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K. K. Golovkin and V. A. Solonnikov
EMBEDDING THEOREMS FOR FRACTIONAL SPACES
(Presented by Academician V. I. Smirnov, 15 XI 1961)
In the present paper direct and inverse embedding theorems are given for a broad class of functional spaces that contains a number of previously studied spaces. These theorems are a development of the results of S. M. Nikol’skii \((^1)\), O. V. Besov \((^2)\), and the authors \((^{3,4})\). In method the present paper is closely connected with the works of the authors \((^{3,4})\).
We study spaces of functions whose norm has the form
\[ \sum_{i=1}^{n} \left\{ I\left[ \frac{1}{h^{r_i-r'_i}} \left\| \Delta_{h,i}^{\,r''_i-r'_i} \frac{\partial^{r'_i}u}{\partial x_i^{r'_i}} \right\|_{p_i,s} \right] + \|u\|_{p_i,n} \right\} \qquad (p_i\geq 1). \tag{1} \]
Here \(r_i\) are positive numbers; \(r'_i\) is the greatest integer less than \(r_i\); \(r''_i\) is the least integer greater than \(r_i\); \(\|\cdot\|_{p,n}\) is the norm in \(L_p\) over \(n\)-dimensional space. \(\|u\|_{p,s}\) is defined as
\[ \sup_{x_{s+1},\ldots,x_n} \left\{ \int |u(x_1,\ldots,x_s,x_{s+1},\ldots,x_n)|^p\,dx_1\ldots dx_s \right\}^{1/p}. \]
\(I\) is a functional of maximization type (abbreviated f.m.t.). This notion was introduced in \((^3)\). Here we formulate another definition, equivalent to the previous one.
We shall say that \(I[\psi(h)]\) is an f.m.t. if it is defined on nonnegative functions \(\psi(h)\) of the positive argument \(h\) by the equality
\[ I[\psi(h)] = N[u(x)], \]
where \(u(x)=\psi(e^x)\); \(N\) is a norm defined on functions given on the whole line and possessing the following properties: 1) \(N[u(x+\xi)] = N[u(x)]\), \(-\infty<\xi<\infty\); 2) from the inequality \(u_1(x)\geq u_2(x)\geq 0\) it follows that \(N[u_1(x)]\geq N[u_2(x)]\).
Examples of f.m.t.’s are
\[ I_{\rho,p}[\psi(h)] = \sup_{s>0} \left\{ \int_{0}^{\infty} \frac{\rho(hs)\psi^p(h)}{h}\,dh \right\}^{1/p}, \tag{2} \]
where \(\rho(h)\) is a measurable function satisfying the inequalities \(1\geq \rho(h)\geq 0\). In the case when \(\rho(h)\equiv 1\), we shall denote the functionals (2) simply by \(I_p[\psi(h)]\).
Other examples of f.m.t.’s may be obtained by taking Orlicz norms for \(N\). An important subclass of f.m.t.’s is formed by those that are generated by \(N\)-norms invariant with respect to any transformation of \(x\) preserving measure. We shall say that these f.m.t.’s have property \(T\). The \(I_p\) have property \(T\). If an f.m.t. \(I\) has property \(T\), then
\[ I[\psi(h^\alpha)] \sim I[\psi(h)] \quad \text{for any } \alpha>0. \]
The main role in the proof of the direct embedding theorems is played by Theorem 1 of [1] and by the following easily proved lemma.
Lemma. If \(I\) has property \(T\), and \(\|\cdot\|\) is invariant with respect to arbitrary shifts of the arguments of the function \(u(x_1,\ldots,x_n)\), then for sufficiently small \(\delta\) the inequality
\[ \sum_{i=1}^{n} I\left[\frac{1}{h^{r_i}}\left\|\Delta_{h,i}^{l}u\right\|\right] \le C \sum_{i=1}^{n} I\left[\frac{1}{h^{r_i}}\left\|\Delta_{h,i}^{l}u_{\delta h x_1^i,\ldots,\delta h x_n^i}^{x_i}\right\|\right], \]
holds, where \(x_j^i=r_i/r_j\); \(u_{\eta_1,\ldots,\eta_n}\) is an average of \(u(x_1,\ldots,x_n)\) constructed with a kernel having a sufficiently large number of zero moments (beginning with the first) and with parameters \(\eta_1,\ldots,\eta_n\).
Such “high-quality” averages of functions play in our method the same role as approximations by entire functions of finite degree in the method of S. M. Nikol’skii.
Denote by \(H_{J;p_1,\ldots,p_n;s}^{r_1,\ldots,r_n}\) \((s\le n)\) the closure, in the norm (1), of sufficiently smooth functions defined in the whole space. In the case \(p_1=p_2=\cdots=p_n=p\) we denote this space by \(H_{J;p;s}^{r_1,\ldots,r_n}\).
We proceed to the formulation of the embedding theorems proved by us.
Theorem 1. Let \(u(x)\in H_{J;p_1,\ldots,p_n}^{r_1,\ldots,r_n}\); let \(I\) have property \(T\), and let the numbers
\[ x_i=1-\sum_{j=1}^{n}\left(\frac{1}{p_j}-\frac{1}{p_i}\right)\frac{1}{r_j} \]
be positive. Then, if \(q\ge p_i\) and
\[ x=1-\sum_{j=1}^{n}\left(\frac{1}{p_j}-\frac{1}{q}\right)\frac{1}{r_j}>0, \]
then \(u\in H_{J;q;n}^{r_1',\ldots,r_n'}\), where \(r_i'=\dfrac{x r_i}{x_i}\).
Remark. This theorem remains valid if the metrics \(L_{p_i}\) and \(L_q\) are replaced respectively by
\[ \sup_{-\infty<z_1,\ldots,z_n<\infty}\|v(x+z)\chi_M(x)\|_{L_{p_i}}, \qquad \sup_{-\infty<z_1,\ldots,z_n<\infty}\|v(x+z)\chi_M(x)\|_{L_p}, \]
where \(\chi_M(x)\) is the characteristic function of an arbitrary open set \(M\).
Theorem 2. Let \(u(x)\in H_{J;p;n}^{r_1,\ldots,r_n}\); let \(I\) have property \(T\). Then, if
\[ x=1-\frac{1}{p}\sum_{j=1}^{n}\frac{1}{r_j} +\frac{1}{q}\sum_{j=1}^{n}\frac{1}{r_j} \]
lies between \(0\) and \(1\) and \(q\ge p\), then, for \(r_i'=x r_i\),
\[ u\in H_{I;q;s}^{r_1,\ldots,r_n}. \]
Remark. If in Theorems 1 and 2 \(r_1=r_2=\cdots=r_n\) and \(p_1=\cdots=p_n\), then \(I\) need not have property \(T\).
It seems necessary also to clarify which mixed partial derivatives exist for functions to which Theorems 1 and 2 are applicable.
This question is resolved by the following theorem.
Theorem 3. Let
\[ I\left[\frac{1}{h^{r_i}}\left\|\Delta_{h,i}^{m_i}u\right\|\right]<\infty, \]
where \(m_i>r_i>0\); \(I\) is an arbitrary f.t.m. having property \(T\); the norm \(\|\cdot\|\) is the same as in the lemma. Let the point \((l_1,\ldots,l_n)\) lie in the space \(E_n\) on the same side of the plane intersecting the coordinate axes at the points \(r_i\) as the origin of coordinates. Put
\[ s_i=\left(1-\sum_{j=1}^{n}\frac{l_j}{r_j}\right)r_i. \]
Then, if
\[ I\left[\frac{\varepsilon(h)}{h^{s_i+l_1+\cdots+l_n}} \left\|\Delta_{h,1}^{l_1}\cdots\Delta_{h,n}^{l_n}\Delta_{h,i}^{k_i}u\right\|\right]<\infty, \]
where
\[
\varepsilon(h)=
\begin{cases}
1, & h\geqslant 1,\\
0, & h<1,
\end{cases}
\]
and, by assumption, \(k_i>s_i\), the estimate holds
\[
\sum_{i=1}^{n} I\left[\frac{1}{h^{s_i}}\left\|\Delta_{h,i}^{k_i}
\frac{\partial^{l_1+\cdots+l_n}u}{\partial x_1^{l_1}\cdots \partial x_n^{l_n}}\right\|\right]
\leqslant
C\sum_{i=1}^{n} I\left[\frac{1}{h^{r_i}}\left\|\Delta_{h,i}^{m_i}u\right\|\right].
\]
Corollary. If \(u\in H_{I;p;s}^{r_1,\ldots,r_n}\) and
\[
\sum_{j=1}^{n}\frac{l_j}{r_j}<1,
\]
then
\[
\partial^{l_1+\cdots+l_n}u/\partial x_1^{l_1}\cdots \partial x_n^{l_n}
\in H_{J;p;s}^{s_1,\ldots,s_n},
\]
where
\[
s_k=r_k\left(1-\sum_{j=1}^{n}\frac{l_j}{r_j}\right).
\]
Finally, let us give an embedding theorem for functional spaces that differ only in the functional of maximization type.
Theorem 4. If \(J\) is an arbitrary f.t.m., and the norm \(\|\cdot\|\) is the same as in the lemma, then the estimate holds
\[
J\left[\frac{1}{h^{r-r'}}\left\|\Delta_{h,i}^{r''-r'}
\frac{\partial^{r'}u}{\partial x_i^{r'}}\right\|\right]
\leqslant
CI_1\left[\frac{1}{h^{r-r'}}\left\|\Delta_{h,i}^{r''-r'}
\frac{\partial^{r'}u}{\partial x_i^{r'}}\right\|\right],
\]
\[
C=28I[\chi(h,1,e)];\qquad
\chi(h,1,e)=
\begin{cases}
1, & h\in(1,e),\\
0, & h\bar{\in}(1,e).
\end{cases}
\]
If, in addition, \(I\) has property \(T\), then
\[
I_{\infty}\left[\frac{1}{h^{r-r'}}\left\|\Delta_{h,i}^{r''-r'}
\frac{\partial^{r'}u}{\partial x_i^{r'}}\right\|\right]
\leqslant
CI\left[\frac{1}{h^{r-r'}}\left\|\Delta_{h,i}^{r''-r'}
\frac{\partial^{r'}u}{\partial x_i^{r'}}\right\|\right],
\]
\[
C=15I[\chi(h,1,2)].
\]
Corollary. If \(I\) is any f.t.m., then
\[
H_{I;p;s}^{r_1,\ldots,r_n}\subset H_{I;p;r}^{r_1,\ldots,r_n}.
\]
If \(I\) is an f.t.m. with properties \(T\), then
\[
H_{I;p;s}^{r_1,\ldots,r_n}\subset H_{I_\infty;p;s}^{r_1,\ldots,r_n}.
\]
We now pass to the formulation of the extension theorem and the inverse embedding theorems.
Let the function \(u(x)\) be given in the half-space \(x_i\geqslant 0\) and belong there to the space
\[
H_{I;p_1,\ldots,p_n}^{r_1,\ldots,r_n},
\]
i.e.,
\[
\sum_{j=1}^{n}\left\{
I\left[\frac{1}{h^{r_j-r'_j}}\left\|
\Delta_{h,j}^{r''_j-r'_j}
\frac{\partial^{r'_j}u}{\partial x_j^{r'_j}}
\right\|_{L_{p_j}(x_j\geqslant 0)}\right]
+\|u\|_{L_{p_j}(x_j\geqslant 0)}
\right\}<\infty.
\]
Theorem 5. There exists an extension of the function \(u(x)\) to the whole space \(E_n\) such that
\[
\sum_{j=1}^{n}\left\{
I\left[\frac{1}{h^{r_j-r'_j}}\left\|
\Delta_{h,j}^{r''_j-r'_j}
\frac{\partial^{r'_j}u}{\partial x_j^{r'_j}}
\right\|_{L_{p_j}(x_j\geqslant 0)}\right]
+\|u\|_{L_{p_j}(x_j\geqslant 0)}
\right\}
\leqslant
\]
\[
\leqslant
C\left\{
\sum_{j=1}^{n}
I\left[\frac{1}{h^{r_j-r'_j}}\left\|
\Delta_{h,j}^{r''_j-r'_j}
\frac{\partial^{r'_j}u}{\partial x_j^{r'_j}}
\right\|_{L_{p_j}(E_n)}\right]
+\|u\|_{L_{p_j}(E_n)}
\right\}.
\]
This extension can be constructed by the well-known method of Hestenes and Whitney.
Theorem 6. Let \(0<s<n\), \(p>1\). Let also a sequence of positive numbers \(l_k\) \((k=1,\ldots,n)\) and \(N\) distinct
sequences of positive integers \(\nu_j^{(i)}\) \((j=s+1,\ldots,n;\ i=1,\ldots,N)\) such that
\[ \mu_i=1-\sum_{j=s+1}^{n}\frac{\nu_j^{(i)}}{l_j} -\frac{1}{p}\sum_{j=s+1}^{n}\frac{1}{l_j}>0. \]
Then, for the functions \(\varphi^{(i)}(x_1,\ldots,x_s)\in H_{l;p;s}^{l_{1,i},\ldots,l_{s,i}}\) prescribed on the hyperplane \(E_s\) \((x_{s+1}=\cdots=x_n=0)\), one can construct a function \(\varphi(x_1,\ldots,x_n)\) in the space \(E_n\) such that \(\varphi\in H_{l;p;n}^{l_1,\ldots,l_n}\),
\[ \left. \frac{\partial^{\nu_{s+1}^{(i)}+\cdots+\nu_n^{(i)}}\varphi} {\partial x_{s+1}^{\nu_{s+1}^{(i)}}\cdots \partial x_n^{\nu_n^{(i)}}} \right|_{x\in E_s} = \varphi^{(i)}(x_1,\ldots,x_s), \]
\[ \sum_{j=1}^{n} I\left[ \frac{1}{h^{l_j}}\left\|\Delta_{h,j}^{m}\varphi\right\|_{L_p(E_n)} \right] +\|\varphi\|_{L_p(E_n)} \leq \]
\[ \leq C\sum_{i=1}^{N} \left\{ \sum_{j=1}^{n} I\left[ \frac{1}{h^{l_{j,i}}} \left\|\Delta_{h,j}^{m}\varphi^{(i)}\right\|_{L_p(E_s)} \right] + \left\|\varphi^{(i)}\right\|_{L_p(E_s)} \right\}, \]
where \(l_{j,i}=l_j\mu_i\), and \(I\) is a functional of maximization type possessing property \(T\).
Theorem 6 reduces to its particular case \(s=n-1\). Further, by virtue of Theorem 5, the function \(\varphi\) can be constructed not in the whole space \(E_n\), but in the half-space \(x_n\geq 0\). This construction can be carried out in explicit form, with the function \(\varphi\) expressed as a sum of integral operators of the form
\[ \frac{1}{x_n^{\sigma_i}} \int_{0}^{x_n^{\alpha_1}}\cdots \int_{0}^{x_n^{\alpha_{n-1}}} K_i(y,x_n)\,\varphi_i(x+y)\,dy. \]
Leningrad Branch
of the V. A. Steklov Mathematical Institute
Academy of Sciences of the USSR
Received
18 X 1961
REFERENCES
- S. M. Nikol’skii, Tr. Mat. Inst. im. V. A. Steklova AN SSSR, 38, 244 (1951).
- O. V. Besov, DAN, 126, No. 6 (1959).
- K. K. Golovkin, DAN, 139, No. 3 (1961).
- V. P. Il’in, V. A. Solonnikov, DAN, 136, No. 3 (1961).