Doklady of the Academy of Sciences of the USSR
1964. Volume 156, No. 5

MATHEMATICS

A. Kh. Gudiev

EMBEDDING THEOREMS FOR SOME CLASSES OF ABSTRACT FUNCTIONS

(Presented by Academician S. L. Sobolev on 23 I 1964)

We denote each point \(\mathbf{x}\) of the \(n\)-dimensional Euclidean space \(R^n\) in the form of a pair \((\mathbf{x}_{n-s}, \mathbf{x}_s)\), where \(\mathbf{x}_{n-s}(x_1, x_2, \ldots, x_{n-s})\), \(\mathbf{x}_s(x_{n-s+1}, x_{n-s+2}, \ldots, x_n)\). In addition, let \(R^{n-s}\) be the \((n-s)\)-dimensional space of vectors \(\mathbf{x}_{n-s}\); \(R^s\) be the \(s\)-dimensional space of vectors \(\mathbf{x}_s\); \(\bar R^n_0\) be the half-space \(x_n \geqslant 0\); \(x(x_1, x_2, \ldots, x_{n-1}) \in R^{n-1}\); \(\bar R^s_0\) be the half-space of points \(x_n \geqslant 0\); \((x_{n-s+1}; x_{n-s+2}, \ldots, x_{n-1}) \in R^{s-1}\).

We shall regard each function \(f(\mathbf{x})\), defined in \(R^n\), as a function of the vector variables \(\mathbf{x}_{n-s}\) and \(\mathbf{x}_s\). Under such consideration, to almost every vector \(\mathbf{x}^{(0)}_{n-s}\) there corresponds an element of a certain abstract space—the function \(f(\mathbf{x}^{(0)}_{n-s}, \mathbf{x}_s)\) of the variable vector \(\mathbf{x}_s\).

From the collection of abstract functions defined in this way one constructs the abstract classes:
\[ L^l_{\alpha;(p_1,p_2)}(\bar R^n_0),\quad W^l_{\alpha;(p_1,p_2)}(\bar R^n_0),\quad W^r_{(p_1,p_2)}(R^n),\quad B^r_{(p_1,p_2)}(R^n) \]
—analogues of the known classes \(L^l_{\alpha;p}\), \(W^l_{\alpha;p}\), \(W^r_p\), \(B^r_p\) \((^{1-7})\).

I. Definition of the classes \(L^l_{\alpha;(p_1,p_2)}(\bar R^n_0)\) (\(l\) an integer). We shall say that a function \(f(\mathbf{x})\) belongs to the class \(L^l_{\alpha;(p_1,p_2)}(\bar R^n_0)\) if:

a) \(f(\mathbf{x})\) has in \(\bar R^n_0\) all generalized derivatives of order \(l\);

b)
\[ \|f\|_{L^l_{\alpha;(p_1,p_2)}(\bar R^n_0)} = \left\{ \int_{R^{n-s}} \left[ \int_{\bar R^s_0} x_n^\alpha \sum_{|\bar\alpha|=l} |D^\alpha f|^{p_1}\,d\bar R^s_0 \right]^{p_2/p_1} dR^{n-s} \right\}^{1/p_2} <\infty . \]

II. Definition of the classes \(W^l_{\alpha;(p_1,p_2)}(\bar R^n_0)\). We shall say that a function \(f(\mathbf{x})\) belongs to \(W^l_{\alpha;(p_1,p_2)}(\bar R^n_0)\) if:

a) \(f(\mathbf{x}) \in L^l_{\alpha;(p_1,p_2)}(\bar R^n_0)\);

b) \(f(\mathbf{x}) \in L_{(p_1,p_2)}(\bar R^n_0)\).

The norm in \(W^l_{\alpha;(p_1,p_2)}(\bar R^n_0)\) is defined by the equality
\[ \|f\|_{W^l_{\alpha;(p_1,p_2)}(\bar R^n_0)} = \|f\|_{L_{(p_1,p_2)}(\bar R^n_0)} + \|f\|_{L^l_{\alpha;(p_1,p_2)}(\bar R^n_0)} . \]

III. Definition of the classes \(W^r_{(p_1,p_2)}(R^n)\).

1) \(r\) an integer. Put \(\mathbf{r}(r,r,\ldots,r)\). We shall say that a function
\[ f(\mathbf{x}) \in W^r_{(p_1,p_2)}(R^n) \]
if:

a) \(f(\mathbf{x}) \in L_{(p_1,p_2)}(R^n)\);

b) \(f(\mathbf{x}) \in L^r_{0;(p_1,p_2)}(R^n)\).

The norm in \(W^r_{(p_1,p_2)}(R^n)\) is defined by the equality
\[ \|f\|_{W^r_{(p_1,p_2)}(R^n)} = \|f\|_{L_{(p_1,p_2)}(R^n)} + \|f\|_{L^r_{0;(p_1,p_2)}(R^n)} . \]

2) \(r\) nonintegral. Let \(r=\bar r+\alpha\), where \(0<\alpha<1\); \(1<p_1<\infty\). Put
\[ \mathbf{r}(\underbrace{r,\ldots,r}_{n-s},\underbrace{\bar r,\ldots,\bar r}_{s}). \]
We shall say that a function \(f(\mathbf{x})\in W^r_{(p_1,p_2)}(R^n)\),

if

a) \(f(x)\in W_{(p_1,p_2)}^{\bar r}(R^n)\), where \(\bar r(\underbrace{\bar r,\bar r,\ldots,\bar r}_{n})\);

b)
\[ \left\|\vec D^{\beta}f\right\|_{\bar W_{(p_1,p_2)}^{-\alpha}(R^n)} = \sum_{j=n-s+1}^{n} \left\{ \int_{R^{\,n-s}} \left[ \int_{R^s} \left( \int_{0}^{\infty} \frac{\left|\Delta_{j,h}D^{\vec\beta}f\right|^{p_1}}{h^{1+p_1\alpha}}\,dh \right)dR^s \right]^{p_2/p_1} dR^{\,n-s} \right\}^{1/p_2}<\infty, \]
where
\[ \Delta_{j,h}D^{\vec\beta}f(x_1,\ldots,x_j+h,\ldots,x_n) - D^{\vec\beta}f(x_1,\ldots,x_j,\ldots,x_n); \qquad |\vec\beta|=\bar r. \]

We define the norm in \(W_{(p_1,p_2)}^{r}(R^n)\) by the equality
\[ \|f\|_{W_{(p_1,p_2)}^{r}(R^n)} = \|f\|_{W_{(p_1,p_2)}^{\bar r}(R^n)} + \sum_{|\vec\beta|=\bar r} \left\|D^{\vec\beta}f\right\|_{\bar W_{(p_1,p_2)}^{-\alpha}(R^n)}. \]

IV. Definition of the classes \(B_{(p_1,p_2)}^r(R^n)\).

1) \(r\) an integer. We shall say that \(f(x)\in B_{(p_1,p_2)}^r(R^n)\) if:

a) \(f(x)\in W_{(p_1,p_2)}^{\vec\gamma}(R^n)\), where \(\vec\gamma(\underbrace{r-1,r-1,\ldots,r-1}_{n})\);

b)
\[ \left\|D^{\vec\beta}f\right\|_{\bar B_{(p_1,p_2)}^{1}(R^n)} = \sum_{j=n-s+1}^{n} \left\{ \int_{R^{\,n-s}} \left[ \int_{R^s} \left( \int_{0}^{\infty} \frac{\left|\Delta_{j,h}^{2}D^{\vec\beta}f\right|^{p_1}}{h^{1+p_1}}\,dh \right)dR^s \right]^{p_2/p_1} dR^{\,n-s} \right\}^{1/p_2}<\infty, \]
where \(|\vec\beta|=r-1\);
\[ \Delta_{j,h}^{2}D^{\vec\beta}f = D^{\vec\beta}f(x_1,\ldots,x_j-h,\ldots,x_n) - 2D^{\vec\beta}f(x_1,\ldots,x_j,\ldots,x_n) + D^{\vec\beta}f(x_1,\ldots,x_j+h,\ldots,x_n). \]

We define the norm in \(B_{(p_1,p_2)}^r(R^n)\) by the equality
\[ \|f\|_{B_{(p_1,p_2)}^r(R^n)} = \|f\|_{W_{(p_1,p_2)}^{\vec\gamma}(R^n)} + \sum_{|\vec\beta|=r-1} \left\|D^{\vec\beta}f\right\|_{\bar B_{(p_1,p_2)}^{1}(R^n)}. \]

2) \(r\) noninteger. Put
\[ r(\underbrace{\bar r,\bar r,\ldots,\bar r}_{n-s}, \underbrace{r,r,\ldots,r}_{s}). \]
In this case we shall assume that \(f(x)\in B_{(p_1,p_2)}^r(R^n)\) if \(f(x)\in W_{(p_1,p_2)}^{r}(R^n)\).

Theorem 1. If \(\alpha-kp_1+1>0\), then
\[ W_{\alpha;(p_1,p_2)}^{l}(\bar R_0^n) \to W_{\alpha-kp_1;(p_1,p_2)}^{\,l-k}(\bar R_0^n). \]

Theorem 2. If \(\alpha-p_1l+1<0\), then
\[ W_{\alpha;(p_1,p_2)}^{l}(\bar R_0^n) \to B_{(p_1,p_2)}^{\vec\gamma}(R^{n-1}), \]
where
\[ \vec\gamma\left( \underbrace{\left[l-\frac{\alpha+1}{p_1}\right],\ldots, \left[l-\frac{\alpha+1}{p_1}\right]}_{n-s}, \underbrace{l-\frac{\alpha+1}{p_1},\ldots,l-\frac{\alpha+1}{p_1}}_{s-1} \right), \]
\([\alpha]\) is the integer part of the number \(\alpha\).

Theorem 3. Let \(1\le m\le n-1,\quad 1<p<p_1<p_2<\infty\),
\[ r=l-\frac{n-1}{p}+\frac{m}{p_1}+\frac{n-m-1}{p_2}; \]
then
\[ W_p^l(R^{n-1})\to W_{(p_1,p_2)}^r(R^{n-1}), \]

where \(r(\underbrace{\bar r,\ldots,\bar r}_{n-s},\underbrace{r,\ldots,r}_{s-1})\), \(l\) is nonintegral \(\bigl(W_{\alpha_1}^{\beta_1}\to W_{\alpha_2}^{\beta_2}\) denotes the embedding of \(W_{\alpha_1}^{\beta_1}\) in \(W_{\alpha_2}^{\beta_2}\) (see the survey article of S. M. Nikol’skii \((^8)\)\()\).

Theorems 1 and 2, as is not difficult to see, can be obtained from the corresponding theorems of S. V. Uspenskii \((^4)\).

We shall outline the proof of Theorem 3. Consider the function

\[ F=kx_n\int_{R^{n-1}}\frac{v(t,x_n)\,dt}{[(t-x)^2+x_n^2]^{n/2}}. \]

From the preceding theorems it follows that, in order to prove Theorem 3, it is sufficient to establish the validity of the inequality

\[ \|F\|_{L_{p_1(\nu-r)-1;(p_1,p_2)}^\nu(\bar R_0^n)} \leq c\|f\|_{W_p^l(R^{n-1})}. \tag{1} \]

Using the estimate

\[ |F^\nu(x,x_n)|\leq c \sum_{|\vec\alpha|\leq \nu}\int_{R^{n-1}} \frac{|v^\nu(t,x_n)|\,x_n}{[(t-x)^2+x_n^2]^{n/2}}\,dt+ \]

\[ +\,c\sum_{|\vec\alpha|\leq \nu-1}\int_{R^{n-1}} \frac{|v^{\nu-1}(t,x_n)|\,dt}{[(t-x)^2+x_n^2]^{n/2}} = c\left(\sum_{|\vec\alpha|\leq \alpha} I_{1,\vec\alpha} +\sum_{|\vec\alpha|\leq \alpha-1} I_{2,\vec\alpha}\right) \]

and Minkowski’s inequality, we obtain

\[ \|F\|_{L_{p_1(\nu-r)-1;(p_1,p_2)}^\nu(\bar R_0^n)} \leq c\left( \sum_{|\vec\alpha|\leq \nu} \|I_{1,\vec\alpha}\|_{L_{p_1(\nu-r)-1;(p_1,p_2)}^\nu(\bar R_0^n)} +\right. \]

\[ \left. +\sum_{|\vec\alpha|\leq \nu-1} \|I_{2,\vec\alpha}\|_{L_{p_1(\nu-r)-1;(p_1,p_2)}^\nu(\bar R_0^n)} \right). \tag{2} \]

Let us estimate one of the terms, for example

\[ \|I_{1,\vec\alpha}\|_{L_{p_1(\nu-r)-1;(p_1,p_2)}^\nu(\bar R_0^n)} = \]

\[ = c\left\{ \int_{R^{n-m-1}} \left[ \int_{\bar R_0^{m+1}} x_n^{p_1(\nu-r)-1} \left( \int_{R^{n-1}} \frac{|v^\nu(t,x_n)|\,x_n\,dt}{[(t-x)^2+x_n^2]^{n/2}} \right)^{p_1} d\bar R_0^{m+1} \right]^{p_2/p_1} dR^{n-m-1} \right\}^{1/p_2}. \]

Applying Hölder’s inequality and simplifying, we shall have

\[ \int_{\bar R_0^{m+1}} x_n^{p_1(\nu-r)-1} \left( \int_{R^{n-1}} \frac{|v^\nu(t,x_n)|\,x_n\,dt}{[(t-x)^2+x_n^2]^{n/2}} \,d\bar R_0^{m+1} \right)^{p_1} \leq \]

\[ \leq c\|f\|_{W_p^l}^{p_1-p} \int_{\bar R_0^{m+1}} x_n^k \left( \int_{R^{n-1}} \frac{|v^\nu|^p\,dt}{[(t-x)^2+x_n^2]^{n/2}} \right) d\bar R_0^{m+1}, \tag{3} \]

where \(k=p_1(l-r)+p(\nu-l)-(n-1)(p_1/p-1)\).

Taking (3) into account and making the necessary transformations, we obtain

\[ \|I_{1,\vec\alpha}\|_{L_{p_1(\nu-r)-1;(p_1,p_2)}^\nu(\bar R_0^n)} \leq \]

\[ \leq c\|f\|_{W_p^l}^{\,1-p/p_1} \left\{ \int_{R^{n-m-1}} \left[ \int_{\bar R^{m+1}} x_n^{k-(1-p_1/p_2)(n-m-1)} \left( \int_{R^{n-1}} \frac{|v^\nu|^p\,dt}{[(t-x)^2+x_n^2]^{n/2}} \right) d\bar R_0^{m+1} \right]\times \]

\[ \times \left[ \int_{\bar R_0^n} x_n^{p(\nu-l)-1}|v^\nu|^p\,d\bar R_0^n \right]^{(p_2-p_1)/p_1} dR^{n-m-1} \right\}^{1/p_2}. \]

The estimates hold:
\[ \int_{\overline{R}_0^n} x_n^{p(\nu-l)-1}\, |v^\nu(t,x_n)|^p\, d\overline{R}_0^n \leq c\|f\|_{W_p^l(R^{n-1})}^p, \]
\[ \int_{R^{n-m-1}} \left[ \int_{\overline{R}_0^{m+1}} x_n^{k-(1-p_1/p_2)(n-m-1)} \left( \int_{R^{n-1}} \frac{|v^\nu|^p\,dt}{\left[(t-x)^2+x_n^2\right]^{n/2}} \right) d\overline{R}_0^{m+1} \right] dR^{n-m-1} \leq \]
\[ \leq \int_{\overline{R}_0^n} x_n^{k-(1-p_1/p_2)(n-m-1)-1}\, |v^\nu(t,x_n)|^p\,d\overline{R}_0^n \leq c\|f\|_{W_p^l(R^{n-1})}^p, \]
therefore
\[ \left\| I_{1,\alpha}^{\to}\right\|_{L_{p_1(\nu-r)-1;(p_1,p_2)}^\nu(\overline{R}_0^n)} \leq c\|f\|_{W_p^l(R^{n-1})}. \tag{4} \]

Similarly one can show that
\[ \left\| I_{2,\alpha}^{\to}\right\|_{L_{(p_1(\nu-r)-1;(p_1,p_2))}^\nu(\overline{R}_0^n)} \leq c\|f\|_{W_p^l(R^{n-1})}. \tag{5} \]

From (2), (4), and (5) we obtain (1).

Hence it is not difficult to obtain the required result. Theorem 3 establishes the validity of the hypothesis of S. L. Sobolev and S. M. Nikol’skii for the generalized fractional spaces \(W_p^l\) of S. L. Sobolev. Works \((^9,\ ^{11},\ ^{12})\) are also devoted to this hypothesis.

Institute of Mathematics with Computing Center
of the Siberian Branch of the Academy of Sciences of the USSR

Received
13 I 1964

CITED LITERATURE

\(^1\) S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics, Siberian Branch of the Academy of Sciences of the USSR, 1962.
\(^2\) L. D. Kudryavtsev, Proceedings of the V. A. Steklov Mathematical Institute, 55 (1959).
\(^3\) S. V. Uspenskii, Siberian Mathematical Journal, 3, No. 3 (1962).
\(^4\) S. V. Uspenskii, Proceedings of the V. A. Steklov Mathematical Institute, Academy of Sciences of the USSR, 60, 282 (1961).
\(^5\) L. N. Slobodetskii, Doklady Akademii Nauk SSSR, 118, No. 2, 243 (1958).
\(^6\) O. V. Besov, Doklady Akademii Nauk SSSR, 126, No. 6, 1163 (1959).
\(^7\) V. P. Il’in, Doklady Akademii Nauk SSSR, 129, 983 (1959).
\(^8\) S. M. Nikol’skii, Uspekhi Matematicheskikh Nauk, 16, issue 5 (101) (1961).
\(^9\) S. M. Nikol’skii, Siberian Mathematical Journal, 3, No. 6, 845 (1962).
\(^10\) S. N. Slobodetskii, Uspekhi Matematicheskikh Nauk, 15, No. 3, 117 (1960).
\(^11\) A. Kh. Gudiev, Doklady Akademii Nauk SSSR, 147, No. 4 (1962).
\(^12\) A. Kh. Gudiev, Doklady Akademii Nauk SSSR, 149, No. 3 (1963).