)\,\varepsilon < 2^{-2l}, \tag{6} \]
\[ 2^{l+1}\varepsilon a_6(n,l)<1. \tag{7} \]

We divide the totality \(\sigma_{n+1}\) of the annuloids \(K_i\) into two parts \(\sigma'_{n+1}\cup\sigma''_{n+1}\) in the following way: \(K_i\in\sigma''_{n+1}\) if \(\operatorname{mes}K_i\ge 0.1\,n_i\), and \(\sigma'_{n+1}=\sigma_{n+1}\setminus\sigma''_{n+1}\). The estimate
\[ \sum_{K_i\in\sigma''_{n+1}}\operatorname{mes} S_i \]
is obvious. Let us estimate
\[ \sum_{K_i\in\sigma'_{n+1}}\operatorname{mes} S_i . \]
Choose some direction, for example that of the vector \(x_1\), and let \(\Pi\) be the plane orthogonal to it, while \(R_x\) is the line orthogonal to \(\Pi\) such that \(\Pi\cap R_x=x\). Define the function \(\chi_i(y)\) as follows: if \(K_i\in\sigma'_{n+1}\), \(y\in R_x\cap S_i\), \(\rho(x_i,R_x)\le m_i\), and from the relations \(a_i\in F_i\), \(b_i\in H_i\),
\((R_x\cap \overline K_i,a_i)=(R_x\cap \overline K_i,b_i)\) there follows the inequality
\[ |(R_x\cap \overline K_i,a_i)|\le \frac12 m_i, \]
then \(\chi_i(y)=1\). In the opposite case \(\chi_i(y)=0\). Let
\[ \chi(y)=\sum_i \chi_i(y). \]
In the obvious way the estimate
\[ \sum_{K_i\in\sigma'_{n+1}}\operatorname{mes}S_i \]
is reduced to estimating
\[ \int_{R_n}\chi(y), \]
and the latter to estimating
\[ \int_{R_x}\chi(y)\,dy. \]
We divide the totality \(\sigma'_{n+1}\) of the annuloids \(K_i\) into the following parts:
\[ \sigma'_{n+1}=\bigcup_{k=1}^{\infty}\sigma'_{n+1,k}, \]
and \(K_i\in\sigma'_{n+1,k}\) if
\[ 4^{1-k}H\ge m_i>4^{-k}H, \]
where
\[ H=\max_i m_i. \]
Put
\[ \Delta_i=\{y:(\chi_i(y)=1)\cap R_x\}. \]
From Lemma 7 and relation (7) it follows that

Lemma 8. If \(K_i\in\sigma'_{n+1,k}\), \(K_j\in\sigma'_{n+1,k}\), then for \(i\ne j\)
\[ \Delta_i\cap\Delta_j=0. \]

Put
\[ \chi^k(y)=\sum_{K_i\in\sigma'_{n+1,k}}\chi_i(y). \]
We have
\[ \chi(y)=\sum_{k=1}^{\infty}\chi^k(y) =\sum_k' \chi^k(y)+\sum_k'' \chi^k(y), \]
where \(\sum_k'\) is taken over those \(k\) for which there exist at least two numbers \(i_1,i_2\) such that, for \(y\in R_x\), \(\chi_{i_1}\ne 0\), \(\chi_{i_2}\ne 0\), and \(K_{i_1}\in\sigma'_{n+1,k}\), \(K_{i_2}\in\sigma'_{n+1,k}\). Put
\[ \chi_1^*(y)=\sum_k' \chi^k(y),\qquad \chi_2^*(y)=\sum_k'' \chi^k(y). \]
The estimate
\[ \int_{R_x}\chi_2^*(y)\,dy \]

is obvious. Let, for \(y \in R_x\),

\[ \chi_1^*(y)=\sum_{k\in E}\chi^k(y)=\sum_{k\in E}\sum_{i\in E_k}\chi_i(y). \]

Put

\[ k\in E;\quad D_k=\{y:\chi^k(y)=1\}\cap R_x, \]

and let \(\Delta_i^k\) be the connected components of the set \(D_k\). From the definition of the collection \(\sigma_{n+1}'\) it follows that between two adjacent intervals \(\Delta_i^k,\Delta_j^k\) there exists a point \(z\) at which

\[ |\operatorname{grad} f(z)|\ge 4^{-(l-1)k}H^{(l-1)k}. \]

Put

\[ \Delta_{k,j}=R_x\cap S(z,2^{-k}H). \]

From Lemma 7 and estimates (6) and (7) the following lemmas follow:

Lemma 9. If \(k_1\ll k_2\), then

\[ \Delta_{i,j}^{k_1}\cap \Delta_r^{k_2}=0. \]

Lemma 10. The intervals \(\Delta_{i,j}^k\), for one and the same \(k\), are pairwise disjoint.

Now the proof of Theorem 1 is completed as follows. Each of the intervals \(\Delta_{i,j}^{k_1}\) can intersect no more than one of the intervals \(\Delta_{i,j}^{k_2}\) for \(k_2>k_1\). Hence we obtain the estimate

\[ \int_{R_x}\chi_1^*(y)\,dy\le 2\sum_{i,j,k}|\Delta_{i,j}^k|\le 8R. \]

Integrating this inequality over the plane \(\Pi\), we obtain the required result. Theorem 1 is proved.

Theorem 2 is easily obtained from Theorem 1, Lemma 1, and the theorem of E. M. Landis, which asserts that if \(l\ge n\), then

\[ \operatorname{mes}\{t:t=f(x),\ x\in N\}=0. \]

The author expresses gratitude to A. G. Vitushkin for posing the problem and for discussing the work.

REFERENCES

1. A. G. Vitushkin, On multidimensional variations, Moscow, 1955.