MATHEMATICS

S. M. NIKOLSKII

EMBEDDING THEOREMS FOR FUNCTIONS WITH PARTIAL DERIVATIVES CONSIDERED IN DIFFERENT METRICS

(Presented by Academician S. L. Sobolev on 25 VI 1957)

In our papers (see, in particular, (\left({}^{5,6}\right))) we obtained a generalization of the theorem of S. L. Sobolev (\left({}^{8,9}\right)) and the theorem of V. I. Kondrashov (\left({}^{4,9}\right)) on the embedding of classes of differentiable functions of many variables. From this point of view we considered classes of functions (H_p^{(r_1,\ldots,r_n)}), where the positive numbers (r_i) indicate the differential properties of the function (f \in H_p^{(r_1,\ldots,r_n)}) with respect to each variable (x_i) separately ((i=1,\ldots,n)), and the number (p) indicates the metric (L_p) by means of which the function (f) and its partial derivatives are normed.

However, one may introduce more general classes (H_{p_1,\ldots,p_n}^{(r_1,\ldots,r_n)}) of functions (f), where the numbers (p_i), generally speaking different, indicate that the partial derivatives (not mixed) of the function (f) with respect to each variable (x_i) are considered in their own metric (L_{p_i}) ((i=1,\ldots,n)). For these classes we have succeeded in obtaining the corresponding embedding theorems, which generalize our previous theorems.

We shall consider real-valued functions (f=f(x_1,\ldots,x_n)) defined on the (n)-dimensional space (R_n). Put

[
|f|{L_p^{(m)}}=
\left(
\int,\ldots,x_n)|^p\,dx_1\ldots dx_m} |f(x_1,\ldots,x_m,x_{m+1
\right)^{1/p}
\quad (m=1,\ldots,n).
]

Thus (|f|{L_p^{(m)}}) for (m<n) depends on (x,\ldots,x_n).

Let positive numbers (M), (r_i), and (p_i) be given, where (1\le p_i\le\infty). Thus, in particular, the (p_i) may be equal to (+\infty). By definition, if a function (f) belongs simultaneously to the classes* (H_{p_i x_i}^{(r_i)}(M)), then we shall say that it belongs to the class (H_{p_1,\ldots,p_n}^{(r_1,\ldots,r_n)}(M)). If (p_1=\cdots=p_n=p), then we put (H_{p_1,\ldots,p_n}^{(r_1,\ldots,r_n)}=H_p^{(r_1,\ldots,r_n)}(M)).

* A function (f\in H_{p x_1}^{(r)}(M)) (see (\left({}^{5}\right)) or (\left({}^{6}\right))) if it is integrable to the (p)-th power on (R_n) together with its partial derivatives (\partial^k f/\partial x_1^k) ((k=0,1,\ldots,\bar r)), where (r=\bar r+\alpha), (\bar r) is an integer and (0<\alpha\le1), and, moreover,

[
\left| f_{x_1}^{(\bar r)}(x_1+h,x_2,\ldots,x_n)
-
f_{x_1}^{(\bar r)}(x_1,\ldots,x_n)
\right|_{L_p^{(n)}}
\le M |h|^\alpha,
\quad \text{if } \alpha<1;
]

[
\left| f_{x_1}^{(\bar r)}(x_1+h,x_2,\ldots,x_n)
-
2f_{x_1}^{(\bar r)}(x_1,\ldots,x_n)
+
f_{x_1}^{(\bar r)}(x_1-h,x_2,\ldots,x_n)
\right|_{L_p^{(n)}}
\le M |h|,
\quad \text{if } \alpha=1.
]

Theorem 1. Let, for the numbers considered below, the inequalities
(r_i>0;\quad 1\le p_i\le q\le\infty;\quad n,\ m) be natural numbers,
(1\le m\le n), hold,

[
\rho_i=\frac{r_i\chi}{\chi_i}>0\qquad (i=1,\ldots,n),
\tag{1}
]

where

[
\chi=
\left|
\begin{array}{cc}
\displaystyle 1-\sum_{1}^{n}\frac{\frac1{p_l}-\frac1q}{r_l}
&
\displaystyle -\frac1q\sum_{1}^{n}\frac1{r_l}
\[1.2em]
\displaystyle -\sum_{m+1}^{n}\frac{\frac1{p_l}-\frac1q}{r_l}
&
\displaystyle 1-\frac1q\sum_{m+1}^{n}\frac1{r_l}
\end{array}
\right|,
\qquad
\chi_i=1-\sum_{l=1}^{n}\frac{\frac1{p_l}-\frac1{p_i}}{r_l}.
]

Let, moreover, a function defined in the (n)-dimensional space (R_n) satisfy
(f\in H_{p_1,\ldots,p_n}^{(r_1,\ldots,r_n)}(M)). Then for any fixed ((x_{m+1},\ldots,x_n)), (f), as a function of (x_1,\ldots,x_m), belongs to the class (H_q^{(\rho_1,\ldots,\rho_m)}(\overline M)). In this case the inequality

[
|f|{L_q^{(m)}}+\overline M<c\left(\min_i |f|+M\right),}^{(n)}
\tag{2}
]

holds, where the constant (c) does not depend on the set standing in the series.

If not only (\chi>0), but also* (1-\sum_{1}^{n}\frac1{p_k r_k}>0), then for any (\varepsilon>0) there exists a function
(f\in H_{p_1,\ldots,p_n}^{(r_1,\ldots,r_n)}(M)), which, for fixed (x_{m+1},\ldots,x_n), as a function of (x_1,\ldots,x_m), does not belong to the class
(H_q^{(\rho_1,\ldots,\rho_{i-1},\,\rho_i-\varepsilon,\,\rho_{i+1},\ldots,\rho_n)}(M_1)) for any (M_1).

This theorem is proved on the basis of approximating functions of the named classes by entire functions

[
g=g_{\nu_1,\ldots,\nu_n}=g_{\nu_1,\ldots,\nu_n}(x_1,\ldots,x_n)
\tag{3}
]

of degrees (\nu_1,\ldots,\nu_n) respectively in the variables (x_1,\ldots,x_n). In doing so, the following theorem proved useful; it is a generalization of Jackson’s theorem.

Theorem 2. For any positive (r_i) and (p_i) there exists a constant (c), depending on these numbers, such that, whatever the function
(f\in H_{p_1,\ldots,p_n}^{(r_1,\ldots,r_n)}(M)), there can be found for it a system of functions (3) ((1\le \nu_i\le\infty,\ i=1,\ldots,n)) for which the inequalities** hold:

[
\left|f-g_{\nu_1,\infty,\ldots,\infty}\right|{L,}^{(n)}}<\frac{cM}{\nu_1^{r_1}
]

[
\left|g_{\nu_1,\infty,\ldots,\infty}
-
g_{\nu_1,\nu_2,\infty,\ldots,\infty}\right|{L,}^{(n)}}<\frac{cM}{\nu_2^{r_2}
\tag{4}
]

[
\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots
]

[
\left|g_{\nu_1,\ldots,\nu_{n-1},\infty}
-
g_{\nu_1,\ldots,\nu_n}\right|{L.}^{(n)}}<\frac{cM}{\nu_n^{r_n}
]

* We do not think that the restriction (1-\sum_{1}^{n}\frac1{p_k r_k}>0) is essential. Without it, one should suppose, the corresponding example can be obtained analogously to how, in the case (p_1=\cdots=p_n=p), T. I. Amanov reasoned (¹).

** If (\nu_k=\infty), this means that (g) need not be an entire function with respect to (x_k).

For (p=p_1=\cdots=p_n), Theorem 2 contains our earlier result (((^{5}), p. 267), which for (p=\infty) turns into a result of S. N. Bernstein ((^{2})).

Remark 1. Theorem 1 remains valid without changes for the class (H^{(r_1,\ldots,r_n)*}{p_1,\ldots,p_n}) of functions (f), periodic in each of the variables (x_i), defined similarly to the class (H^{(r_1,\ldots,r_n)}), but where the integrals serving to define the norm (|f|{L_p^{(m)}}) are calculated not over the whole space, but over a period. There is also a corresponding analogue of Theorem 2, where the role of (P) is played by trigonometric polynomials.

Remark 2. Let us note cases when Theorem 1 retains its force for classes of functions (f\in H^{(r_1,\ldots,r_n)}{p_1,\ldots,p_n}(G;M)), given on a domain (G\subset R_n). These classes are defined analogously to (H^{(r_1,\ldots,r_n)}).

a) If (G) is a rectangular parallelepiped with faces parallel to the coordinate axes, then a function (f\in H^{(r_1,\ldots,r_n)}{p_1,\ldots,p_n}(G;M)) can be extended to (R_n) so that the extended function (\bar f\in H^{(r_1,\ldots,r_n)}), in any case if the (r_i) are not integers. For integer (r_i), this is probably also true, but has not yet been proved.

b) On the other hand, if (r_1,\ldots,r_m) are integers and (r_{m+1},\ldots,r_n) are not integers, if by the class (H^{(r_1,\ldots,r_n)}{p_1,\ldots,p_n}(G;M)) one understands the intersection of the classes(^*) (W^{(r_i)}}(G;M)) ((i=1,\ldots,m)) and (H^{(r_i){p_i x_i}(G;M)) ((i=m+1,\ldots,n)), then a function of this class can certainly be extended, preserving these properties, to (R_n) beyond the parallelepiped (G) defined above, and then the extended function (\bar f\in H^{(r_1,\ldots,r_n)}(cM)).

c) Let (G=G'\times G'') be the topological product of domains (G'\subset R_l(x_1,\ldots,x_l)) and (G''\subset R_{n-l}(x_{l+1},\ldots,x_n)), bounded by surfaces continuously differentiable (r+2) times, and let the function (f), given on (G), have partial derivatives

[
\frac{\partial^{r'} f}{\partial x_1^{\alpha_1}\cdots \partial x_l^{\alpha_l}}
\left(\sum_{k=1}^{l}\alpha_k\le r'\right),
]

belonging to (L_{p'}(G)), as well as partial derivatives

[
\frac{\partial^{r''} f}{\partial x_{l+1}^{\beta_{l+1}}\cdots \partial x_n^{\beta_n}}
\left(\sum_{l+1}^{n}\beta_k\le r''\right),
]

belonging to (L_{p''}(G)). Then (f) can be extended beyond (G) to (R_n) with preservation of the indicated differentiability properties. Thus the extended function (\bar f\in H^{(r_1,\ldots,r_n)}{p_1,\ldots,p_n}(cM)), where (r_1=\cdots=r_l=r'), (r=\cdots=p_n=p'').}=\cdots=r_n=r''), (p_1=\cdots=p_l=p'), (p_{l+1

In conclusion we note that, on the basis of Theorem 1, a corresponding compactness theorem can be obtained, analogous to our results published in ((^{7})). This generalizes, for the domains considered by us, the compactness theorems obtained by E. Gagliardo ((^{3})).

Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
20 VI 1957

REFERENCES

(^{1}) G. I. Amanov, Izv. AN SSSR, ser. matem., 19, 17 (1955).
(^{2}) S. N. Bernstein, DAN, 57, 111 (1947).
(^{3}) E. Gagliardo, Univ. di Genova, Publ. del Inst. di Mat., No. 28, 148 (1956).
(^{4}) V. I. Kondrashov, DAN, 48, 563 (1945).
(^{5}) S. M. Nikol’skii, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 38, 244 (1951).
(^{6}) S. M. Nikol’skii, Matem. sborn., 33 (75), 2, 261 (1953).
(^{7}) S. M. Nikol’skii, Izv. AN SSSR, ser. matem., 20, 611 (1956).
(^{8}) S. L. Sobolev, Matem. sborn., 4 (46), 3, 471 (1938).
(^{9}) S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics, L., 1950.

(^*) (f\in W^{(r)}_{p x_1}(G;M)), if (\partial^k f/\partial x^k\in L_p(G)) ((k=0,1,\ldots,r)).