Abstract Generated abstract
The paper develops a class of weighted Sobolev type spaces in Euclidean space defined by radial differential operators, including fractional radial derivatives, with weights determined by distance from a point or submanifold. It proves embedding and norm equivalence results showing that lower order weighted derivatives are controlled by the defining norms, and extends these results to product decompositions of Euclidean space with mixed radial and Cartesian differentiability. The main contribution is a set of trace and extension theorems for functions and derivatives on lower dimensional subspaces, giving precise admissibility conditions and norm estimates for the corresponding boundary data. Analogous results are noted for bounded domains.
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).