MATHEMATICS

O. V. BESOV

ONE EXAMPLE IN THE THEORY OF EMBEDDING THEOREMS*

(Presented by Academician S. L. Sobolev, 27 XI 1961)

Let \(W_p^{(r)}(D)\) \((p>1,\ r>0)\) denote the class of functions
\(f(x)=f(x_1,\ldots,x_n)\), defined on the rectangle
\(D=\{x: 0<x_i<1,\ i=1,\ldots,n\}\), which have there all generalized partial derivatives of orders \(1,2,\ldots,[r]\) and finite norm

\[ \|f\|_{W_p^{(r)}(D)}=\|f\|_{L_p(D)}+\|f\|_{L_p^{(r)}(D)}, \]

where

\[ \|f\|_{L_p^{(r)}(D)} = \sum_{i_1,\ldots,i_r=1}^{n} \left\| \frac{\partial^r f}{\partial x_{i_1}\cdots \partial x_{i_r}} \right\|_{L_p(D)} \]

when \(r\) is an integer,

\[ \|f\|_{L_p^{(r)}(D)} = \sum_{i_1,\ldots,i_{[r]}=1}^{n} \left\{ \iint_{D\ D} \frac{ \left| \frac{\partial^{[r]} f(x)}{\partial x_{i_1}\cdots \partial x_{i_{[r]}}} - \frac{\partial^{[r]} f(y)}{\partial x_{i_1}\cdots \partial x_{i_{[r]}}} \right|^p }{ |x-y|^{\,n+(r-[r])p} } \,dx\,dy \right\}^{1/p} \]

when \(r\) is not an integer.

The spaces \(W_p^{(r)}\) for integral \(r\) were studied by S. L. Sobolev, and for arbitrary \(r>0\) by L. N. Slobodetskii.

Put \(\mu=r-n/p\). We shall consider all possible pairs of numbers \((\mu,r)\) for which the given function \(f\) turns out to belong to \(W_p^{(r)}(D)\). Such points form a certain set \(\Xi^*(f)\). We denote the set of its interior points by \(\Xi(f)\). The sets \(\Xi^*(f)\), \(\Xi(f)\) are contained in the half-strip
\[ \Lambda=\{(\mu,r): r>0,\ \mu<r<\mu+n\}. \]
From the embedding theorems for classes of functions, and also from theorems of the Gagliardo–Nirenberg type on the embedding of the intersection of two classes into a third, it follows that the sets \(\Xi^*(f)\) and \(\Xi(f)\) occupy the lower part of the half-strip \(\Lambda\), separated from its upper part by a nondecreasing convex curve.

S. L. Sobolev put forward the hypothesis that any admissible domain of the half-strip \(\Lambda\) is the domain \(\Xi(f)\) for some function \(f(x)\). In the present note we give a proof of this hypothesis. Namely, the following holds.

Theorem. Let the domain \(H\), which is the lower part of the half-strip \(\Lambda\), be separated from the upper part by a nondecreasing convex curve \(r=r(\mu)\). Then there exists a function \(f(x)\) for which the set \(\Xi(f)\) coincides with \(H\).

We first prove a lemma.

Lemma. The theorem is true in the particular case when \(r=r(\mu)\) is a straight line whose angular coefficient differs from \(-\dfrac{m}{\,n-m\,}\).

* The result of this note was reported at the IV All-Union Mathematical Congress in July 1961.

Proof. Let \(\omega_m(x)\) be a smooth function depending only on \(x_1,\ldots,x_m\) \((1 \le m \le n)\), with support contained in the set

\[ D_m=\{x: 0<x_i<1,\ i=1,2,\ldots,m\}. \]

Consider the function

\[ f(x)=\sum_{k=k_0}^{\infty} k^{-\beta}\omega_m\left[k^\delta\left(x-\frac{e_1}{\ln k}\right)\right] =\sum_{k=k_0}^{\infty}\Omega_k(x). \]

Here \(e_1=(1,0,\ldots,0)\), \(\delta>1\), and \(k_0=k_0(\lambda)\) is chosen so large that the supports of distinct \(\Omega_k(x)\) do not overlap. Let us find the set \(\Xi^*(f)\).

By changes of variables it is easy to see that

\[ \|\Omega_k\|_{L_p^{(r)}(D)}\sim k^\delta{}^{(r-m/p)}. \]

and also that

\[ \|f(x)\|_{L_p^{(r)}(D)}^p\sim \sum_{k=k_0}^{\infty} k^{-\beta p+\delta(rp-m)}. \]

The last series converges if and only if \(-\beta p+\delta(rp-m)<-1\), or

\[ [\delta(n-m)+1]r+(\delta m-1)\mu-\beta n<0. \tag{1} \]

Thus we see that the set \(\Xi^*(f)\) consists of the points lying below the line (1). By choosing \(m\), \(\delta>1\), \(\beta\), this line can assume any admissible position except one parallel to the vectors \((m-n,m)\), where \(m=0,1,\ldots,n\). The lemma is thereby proved.

Proof of the theorem. Suppose now that an admissible domain \(H\) is given, representing the lower part of the half-plane \(\Lambda\), separated from the upper part by a convex nondecreasing curve \(r=r(\mu)\). One can indicate a countable number of lines \(l_i\) with negative angular coefficients (not equal, however, to \(-\frac{m}{n-m}\), where \(m=1,\ldots,n\)), cutting off from below, from the half-plane \(\Lambda\), sets \(\Delta_i\) such that the set of interior points

\[ \bigcap_{i=1}^{\infty}\Delta_i \]

coincides with \(H\). With each line \(l_i\) let us associate, by the method indicated in the lemma, the function

\[ f_i(x)=\sum_{k=k_i}^{\infty} k^{\beta_i}\omega_{m_i}\left[k^{\beta_i}\left(x-\frac{1}{\ln k}-2^{-i}\right)\right]. \]

We choose \(\beta_i\), \(\delta_i\), and \(m_i\) in such a way that the equation

\[ [\delta_i(n-m_i)+1]r+(\delta_i m_i-1)\mu-\beta_i n=0 \tag{2} \]

is the equation of the line \(l_i\). Suppose also that the numbers \(k_i\) are chosen so large that

\[ 2^{-i}+\ln^{-1}k<2^{-i+1}\quad \text{for } k\ge k_i\quad (i=1,2,\ldots). \]

This ensures that the supports of two distinct functions \(f_i\) and \(f_j\) do not intersect. Consider now the function

\[ f(x)=\sum_{1}^{\infty} f_i(x). \tag{3} \]

Whatever point \(\xi\) of the half-plane \(\Lambda\) may be, \(\xi\notin \overline{H}\), there exists a line \(l_i\) situated below it. In view of the nonintersection of the supports of the terms of the series (2)

one may assert that \(\xi \in \Xi^*(f)\). We shall now show that, for a special choice of the lines \(l_i\) and the numbers \(k_i\), \(\Xi^*(f)=\mathrm H\). Put

\[ -\eta\gamma_i=[\delta_i(n-m_i)+1]r+(\delta_i m_i-1)\mu-\beta_i n<0. \]

Let us note that, for fixed \(\delta_i\) and \(m_i\), the distance from the point \((r,\mu)\) to the line (2) is proportional to \(\gamma_i\) and depends on \(\beta_i\) and the point \((r,\mu)\). Let \(\sigma_i=\inf \gamma_i\) for \((r,\mu)\in \mathrm H\). The lines \(l_i\) can always be chosen so that \(\sigma_i=\sigma_i(\delta_i,m_i,\beta_i)\) tend to zero sufficiently slowly. In this case, by means of estimates analogous to those indicated in the lemma, we shall have:

\[ \|f\|_{L_p^r(D)}^p \leq A(\mu,p)\sum_{i=1}^{\infty}\sum_{k=k_i}^{\infty} k^{-1-p\gamma_i}\leq \]

\[ \leq A_1(\mu,p)\sum_{i=1}^{\infty}\frac{1}{p\gamma_i k_i^{p\gamma_i}} \leq A_1(\mu,p)\sum_{i=1}^{\infty}\frac{1}{\sigma_i k_i^{\sigma_i}}. \]

Here \(\sigma_i\) does not depend on the point \((r,\mu)\in \mathrm H\), and therefore it is possible to specify a sequence \(k_i\) for which the last series converges. Thus the proof is complete.

Remark. Denote by \(\Xi^{**}(f)\) the set of points of \(\Xi(f)\) and the curve \(r=r(\mu)\) (separating the set \(\Xi(f)\) from the upper part of the half-plane \(\Lambda\)). By the same method it is shown that an arbitrary admissible set \(\mathrm H^{**}\) is the set \(\Xi^{**}(f)\) for some function \(f(x)\).

It would be interesting to establish whether every (admissible) set \(\mathrm H^*\) is the set \(\Xi^*(f)\) for some function \(f(x)\).

Mathematical Institute
named after V. A. Steklov
Academy of Sciences of the USSR

Received
17 XI 1961