Full Text
I. A. Kipriyanov
On a Certain Class of Weighted Embedding Theorems
(Presented by Academician S. L. Sobolev, 13 VI 1962)
Let \(R_n\) denote the \(n\)-dimensional Euclidean space of points
\(x=(x_1,\ldots,x_n)\). Introduce in \(R_n\) spherical coordinates with origin at a certain point
\(x_0=(x_1^{(0)},\ldots,x_n^{(0)})\). Then a function \(f(x)\), defined in the space \(R_n\), may be regarded as a function of \(x_0,\rho,\omega_n\), where by \(\rho\) we denote the distance between the points \(x_0\) and \(x\), and by \(\omega_n\) the collection of angular spherical coordinates. Consider the collection of all functions \(f(x)\) that are finite in the domain \(R_n\) with the point \(x_0\) removed. We call a function \(f(x)\) finite in the domain \(R_n\) with the point \(x_0\) removed if it is continuously differentiable in \(R_n\) and vanishes not only outside some bounded domain (its own for each function \(f(x)\)) for which the point \(x_0\) is an interior point, but also in a neighborhood (its own for each function \(f(x)\)) of the point \(x_0\) itself.
On the collection thus introduced define the differential operator
\[ D_\rho f=\frac{1}{\rho}\sum_{i=1}^{n}\frac{\partial f}{\partial x_i}\cos(\rho,x_i) \tag{1} \]
and powers of this operator
\[ D_\rho^l f=\frac{\partial}{\rho\partial\rho}\left(\frac{\partial^{\,l-1}f}{(\rho\partial\rho)^{l-1}}\right) \qquad (l=2,3,\ldots). \tag{2} \]
Define the functional space \(W_{\rho,2}^{(l)}(R_n)\) as the closure of the set of functions \(f(x)\), finite in the domain \(R_n\) with the point \(x_0\) removed, with respect to the norm
\[ \|f\|_{W_{\rho,2}^{(l)}(R_n)}^{2} = \int_{R_n}|f|^{2}\,dx+ \int_{R_n}\left|\rho^l\frac{\partial^{\,l}f}{(\rho\partial\rho)^l}\right|^{2}\,dx, \tag{3} \]
where \(dx=\rho^{n-1}d\rho\,d\omega_n\), and \(l\) is a positive integer.
Theorem 1. If the function \(f(x)\) belongs to the space \(W_{\rho,2}^{(l)}(R_n)\), then \(\partial^m f/(\rho\partial\rho)^m\) \((m<l)\) is square summable with weight \(\rho^m\) and the inequality
\[ \left\|\rho^m\partial^m f/(\rho\partial\rho)^m\right\|_{L_2(R_n)} \le c_1\|f\|_{W_{\rho,2}^{(l)}(R_n)}. \tag{4} \]
holds.
It follows from this that the norm in \(W_{\rho,2}^{(l)}(R_n)\) defined by the formula
\[ \|f\|_{W_{\rho,2}^{(l)}(R_n)}^{2} = \int_{R_n}|f|^{2}\,dx+ \sum_{k=1}^{l}\int_{R_n}\left|\rho^k\frac{\partial^k f}{(\rho\partial\rho)^k}\right|^{2}\,dx, \tag{5} \]
will be equivalent to the norm introduced above.
On the set of functions \(f(x)\), finite in the domain \(R_n\) with the point \(x_0\) removed, introduce for consideration the operator
\[ D_\rho^{m+\alpha}f = \frac{1}{\rho^{\,n-1-\alpha+m}} \frac{\partial}{\partial\rho} \left[ \frac{1}{\Gamma(1-\alpha)} \int_{0}^{\rho} (\rho^2-\tau^2)^{-\alpha}\tau^m \left( \tau^m \frac{\partial^m f(x_0,\tau,\omega_n)}{(\tau\partial\tau)^m} \right) \tau^{n-1}\,d\tau \right], \tag{6} \]
where \(m\) is a nonnegative integer and \(\alpha\) is a proper fraction \((0<\alpha<1)\).
The functional space \(W_{\rho,2}^{(l+\alpha)}(R_n)\), where \(l\) is a nonnegative integer and \(\alpha\) is a proper fraction \((0<\alpha<1)\), is defined as the closure of the set of functions \(f(x)\), finite in \(R_n\) with the point \(x_0\) removed, with respect to the norm
\[ \|f\|_{W_{\rho,2}^{(l+\alpha)}(R_n)}^{2} = \int_{R_n}|f|^2\,dx + \int_{R_n}|D_{\rho}^{\,l+\alpha} f|^2\,dx . \tag{7} \]
Then the following holds.
Theorem 2. If \(f\in W_{\rho,2}^{(l+\alpha)}(R_n)\), then also \(f\in W_{\rho,2}^{(l)}(R_n)\), and the inequality
\[ \|\rho^l\partial^l f/(\rho\partial\rho)^l\|_{L_2(R_n)} \le c_2\|f\|_{W_{\rho,2}^{(l+\alpha)}(R_n)} \tag{8} \]
is satisfied.
Therefore, in the space \(W_{\rho,2}^{(l+\alpha)}(R_n)\) one may also introduce a norm by the formula
\[ \|f\|_{W_{\rho,2}^{(l+\alpha)}(R_n)}^{2} = \|f\|_{W_{\rho,2}^{(l)}(R_n)}^{2} + \int_{R_n}|D_{\rho}^{\,l+\alpha} f|^2\,dx, \tag{9} \]
which will be equivalent to the norm (7).
Theorem 3. If \(f\in W_{\rho,2}^{(l)}(R_n)\), then also \(f\in W_{\rho,2}^{(l_1)}(R_n)\), where \(l\) and \(l_1\) are positive numbers and \(l_1<l\), and the inequality
\[ \|D_{\rho}^{\,l_1}f\|_{L_2(R_n)} \le c_3\|f\|_{W_{\rho,2}^{(l)}(R_n)} \tag{10} \]
holds.
It follows from the last theorem that in the space \(W_{\rho,2}^{(l+\alpha)}(R_n)\) the norm defined by the formula
\[ \|f\|_{W_{\rho,2}^{(l+\alpha)}(R_n)}^{2} = \|f\|_{W_{\rho,2}^{(l)}(R_n)}^{2} + \sum_{k=0}^{l}\int_{R_n}|D_{\rho}^{\,k+\alpha} f|^2\,dx, \tag{11} \]
will be equivalent to the norm (9). In the case \(l=0\) we understand \(W_{\rho,2}^{(0)}(R_n)\) to mean the space \(L_2(R_n)\).
Represent \(R_n\) as a topological product \(R_n=R_m\times R_{n-m}\). In each of the subspaces \(R_m\) and \(R_{n-m}\) introduce spherical coordinates. Then any point \(x\in R_n\) can be represented as a pair \(x=(x^{(m)},x^{(n-m)})\), and a function \(f(x)\), defined on \(R_n\), can be represented in the form \(f(x)=f(x^{(m)},x^{(n-m)})=f(x_0^{(m)},\rho_1,\omega_m,x_0^{(n-m)},\rho_2,\omega_{n-m})\). Here by \(\rho_1\) we denote the distance between the points \(x_0^{(m)}\) and \(x^{(m)}\) in \(R_m\), by \(\rho_2\) the distance between the points \(x_0^{(n-m)}\) and \(x^{(n-m)}\) in the subspace \(R_{n-m}\), and by \(\omega_m\) and \(\omega_{n-m}\) the collections of angular spherical coordinates in the corresponding subspaces. The points \(x_0^{(m)}\) and \(x_0^{(n-m)}\) are the origins of spherical coordinates, respectively, in the subspaces \(R_m\) and \(R_{n-m}\).
Define the functional space \(W_{\rho_1,2}^{(l_1)}(R_n)\), where \(l_1\) is a nonnegative number, as the closure of the set of functions \(f(x)\), finite in \(R_n\) with the submanifold \(R_{n-m}\) removed, with respect to the norm
\[ \|f\|_{W_{\rho_1,2}^{(l_1)}(R_n)}^{2} = \int_{R_n}|f|^2\,dx + \int_{R_n}|\widetilde D_{\rho_1}^{\,l_1}f|^2\,dx, \tag{12} \]
where, in the case of integer \(l_1\), by \(\widetilde D_{\rho_1}^{\,l_1}f\) one should understand the operator \(\rho_1^{l_1}\partial^{l_1}f/(\rho_1\partial\rho_1)^{l_1}\), while in the case of fractional \(l_1\), by \(\widetilde D_{\rho_1}^{\,l_1}f\) one should understand the operator \(D_{\rho_1}^{\,l_1}f\).
The space \(W_{\rho_2,2}^{(l_2)}(R_n)\) is defined analogously. We shall say that
\[ f(x^{(m)},x^{(n-m)})\in W_{\rho_1,\rho_2,2}^{(l_1,l_2)}(R_n), \]
if it belongs to the intersection of the classes \(W_{\rho_1,2}^{(l_1)}(R_n)\) and \(W_{\rho_2,2}^{(l_2)}(R_n)\). We introduce the norm by the formula
\[ \|f\|_{W_{\rho_1,\rho_2,2}^{(l_1,l_2)}(R_n)}^{2} = \|f\|_{W_{\rho_1,2}^{(l_1)}(R_n)}^{2} + \|f\|_{W_{\rho_2,2}^{(l_2)}(R_n)}^{2}. \tag{13} \]
Theorem 4. Let nonnegative integers \(\alpha\) be given, for which
\[ \mu=\mu(\alpha)=1-2\alpha/l_2-(n-m)/2l_2>0. \tag{14} \]
For such \(\alpha\), define on \(R_m\) functions
\(\varphi^{(\alpha)}(x^{(m)})\in W_{\rho_1,2}^{(\bar l_1)}(R_m)\), \(\bar l_1=\mu l_1\).
There exists a function \(\bar f(x)\in W_{\rho_1,\rho_2,2}^{(l_1,l_2)}(R_n)\) such that
\[ \lim_{\rho_2\to0} \left\| \frac{\partial^\alpha \bar f(x^{(m)},x_0^{(n-m)},\rho_2,\omega_{n-m})} {(\rho_2\partial\rho_2)^\alpha} -\varphi^{(\alpha)}(x^{(m)}) \right\|_{W_{\rho_1,2}^{(\bar l_1)}(R_m)} =0 \tag{15} \]
for all admissible \(\alpha\). Moreover, the inequality
\[ \|\bar f\|_{W_{\rho_1,\rho_2,2}^{(l_1,l_2)}(R_n)} \leq c_4\sum_\alpha \|\varphi^{(\alpha)}(x^{(m)})\|_{W_{\rho_1,2}^{(\bar l_1)}(R_m)} \tag{16} \]
holds, where the positive constant \(c_4\) does not depend on the functions \(\varphi^{(\alpha)}\).
When relation (15) holds, we shall denote the function \(\varphi^{(\alpha)}(x^{(m)})\) by
\(\partial^\alpha f/(\rho_2\partial\rho_2)^\alpha\big|_{\rho_2=0}\).
The converse theorem is also valid:
Theorem 5. Let \(f(x)\in W_{\rho_1,\rho_2,2}^{(l_1,l_2)}(R_n)\), and suppose that for some nonnegative integers \(\alpha\) inequality (14) is satisfied. Then
\(\partial^\alpha f/(\rho_2\partial\rho_2)^\alpha\), as a function of \(x^{(m)}\), with \(\rho_2\) and \(\omega_{n-m}\) fixed in the corresponding manner, belongs to the space
\(W_{\rho_1,2}^{(\bar l_1)}(R_m)\) with \(\bar l_1=\mu l_1\), and the inequality
\[ \left\| \partial^\alpha f/(\rho_2\partial\rho_2)^\alpha\big|_{\rho_2=0} \right\|_{W_{\rho_1,2}^{(\bar l_1)}(R_m)} \leq c_5\|f\|_{W_{\rho_1,\rho_2,2}^{(l_1,l_2)}(R_n)} \tag{17} \]
holds, where \(c_5\) does not depend on \(f\).
Representing \(R_n\) as the topological product \(R_n=R_m\times R_{n-m}\), we introduce spherical coordinates only in the subspace \(R_m\). Then a function \(f(x)\) defined on \(R_n\) is written in the form
\(f(x)=f(x^{(m)},x^{(n-m)})=f(x_0^{(m)},\rho,\omega_m,x_{m+1},\ldots,x_n)\), where \(\rho\) is the distance between \(x_0^{(m)}\) and \(x^{(m)}\) in the subspace \(R_m\), and \(\omega_m\), as usual, denotes the set of angular spherical coordinates in \(R_m\). Let \(l_1,l_{m+1},\ldots,l_n\) be nonnegative numbers. We shall say that
\(f(x)\in W_{\rho,x_{m+1},\ldots,x_n,2}^{(l,l_{m+1},\ldots,l_n)}(R_n)\), if
\(f(x)\in W_{\rho,2}^{(l)}(R_n)\) and if \(f(x)\in W_{x_i,2}^{(l_i)}(R_n)\) for all \(i=m+1,\ldots,n\)
(see definition (1)). As usual, we introduce the norm by the formula
\[ \|f\|_{W_{\rho,x_{m+1},\ldots,x_n}^{(l,l_{m+1},\ldots,l_n)}(R_n)}^2 = \|f\|_{W_{\rho,2}^{(l)}(R_n)}^2 + \sum_{i=m+1}^{n}\|f\|_{W_{x_i,2}^{(l_i)}(R_n)}^2 . \tag{18} \]
Theorem 6. Let nonnegative integers \(r_{m+1},\ldots,r_n\) be given, for which
\[ \mu=\mu(r_{m+1},\ldots,r_n) = 1-\sum_{m+1}^{n}\frac{r_i}{l_i} -\frac{1}{2}\sum_{m+1}^{n}\frac{1}{l_i} >0. \tag{19} \]
For such numbers, define on \(R_m\) functions
\[ \varphi^{(\alpha)}(x^{(m)}) = \varphi^{(r_{m+1},\ldots,r_n)}(x^{(m)}) \in W_{\rho,2}^{(\bar l)}(R_m)\quad \bar l=\mu l. \]
There exists a function
\(\bar f\in W_{\rho,x_{m+1},\ldots,x_n,2}^{(l,l_{m+1},\ldots,l_n)}(R_n)\)
such that
\[ \lim_{x^{(n-m)}\to0} \left\| \partial^\alpha \bar f(x^{(m)},x^{(n+m)})/ \partial x_{m+1}^{r_{m+1}}\cdots \partial x_n^{r_n} - \varphi^{(\alpha)}(x^{(m)}) \right\|_{W_{\rho,2}^{(\bar l)}(R_m)} =0 \tag{20} \]
for all admissible \(r_{m+1},\ldots,r_n\). Moreover,
\[ \|\bar f\|_{W_{\rho,x_{m+1},\ldots,x_n,2}^{(l,l_{m+1},\ldots,l_n)}(R_n)} \leq c_6 \sum_{\alpha}\|\varphi^{(\alpha)}(x^{(m)})\|_{W_{\rho,2}^{(\bar l)}(R_m)}, \tag{21} \]
where \(c_6\) does not depend on \(\varphi^{(\alpha)}\).
The converse theorem is also true:
Theorem 7. Let \(f\in W_{\rho,x_{m+1},\ldots,x_n,2}^{(l,l_{m+1},\ldots,l_n)}(R_n)\), and suppose that for some nonnegative integers \(r_{m+1},\ldots,r_n\) inequality (19) is satisfied. Then the derivatives
\(\partial^\alpha f/\partial x_{m+1}^{r_{m+1}}\cdots \partial x_n^{r_n}\), as functions of \(x^{(m)}\), with the corresponding \(x^{(n-m)}\) fixed in an appropriate way, belong to \(W_{\rho,2}^{(\bar l)}(R_m)\), with \(\bar l=\mu l\). Moreover,
\[ \left\| \frac{\partial^\alpha f}{\partial x_{m+1}^{r_{m+1}}\cdots \partial x_n^{r_n}} \right\|_{W_{\rho,2}^{(\bar l)}(R_m)} \leq c_7 \|f\|_{W_{\rho,x_{m+1},\ldots,x_n,2}^{(l,l_{m+1},\ldots,l_n)}(R_n)}, \tag{22} \]
where the constant \(c_7\) does not depend on \(x^{(n-m)}\) or on \(f(x)\).
The functional space \(W_{x_1,\ldots,x_m,\rho,2}^{(l_1,\ldots,l_m,l)}(R_n)\) is defined analogously for nonnegative numbers \(l_1,\ldots,l_m,l\).
Theorem 8. Let nonnegative integers \(\alpha\) be given for which
\[ \mu=\mu(\alpha)=1-2\alpha/l-(n-m)/2l>0. \tag{23} \]
For such \(\alpha\), prescribe on \(R_m\) functions
\[ \varphi^{(\alpha)}(x^{(m)})\in W_{x_1,\ldots,x_m,2}^{(\bar l_1,\ldots,\bar l_m)}(R_m), \qquad \bar l_i=\mu l_i \quad (i=1,2,\ldots,m). \]
There exists a function \(\bar f\in W_{x_1,\ldots,x_m,\rho,2}^{(l_1,\ldots,l_m,l)}(R_n)\) such that
\[ \lim_{\rho\to 0} \left\| \frac{\partial^\alpha \bar f(x^{(m)},x_0^{(n-m)},\rho,\omega_{n-m})}{(\rho\,\partial\rho)^\alpha} -\varphi^{(\alpha)}(x^{(m)}) \right\|_{W_{x_1,\ldots,x_m,2}^{(\bar l_1,\ldots,\bar l_m)}(R_m)} =0 \tag{24} \]
for all admissible \(\alpha\). Moreover,
\[ \|\bar f\|_{W_{x_1,\ldots,x_m,\rho,2}^{(l_1,\ldots,l_m,l)}(R_n)} \leq c_8 \sum_{\alpha} \|\varphi^{(\alpha)}(x^{(m)})\|_{W_{x_1,\ldots,x_m,2}^{(\bar l_1,\ldots,\bar l_m)}(R_n)}, \tag{25} \]
where the constant \(c_8\) does not depend on \(\varphi^{(\alpha)}(x^{(m)})\).
The converse assertion is also valid:
Theorem 9. Let \(f(x)\in W_{x_1,\ldots,x_m,\rho,2}^{(l_1,\ldots,l_m,l)}(R_n)\), and suppose that for some nonnegative integers \(\alpha\) inequality (23) is satisfied. Then \(\partial^\alpha f/(\rho\partial\rho)^\alpha\), as functions of \(x^{(m)}\), with the corresponding \(\rho\) and \(\omega_{n-m}\) fixed in an appropriate way, belong to
\[ W_{x_1,\ldots,x_m,2}^{(\bar l_1,\ldots,\bar l_m)}(R_m) \quad\text{with}\quad \bar l_i=\mu l_i \quad (i=1,2,\ldots,m). \]
Moreover, the inequality
\[ \left\| \partial^\alpha f/(\rho\partial\rho)^\alpha\big|_{\rho=0} \right\|_{W_{x_1,\ldots,x_m,2}^{(\bar l_1,\ldots,\bar l_m)}(R_n)} \leq c_9 \|f\|_{W_{x_1,\ldots,x_m,\rho,2}^{(l_1,\ldots,l_m,l)}(R_n)} \tag{26} \]
holds, where \(c_9\) does not depend on the function \(f(x)\).
In conclusion, we note that results analogous to those given above also hold for bounded domains in \(R_n\).
Received
8 VI 1962
CITED LITERATURE
- L. N. Slobodetskii, DAN, 118, No. 2 (1958).