UDC 517.51

MATHEMATICS

T. V. TEREKHOVA

ON THE SMOOTHNESS OF A FUNCTION OF THREE VARIABLES OF BOUNDED VARIATION

(Presented by Academician P. S. Novikov on 26 VI 1969)

In this paper, (t = F(\eta)) denotes a function of 3 variables, continuous in (R_3) and equal to zero outside the closed ball (Q). (E_{t_0}) is the level set (t = t_0) of the function (F(\eta)). If (E) is a set in (R_k) ((k \leq 3)), then its (k)-dimensional Lebesgue measure will be denoted by (\mu_k(E)). Let (E \subset R_3) be a point set; by (\operatorname{mes}2 E) we shall denote its area, i.e., its two-dimensional Hausdorff measure. By (P)) we shall denote the plane (x = x_0) (respectively (y = y_0, z = z_0)).}) (respectively (P_{y_0}, P_{z_0

Definitions. Let (U \subset Q) be a closed ball and let (\Sigma) be an arbitrary finite system of pairwise disjoint closed balls (U_1, \ldots, U_n) contained in (Q).

1. Let (h_{0,U}(t)) be the number of components of the level set (E_t) wholly belonging to the ball (U) (it is allowed that (h_{0,U}(t) = +\infty)). The partial lower variation of (F(\eta)) in the ball (U) is

[
V_{0,U}(F) = \int_{-\infty}^{+\infty} h_{0,U}(t)\,dt .
]

The partial lower variation of (F(\eta)) with respect to the system (\Sigma) is

[
V_{0,\Sigma}(F) = \sum_{i=1}^{n} V_{0,U_i}(F).
]

If (U \equiv Q), then (W_0(F) \equiv V_{0,Q}(F)) is called the total lower variation of (F(\eta)).

1. Let (v_U(t,x_0)) be the number of components of the set (E_t \cap P_{x_0}) wholly belonging to the ball (U). The partial (x)-length of the level set (E_t) in the ball (U) is

[
h_{1,x,U}(t) = \int_{-\infty}^{+\infty} v_U(t,x)\,dx .
]

Analogously, (h_{1,y,U}(t)) and (h_{1,z,U}(t)) are defined—the partial (y)- and (z)-lengths of (E_t) in the ball (U). Put

[
h_{1,U}(t) = h_{1,x,U}(t) + h_{1,y,U}(t) + h_{1,z,U}(t).
]

1. The partial middle variation of (F(\eta)) in the ball (U) is

[
V_{1,U}(F) = \int_{-\infty}^{+\infty} h_{1,U}(t)\,dt,
]

and the partial middle variation of (F(\eta)) with respect to the system (\Sigma) is the sum

[
V_{1,\Sigma}(F) = \sum_{i=1}^{n} V_{1,U_i}(F).
]

If (U \equiv Q), then (W_1(F) \equiv V_{1,Q}(F)) will be called the total middle variation of the function (F(\eta)).

1. Put

[
h_{2,U}(t) = \operatorname{mes}_2(E_t \cap U).
]

The upper partial variation of (F(\eta)) in (U) is

[
V_{2,U}(F) = \int_{-\infty}^{+\infty} h_{2,U}(t)\,dt .
]

The upper partial variation with respect to the system (\Sigma) is

[
V_{2,\Sigma}(F) = \sum_{i=1}^{n} V_{2,U_i}(F).
]

For (U \equiv Q), we shall call (W_2(F) \equiv V_{2,Q}(F)) the total upper variation of the function (F(\eta)).

All functions occurring in definitions 1–4 under the integral sign are Lebesgue measurable. The integral is everywhere understood as a Lebesgue integral.

Theorem 1. For any system (\Sigma) of pairwise nonintersecting closed balls (U_i \subset Q) ((i=1,2,\ldots,n)), one has

[
V_{s,\Sigma}(F)\leq W_s(F) \qquad (s=0,1,2).
]

Theorem 2. Let (U\subset Q) and (V\subset Q) be concentric closed balls of radii (\delta) and (8\delta), respectively, and suppose that for (t=t_0), (E_{t_0}\cap U\ne\varnothing). Then at least one of the three inequalities holds:

[
h_{0,V}(t)\geq 1,\qquad h_{1,V}(t_0)\geq \delta/2,\qquad h_{2,V}(t_0)\geq \delta^2/4.
]

In the proof of Theorem 2 the following lemmas are used:

Lemma 1. Under the assumptions of Theorem 2, let (L\subset V) denote a continuum intersecting both (U) and the sphere (S_V) of the ball (V). Let (K_\chi) denote the surface of the cube whose faces are parallel to the coordinate planes, whose center coincides with the center of the balls (U) and (V), and whose edge is equal to (2\chi).

By a marked plane (\pi_\chi) we shall mean any plane that contains a face of (K_\chi) intersecting (L). Then the marked planes (\pi_\chi), for (\chi\in[\delta,4\delta]), cut out on the coordinate axes closed sets (F_x,F_y), and (F_z), respectively. For at least one of these sets the linear Lebesgue measure is not less than (\delta).

Lemma 2. Let (F_x) be the set defined under the assumptions of Lemma 1, with (\mu_1(F_x)\geq \delta). Denote by (E) the set of all such (x_0\in F_x) for each of which there exists at least one component of the set (L\cap P_{x_0}) that does not intersect (S_V). Then at least one of the inequalities is true:

[
\mu_1(E)\geq \delta/2,\qquad \operatorname{mes}_2 L\geq \delta^2/4.
]

A result well known from topology is used essentially in the work:

Separation theorem. If a closed set (F) separates the points (a) and (\beta) in the ball (U), then there exists at least one component of the set (F) separating these points in (U).

Theorem 3 (on the differentiability of a function of bounded variation). Let all three variations (W_0(F)), (W_1(F)), and (W_2(F)) of the function (F(\eta)) be bounded. Then (F(\eta)) has a complete differential almost everywhere.

Proof. Let (E) be the set of points of nondifferentiability of the function (F(\eta)). Suppose that (\mu_3(E)=\tau>0) (the set (E) is measurable (\mathrm{B})). Let (E^*) be the set of all points (\alpha\in E) for which

[
\sup_{\eta\in Q}\left|\frac{F(\eta)-F(\alpha)}{\rho(\eta,\alpha)}\right|=+\infty.
]

By a theorem of V. V. Stepanov, (\mu_3(E^)=\tau). Fix (k>0). From the continuity of (F(\eta)) it follows that in any neighborhood of a point (\alpha\in E^) there exists a point (\beta) such that

[
|F(\beta)-F(\alpha)|>k\rho(\beta,\alpha).
\tag{*}
]

For each point (\alpha\in E^) construct a sequence of points ({\eta_{n,\alpha}}) converging to (\alpha), satisfying relation (()), with (\rho(\alpha,\eta_{n,\alpha})<1).

The system of balls ({V_{n,\alpha}}) with centers at (\alpha\in E^) and radii (8\rho(\alpha,\eta_{n,\alpha})) forms a Vitali covering of the set (E^). From it select a system (\Sigma) of pairwise nonintersecting closed balls ({U_i}) ((i=1,2,\ldots,l)) contained in (Q), with centers at the points (\alpha_i) and radii (8\delta_i), where (\delta_i=\rho(\alpha_i,\beta_i)), and the points (\alpha_i,\beta_i) satisfy relation ((*)), such that

[
\mu_3\left(E^*\cap \sum_{i=1}^l U_i\right)> \frac13\tau.
]

Then we have:

[
\frac43\pi\sum_{i=1}^l (8\delta_i)^3>\frac13\tau,\qquad
\sum_{i=1}^l(\delta_i)^3>\frac{1}{8000}\tau,\qquad
|F(\beta_i)-F(\alpha_i)|>k\delta_i>k\delta_i^2.
]

Let (T_i) be the set of all (t) lying between (F(\alpha_i)) and (F(\beta_i)). For each (t\in T_i), the set (E_t\cap U_i) separates the points (\alpha_i) and (\beta_i) in the ball (U_i). Consequently, in the set (E_t\cap U_i) there exists at least one component (K_i) separating (\alpha_i) and (\beta_i) in the ball (U_i). Let (S_U) be the sphere of the ball (U_i). If (K_i\cap S_{U_i}=\varnothing), then assign (K_i) to the 1st class. If (K_i\cap S_{U_i}\ne\varnothing) and (\operatorname{mes}_2 K_i<)

(< \frac14\delta_i^2), then we assign (K_i) to the 2nd class. If (K_i\cap S_{U_i}\ne \varnothing) and (\operatorname{mes}_2 K_i\ge \frac14\delta_i^2), then we assign (K_i) to the 3rd class.

Let (T_{1,i}) be the set of all (t\in T_i) for which at least one component of the set (E_t\cap U_i) separating (\alpha_i) and (\beta_i) in the ball (U_i) belongs to the 1st class.

Denote by (T_{2,i}) the set of all (t\in T_i) for which in the set (U_i\cap E_t) there exists at least one component of the 2nd class separating (\alpha_i) and (\beta_i). Put (T_{3,i}=T_i\setminus (T_{1,i}\cup T_{2,i})). The sets (T_{1,i}, T_{2,i}), and (T_{3,i}) are (B)-measurable, and the measure of at least one of them is not less than (\frac13 k\delta_i). Then, by Theorem 2, we have:
[
V_{0,\Sigma}(F)\ge \sum_{i=1}^{l}\mu_1(T_{1,i})

\sum_{i=1}^{l}\mu_1(T_{1,i})\delta_i^2,\qquad
V_{1,\Sigma}(F)\ge \frac12 \sum_{i=1}^{l}\mu_1(T_{2,i})\delta_i >
]
[
\frac12 \sum_{i=1}^{l}\mu_1(T_{2,i})\delta_i^2,\qquad
V_{2,\Sigma}(F)\ge \frac14 \sum_{i=1}^{l}\mu_1(T_{3,i})\delta_i^2.
]

Hence, taking Theorem 1 into account, we have
[
W_0(F)+W_1(F)+W_2(F)\ge
V_{0,\Sigma}(F)+V_{1,\Sigma}(F)+V_{2,\Sigma}(F)\ge
]
[
\ge \frac1{12}k\sum_{i=1}^{l}\delta_i^3>\frac{k\tau}{10^5}.
]

Since (k>0) was chosen arbitrarily, we conclude that at least one of the three variations (F(\eta)) is unbounded. This contradicts the conditions of the theorem. We shall show that boundedness of any two variations of a continuous function (F(\eta)) is not a sufficient condition for differentiability almost everywhere of this function.

In the examples considered below, we shall regard (F(\eta)) as defined in the closed ball (R) of unit radius with center at the origin (O).

Example 1. Inscribe in (R) a cube (Q), into which we place an open ball (U_0) of radius (2^{-2}). Divide (Q) into a system of equal cubes of the 1st rank so that the sum of the volumes of the cubes of the 1st rank intersecting (U_0) does not exceed (\frac83\pi\cdot 2^{-6}).

Let (Q_{1,1},\ldots,Q_{1,k_1}) be the cubes of the 1st rank not intersecting (U_0). In each cube (Q_{1,i}) place an equal open ball (U_{1,i}) of radius not exceeding (k_1^{-1}), and such that the sum of the volumes of the balls (U_{1,i}) does not exceed (\frac43\pi\cdot 2^{-9}). Let
[
\sigma_1=\bigcup_{i=1}^{k_1} U_{1,i}
\quad\text{and}\quad
P_1=\bigcup_{i=1}^{k_1} Q_{1,i}.
]

Divide (P_1) into a system of equal cubes of the 2nd rank so that the sum of the volumes of the cubes of the 2nd rank intersecting (\sigma_1) does not exceed (\frac83\pi\cdot 2^{-9}). Let (Q_{2,1},\ldots,Q_{2,k_2}) be the cubes of the 2nd rank not intersecting (\sigma_1). In each cube (Q_{2,i}) place an equal open ball (U_{2,i}) of radius not exceeding (k_2^{-1}), and such that the sum of the volumes of the balls (U_{2,i}) does not exceed (\frac43\pi\cdot 2^{-12}). Let
[
\sigma_2=\bigcup_{i=1}^{k_2} U_{2,i}
\quad\text{and}\quad
P_2=\bigcup_{i=1}^{k_1} Q_{2,i}.
]
Continue this process without bound. Define (F(\eta)) as follows: 1) (F(\eta)=0) for
(\eta\in R\setminus \bigcup_{n=1}^{\infty}\sigma_n);

2) (F(\eta)=2^{-n}), if (\eta) is the center of a ball of the system (\sigma_n) ((n=1,2,\ldots)); 3) (F(\eta)) is linear on the radii of each of the balls of these systems. It is not difficult to see that (F(\eta)) is continuous, while the variations (W_1(F)) and (W_2(F)) are bounded. Let
[
\Omega=\bigcap_{n=1}^{\infty} P_n.
]
Then (\mu_3(\Omega)>0). At the points of the set (\Omega) the function (F(\eta)) is not differentiable.

Example 2. Let (Q_1) and (Q_2) be parts of right circular cylinders of radii (\frac12) and (\frac14), respectively, with generators parallel to the (z)-axis, bounded by the planes (z=-\frac12) and (z=\frac12), and with the centers of symmetry of (Q_1)

and (Q_2) coincide with (O). Denote by (A_0) the interior of the disk (A)—the intersection of (Q_2) with the plane (z=0).

Draw in the disk (A_0) a simple arc (L_1) so that in the neighborhood of any point (a\in A_0) of radius (10^{-2}) there exists at least one point of (L_1). Let (P_1\supset L_1) be such an open strip that (\mu_2(P_1)<10^{-2}) and (\overline P_1\subset A_0).

Put (M_1=A_0\setminus \overline P_1). Draw in (M_1) a simple arc (L_2) so that in the neighborhood of any point of the set (M_1) of radius (10^{-4}) there exists at least one point of (L_2). Let (P_2\supset L_2) be such an open strip that (\mu_2(P_2)<10^{-4}) and (\overline P_2\subset M_1). Put (M_2=M_1\setminus \overline P_2).

On the disk (A) define the function (\varphi(x,y)) as follows: 1) (\varphi(x,y)=0) outside (\bigcup_{n=1}^{\infty}P_n); 2) (\varphi(x,y)=2^{-(n+1)}) on (L_n) ((n=1,2,\ldots)), and extend (\varphi(x,y)) in each strip (P_n) continuously so that in (P_n) one has (0\leq \varphi(x,y)\leq 2^{-(n+1)}), and each level set of (\varphi(x,y)) in the strip (P_n) consists of a single component. Let (I) be the graph of the function (\varphi(x,y)), (I_1) the graph of the function (\varphi(x,y)-1/4), and (I_2) the graph of the function (\varphi(x,y)+1/4).

Define the function of three variables (F(\eta)) as follows: 1) (F(\eta)=0) for (\eta\in I_1\cup I_2\cup (R\setminus Q_1)), and also in the cylinder (Q_2) between the plane (z=-1/2) and the surface (I_1), and between the surface (I_2) and the plane (z=1/2); 2) (F(\eta)=1), if (\eta\in I), and it is linear in (z) between the surfaces (I_1) and (I) and between (I) and (I_2); 3) in the cylindrical coordinate system (\varphi,r,z), the function (F(\eta)) is linear in (r) for (1/4\leq r\leq 1/2). (F(\eta)) is continuous and the variations (W_0(F)) and (W_1(F)) are bounded. Let

[
W=\bigcap_{n=1}^{\infty} M_n\subset A_0.
]

Then (\mu_2(W)>0).

Denote by (\Omega) the part of the cylinder over (W), with generator parallel to the (z)-axis, enclosed between (I_1) and (I_2). Then (\mu_3(\Omega)>0), and at the points of (\Omega) the function (F(\eta)) is not differentiable.

Example 3. Let (M) be the sphere of radius (2^{-1}) with center at (O). We shall consider a partition of (M) into (n^2) spherical polygons by a uniform net of (n) parallels and meridians. Divide (M) into 4 polygons (M_1,M_2,M_3,M_4). In each (M_i) choose a spherical segment (S_i) so that the sum of the areas of these segments does not exceed (2\pi\cdot 10^{-8}). Let (\sigma_0=\bigcup S_i). Divide (M) into polygons of the 1st rank so that the sum of the areas of the polygons of the 1st rank intersecting (\sigma_0) does not exceed (4\pi\cdot 2^{-8}), and the maximal diameter of these polygons is not greater than (10^{-1}).

Let (M_{1,1},M_{1,2},\ldots,M_{1,k_1}) be the spherical polygons of the 1st rank not intersecting (\sigma_0). In each (M_{1,i}) choose a segment (\sigma_{1,i}) so that the sum of the areas of the segments (S_{1,i}) does not exceed (2\pi\cdot 2^{-10}). Let

[
P_1=\bigcup_{i=1}^{k_1} M_{1,i}
\quad\text{and}\quad
\sigma_1=\bigcup_{i=1}^{k_1} S_{1,i}.
]

Divide (P_1) into polygons of the 2nd rank so that the sum of the areas of the polygons of the 2nd rank intersecting (\sigma_1) does not exceed (4\pi\cdot 2^{-10}), and the maximal diameter of a polygon of the 2nd rank does not exceed (10^{-2}). Continue this process indefinitely.

On each of the disks cutting off the segments of the systems (\sigma_0,\sigma_1,\ldots), as on bases, construct cones whose vertices are located outside the ball bounded by the sphere (M), and the height of the cone supported on a segment from (\sigma_l) is equal to (2^{-(l+1)}). Denote by (I) the surface thus obtained, and by (I^) the open body bounded by the surface (I). Define (F(\eta)) as follows: 1) (F(\eta)=0), if (\eta\in R\setminus I^); 2) (F(\eta)=1) at the origin (O); 3) (F(\eta)) is linear on each ray emanating from (O); 4) (F(\eta)) is continuous, and the variations (W_0(F)) and (W_2(F)) are bounded. Let

[
W=M\setminus \bigcap_{n=1}^{\infty} P_n.
]

Then (\operatorname{mes}_2 W>0). Denote by (\Omega) the intersection of (R) and the cone over (W) with vertex at (O). (F(\eta)) is not differentiable at the points of (\Omega), and (\mu_3(\Omega)>0).

Moscow Pedagogical Institute
named after V. I. Lenin

Received
24 VI 1969

REFERENCES

1. A. S. Kronrod, UMN, 5, no. 1 (1950).
2. S. Saks, Theory of the Integral, IL (1949).
3. V. V. Stepanov, Math. Ann., 90 (1923).