Reports of the Academy of Sciences of the USSR
1960. Volume 135, No. 3

MATHEMATICS

V. P. IL'IN

ON THE COMPLETE CONTINUITY OF THE EMBEDDING OPERATOR FOR THE CASE OF AN UNBOUNDED DOMAIN

(Presented by Academician S. L. Sobolev on 17 VI 1960)

In the present note we shall give some sufficient conditions for the complete continuity of the embedding operator for the case when the functions of the corresponding functional classes are defined on the whole (n)-dimensional space (E_n) or on a subspace (E_m) of dimension (m<n). Questions of a similar kind for the special case (embedding in (L_2(E_n))) were studied by A. M. Molchanov ((^1)), M. Sh. Birman ((^2)), and M. Sh. Birman and B. S. Pavlov*. The results of these authors have the character of necessary and sufficient conditions.

Let (E_n) be (n)-dimensional Euclidean space, and let (\omega(X)=\omega(x_1,\ldots,x_n)) be a positive measurable function given in (E_n). Let (p>1).

Denote by (L_p(\omega;E_n)) the set of functions (f(X)), defined in (E_n), for which

[
|f|{L_p(\omega;E_n)}
=
\left[
\int\limits \cdots \int \omega |f|^p\, dX
\right]^{1/p}
<\infty .
\tag{1}
]

Let (l) be an arbitrary positive number, (\bar l=[l]). Suppose that (f(X)) has continuous derivatives of order (\bar l). Put

[
|f|{L_p^{(l)}(E_n)}
=
\sum}=1}^{n
\left[
\int\limits_{(E_n)} \cdots \int
\left(
\int\limits_{(E_n)} \cdots \int
\frac{
\left|
\dfrac{\partial^{\bar l} f(X)}{\partial x_{i_1}\cdots \partial x_{i_{\bar l}}}
-
\dfrac{\partial^{\bar l} f(Y)}{\partial x_{i_1}\cdots \partial x_{i_{\bar l}}}
\right|^p
}{
|X-Y|^{\,n+(l-\bar l)p}
}
\, dY
\right)
dX
\right]^{1/p}
\tag{2}
]

when (l) is not an integer, and

[
|f|{L_p^{(l)}(E_n)}
=
\sum}^{n
\left[
\int\limits_{(E_n)} \cdots \int
\left|
\frac{\partial^l f(X)}{\partial x_{i_1}\cdots \partial x_{i_l}}
\right|^p
\, dX
\right]^{1/p}
\tag{2'}
]

when (l) is an integer, if the right-hand sides of (2) or, respectively, ((2')) are finite. Further, put

[
|f|{W_p^{(l)}(\omega;E_n)}
=
|f|
+
|f|_{L_p^{(l)}(E_n)} .
\tag{3}
]

If (\omega\equiv 1), we shall write (W_p^{(l)}(E_n)), respectively (L_p(E_n)). We define the space (W_p^{(l)}(\omega;E_n)) as the closure, in the norm (3), of the set of (\bar l)-times continuously differentiable functions for which the right-hand side of (3) is finite.

* Unpublished.

Further, by (\Ш_R^{(n)}(X)) we shall denote the (n)-dimensional ball of radius (R) with center at (X); in particular, (\Ш_R^{(n)}(0)) denotes the ball with center at the origin; (E_n-\Ш_R^{(n)}(0)) is the complement of (\Ш_R^{(n)}(0)) in (E_n); (|X|=\sqrt{x_1^2+\cdots+x_n^2}) is the distance from (X) to the origin.

Theorem 1. Let (F) be a set of functions (f(X)\in W_p^{(l)}(\omega;E_n)) such that

[
\sup_{f\in F}|f|_{W_p^{(l)}(\omega;E_n)}\leq M,
\tag{4}
]

where (\omega(X)) satisfies the conditions:

[
\sup_{Y\in E_n}\int_{\Ш_H^{(n)}(Y)}\cdots\int \omega^{-p'/p}(X)\,dX<\infty;\qquad
\int_{\Ш_H^{(n)}(Y)}\cdots\int \omega^{-p'/p}(X)\,dX\to 0\quad \text{as } |Y|\to\infty,
\tag{5}
]

where (H>0) is a fixed number.

Then, if

[
0\leq s1,\qquad m\text{ is an integer},\qquad 1\leq m\leq n,\qquad
l-s+\frac{m}{q}-\frac{n}{p}>0,
\tag{6}
]

the functions (f\in F), considered on the hyperplane (E_m) of dimension (m), form a set (F') compact in (W_q^{(s)}(E_m)).

Proof. Starting from the inequality

[
\left|\frac{\partial^k f(X)}{\partial x_{i_1}\cdots \partial x_{i_k}}\right|
\leq
\frac{C_1}{h^{n+k}}
\int_{\Ш_h^{(n)}(X)}\cdots\int |f(Y)|\,dY+
]

[
+\sum_{i_1,\ldots,i_l=1}^{n}
C_{i_1\ldots i_l}
\int_0^h \frac{dv}{v^{\,l+n+k-l}}
\left[
\int_{\Ш_v^{(n)}(X)}\cdots\int
\left(
\int_{(E_n)}\cdots\int
\left|
\frac{
\frac{\partial^l f(Y)}{\partial x_{i_1}\cdots \partial x_{i_l}}
-
\frac{\partial^l f(Z)}{\partial z_{i_1}\cdots \partial z_{i_l}}
}{
|Y-Z|^{\,n+(l-l)/p}
}
\right|^p
dZ
\right)^{1/p}
dY
\right]
]

(for integral (l) the second term is written more simply), which is valid for continuously differentiable functions for every integer (k<l), where (h) is an arbitrary positive number, the (C_i) do not depend on (f) and (h), and also from analogous inequalities for differences of derivatives of order (k), it is not difficult to prove that:

1) For any domain (\Omega_n\subset E_n) (in particular, for (\Omega_n=E_n)) the inequality

[
|f|{L_p(\Omega_n)}
\leq
C_2 h^{-n/p'}|f|
\sup_{X\in\Omega_n}
\left(
\int_{\Ш_h^{(n)}(X)}\cdots\int \omega^{-p'/p}(Y)\,dY
\right)^{1/p'}
+
C_3 h^l|f|_{L_p^{(l)}(E_n)}
\tag{7}
]

holds;

2)

[
|f|{W_q^{(s)}(E_m)}
\leq
C_4\left(
|f|\,}^{\,l-s+m/q-n/p
|f|_{W_p^{(l)}(\omega;E_n)}^{\,n/p+s-m/q}
\right)^{1/l}.
\tag{8}
]

It follows from (8) (\bigl(l-s+m/q-n/p>0\bigr)) that it is sufficient to prove the compactness of the set (F) in (L_p(E_n)); and for this it is necessary and sufficient that, for every (R>0), (F) be compact in the metric (L_p(\Ш_R^{(n)}(0))) and that, as (R\to\infty),

[
\sup_{f\in F}|f|_{L_p(E_n-\Ш_R^{(n)}(0))}\to 0.
\tag{9}
]

The first assertion is proved in the usual way, and the second follows from (8) for (\Omega_n=E_n-\Pi_R^{(n)}(0)). Indeed, taking (h\leqslant H), on the basis of (4) we have:

[
|f|{L}(E_n-\Pi_R^{(n)}(0))
\leqslant
C_5 M\left[
h^{-n/p'}\sup_{X\in E_n-\Pi_R^{(n)}(0)}
\left(
\int_{\Pi_H^{(n)}(X)}\cdots\int
\omega^{-p'/p}(Y)\,dY
\right)^{1/p'}
+h^l
\right],
]

whence, by virtue of (5) and the arbitrariness of (h), we obtain (9).

Remark. Put, for any positive number (s),

[
|f|{C^{(s)}(E_n)}
=
\sup|f(X)|
+
\sup_{X\in E_n}|f'(X)|
+\cdots+
\sup_{X\in E_n}|f^{[s]}(X)|
+
\sup_{X,Y\in E_n}
\frac{|f^{[s]}(X)-f^{[s]}(Y)|}{|X-Y|^{\,s-[s]}}.
]

Then, if condition (6) of Theorem 1 is fulfilled in the form

[
0\leqslant s0,
\tag{6'}
]

then the set (E) is compact in (C^{(s)}(E_n)).

Theorem 2. Let (F) be a set of functions (f(X)\in W_p^{(l)}(E_n)) for which

[
|f|_{W_p^{(l)}(E_n)}\leqslant M.
\tag{5'}
]

Let (\bar s, m) be integers, (0\leqslant \bar s1), (l-\bar s+m/q-n/p>0). Further, let (E_m) be a hyperplane of dimension (m); let (\omega(X)) be a positive function defined on (E_m), satisfying the conditions:

[
\text{1) }\quad
\sup_{Y\in E_m}
\int_{\Pi_H^{(m)}(Y)}\cdots\int
\omega(X)\,dX<\infty,
\qquad
\int_{\Pi_H^{(m)}(Y)}\cdots\int
\omega(X)\,dX\to 0
\quad\text{as } |Y|\to\infty;
\tag{10}
]

[
\text{2) there exists a number } \delta>0 \text{ such that}
]

[
\sup_{Y\in E_m}
\left(
\int_{\Pi_H^{(m)}(Y)}\cdots\int
\frac{\omega(X)\,dX}{|X-Y|^{[\,n/p+s+\delta-l\,]q}}
\right)<\infty,
\tag{11}
]

where (H>0) is a fixed number.

Then the derivatives of order (\bar s) of the functions (f\in F), considered on the hyperplane (E_m), form a set (F') compact in (L_q(\omega;E_m)).

For the proof the same methods as above are applied.

Remark. If (l-\bar s-n/p>0), then condition (11) is superfluous (one may put (\delta=l-\bar s-n/p>0), and then (11) coincides with the first of conditions (10)). In this case M. Sh. Birman and B. S. Pavlov obtained necessary and sufficient conditions.

Leningrad Branch
of the V. A. Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
14 VI 1960

REFERENCES

1. A. M. Molchanov, Tr. Mosk. matem. obshch., 2, 169 (1953).
2. M. Sh. Birman, DAN, 125, No. 3 (1959).