Abstract Generated abstract
The paper introduces and studies anisotropic function spaces defined by finite differences with fractional smoothness indices, including variants of Besov type and simultaneous spaces formed by intersecting over coordinate subsets. It proves direct embedding theorems for derivatives of functions in these spaces, giving conditions under which restrictions to lower-dimensional variables belong to corresponding spaces with new integrability and smoothness parameters. The paper also defines weighted fractional spaces on the half-space and establishes trace and extension theorems for restrictions to coordinate hyperplanes, with norm estimates. These results extend earlier embedding, trace, and extension results associated with Nikol’skii, Besov, Il’in, Solonnikov, and Kudryavtsev.
Full Text
A. D. DZHABRAILOV
ON CERTAIN FUNCTION SPACES
DIRECT AND INVERSE EMBEDDING THEOREMS
(Presented by Academician I. M. Vinogradov on 26 VI 1964)
We shall assume (following the notation of S. M. Nikol’skii \((^3,^8)\)) that \(e\) is a subset of the natural numbers \(e_n=\{1,\ldots,n\}\). If \(K=(k_1,\ldots,k_n)\) is a given vector, then let \(K^e=(k_1^e,\ldots,k_n^e)\), where
\[ k_j^e= \begin{cases} k_j, & \text{for } j\in e,\\ 0, & \text{for } j\in e_n\setminus e. \end{cases} \]
We shall denote by \(e_k\) the support of the vector \(K\), i.e. the smallest subset \(e\subset e_n\) for which \(K^e=K\). Let a vector \(r=(r_1,\ldots,r_n)\) with nonnegative components be given. Put \(r_j=\bar r_j+\alpha_j\), where \(\bar r_j\) is the greatest integer less than \(r_j\), so that \(0<\alpha_j\leq 1\), and if \(r_j=0\), then \(\bar r_j=0\). Thus the vector \(r\) uniquely determines the vector \(\bar r=(\bar r_1,\ldots,\bar r_n)\).
Let \(f(x)\) be a smooth function defined in \(E^n=\{x=(x_1,\ldots,x_n)\}\). Denote by \(f^{(\bar r_\tau,\bar r^e)}(x)\) the partial derivative of the function \(f(x)\) of order \((\bar r_\tau,\bar r^e)\) with respect to the variables \((x_\tau,x^e)\), respectively, where \(e\) is any fixed subset of the set \(e_r\), and \(\tau\in e_r\setminus e\). The order of differentiation is arbitrary, for example:
\[ f^{(\bar r_\tau,\bar r^e)}(x) = \frac{\partial^{\bar r_\tau}}{\partial x_\tau^{\bar r_\tau}} \frac{\partial^{\bar r_1^e}}{\partial x_1^{\bar r_1^e}} \cdots \frac{\partial^{\bar r_n^e}}{\partial x_n^{\bar r_n^e}} f(x). \]
For any \(\tau\in e_r\setminus e\) put
\[ \|f(x)\|_{\mathcal L^{\bar r_\tau,\bar r^e}_{p,x_\tau,x^e}(E^n)} = \left( \int_0^\infty \frac{dt_\tau}{t_\tau^{1+p\alpha_\tau}} \prod_{j\in e} \int_0^\infty \frac{ \left\| \Delta_{\tau}^{2,\omega^e}\Delta_{t_\tau,t^e}^{(\bar r_\tau,\bar r^e)} f(x) \right\|_{L_p(E^n)}^p }{ t_j^{1+p\alpha_j} } \,dt_j \right)^{1/p}, \]
where \(\omega=(1,\ldots,1)\) is a vector whose components consist only of ones, \(\Delta_{\tau}^{2,2\omega^e}\Delta_{t_\tau,t^e} f\) is the finite difference of order \(2\) with respect to the variable \(x_\tau\) and of order \(2\omega^e\) with respect to the variables \(x^e\), with steps \(t_\tau\) and \(t^e\), respectively; \(1\leq p\leq\infty\).
Definition 1. The space \(\mathcal L_p^{(r)}(e;E^n)\) will mean the closure of the set of smooth finite functions in the norm
\[ \|f(x)\|_{\mathcal L_p^{(r)}(e;E^n)} = \sum_{\tau\in e_r\setminus e} \|f(x)\|_{\mathcal L^{\bar r_\tau,\bar r^e}_{p,x_\tau,x^e}(E^n)}. \tag{1} \]
If \(e_0\) is the empty set and \(e_r=e_n\), then the space \(\mathcal L_p^{(r)}(e_0,E^n)\) is the space \(\mathcal L_p^{(r_1,\ldots,r_n)}(E^n)\), which was considered earlier by V. P. Il’in, V. A. Solonnikov \((^6)\), and others.
Definition 2. The space \(B_{p_0,p}^{(r)}(e;E^n)\) will mean the closure of the set of smooth finite functions in the norm
\[ \|f(x)\|_{B_{p_0,p}^{(r)}(e;E^n)} = \|f(x)\|_{L_{p_0}(E^n)} + \|f(x)\|_{\mathcal L_p^{(r)}(e;E^n)}, \tag{2} \]
where \(\|f(x)\|_{L_{p_0}(E^n)}\) is the norm of the function in \(L_{p_0}(E^n)\) \((1\leq p_0\leq\infty)\).
If \(e_0\) is the empty set and \(e_r=e_n\), then \(B_{p_0,p}^{(r)}(e_0;E^n)\) coincides with the well-known space
\(B_{p_0,p,\ldots,p}^{r_1,\ldots,r_n}(E^n)\), defined for \(p_0=p\) by O. V. Besov \((^7)\), the theory of which was later developed by V. P. Il’in \((^5)\), and others.
Definition 3. We shall say that a function \(f(x)\) belongs to the space \(S_p^{(r)}\mathcal L(E^n)\) if it belongs simultaneously to all \(\mathcal L_p^{(r)}(e;E^n)\) for every \(e\) from \(e_r\). We define the norm in this space as follows:
\[
\|f(x)\|_{S_p^{(r)}\mathcal L(E^n)}
=
\sum_{e\subset e_r}\|f(x)\|_{\mathcal L_p^{(r)}(e;E^n)}.
\tag{3}
\]
Definition 4. We shall say that a function \(f(x)\) belongs to the space \(S_{p_0,p}^{(r)}B(E^n)\) if it belongs simultaneously to all \(B_{p_0,p}^{(r)}(e;E^n)\) for every \(e\) from \(e_r\). We define the norm in this space as follows:
\[
\|f(x)\|_{S_{p_0,p}^{(r)}B(E^n)}
=
\sum_{e\subset e_r}\|f(x)\|_{B_{p_0,p}^{(r)}(e;E^n)}.
\tag{4}
\]
It is obvious that the norm (4) and the following norm are equivalent:
\[
\|f(x)\|_{S_{p_0,p}^{(r)}B(E^n)}^{*}
=
\|f(x)\|_{L_{p_0}(E^n)}
+
\|f(x)\|_{S_p^{(r)}\mathcal L(E^n)}.
\tag{5}
\]
Put \(B_p^{(r)}(e;E^n)=B_{p,p}^{(r)}(e;E^n)\), \(S_p^{(r)}B(E^n)=S_{p,p}^{(r)}B(E^n)\). Some embedding theorems have been obtained for the classes \(B_p^{(r)}(e;E^n)\) and \(S_p^{(r)}B(E^n)\), which are a development of S. M. Nikol’skii’s results for the classes \(S_p^{(r)}H\) and \(S_p^{(r)}W\).
Theorem 1. Let \(f(x)\in S_p^{(r)}B(E^n)\), where \(r_i>0\) \((i=1,\ldots,n)\), \(p>1\). Let natural numbers \(\nu_i\) \((i=1,\ldots,n)\) and \(m\) \((0<m\le n)\) be given, as well as numbers \(q\) \((p\le q\le\infty)\), \(\rho_s\) \((s=1,\ldots,m)\), and suppose that the conditions
\[
\varepsilon
=
1-\frac1p\sum_{j=1}^n\frac1{r_j}
-\sum_{j=1}^n\frac{\nu_j}{r_j}
+\frac1q\sum_{j=1}^m\frac1{r_j}
>0,
\tag{A}
\]
\[
0<\rho_s\le \varepsilon r_s
\qquad (s=1,\ldots,m).
\tag{B}
\]
are satisfied.
Let \(e\) be any fixed subset of the set \(e_n\). Then, for any fixed \(x_{m+1},\ldots,x_n\), the function \(f^{(\nu)}(x)\in \mathcal L_q^{(\rho)}(e\cap e_m;E^m)\) and the inequality
\[
\|f^{(\nu)}(x)\|_{\mathcal L_q^{(\rho)}(e\cap e_m;E^m)}
\le
C\|f\|_{S_p^{(r)}B(E^n)}
\tag{6}
\]
holds, where \(\nu=(\nu_1,\ldots,\nu_n)\), \(e_m=\{1,\ldots,m\}\).
Theorem 2. Let \(f(x)\in S_p^{(r)}B(E^n)\), where \(r_i>0\) \((i=1,\ldots,n)\), \(p>1\). Let natural numbers \(\nu_i\) \((i=1,\ldots,n)\) and \(m\) \((0<m\le n)\) be given, as well as numbers \(q\) \((p\le q\le\infty)\), \(\rho_s\) \((s=1,\ldots,m)\), and let the conditions (A), (B) of Theorem 1 be satisfied. Then, for any fixed \(x_{m+1},\ldots,x_n\), the function \(f^{(\nu)}(x)\in S_q^{(\rho)}\mathcal L(E^m)\) and the inequality
\[
\|f^{(\nu)}(x)\|_{S_q^{(\rho)}\mathcal L(E^m)}
\le
C\|f\|_{S_p^{(r)}B(E^n)}
\tag{7}
\]
holds.
Theorem 3. Let \(f\in S_p^{(r)}B(E^n)\), where \(r_i\) \((i=1,\ldots,n)\), \(p>1\). Let natural numbers \(\nu_i\) \((i=1,\ldots,n)\) and \(m\) \((0<m\le n)\) be given, as well as numbers \(q\) \((p\le q\le\infty)\), \(\rho_s\) \((s=1,\ldots,m)\), and let the conditions (A), (B) of Theorem 1 be satisfied. Further, let \(e\) be any fixed subset of the set \(e_n\). Then the function \(f^{(\nu)}(x)\in B_q^{(\rho)}(e\cap e_m;E^m)\) and the inequality
\[
\|f^{(\nu)}(x)\|_{B_q^{(\rho)}(e\cap e_m;E^m)}
\le
C\|f\|_{S_p^{(r)}B(E^n)}
\tag{8}
\]
holds.
Theorem 4. Let \(f \in S_p^{(r)}B(E^n)\), where \(r_i>0\) \((i=1,\ldots,n)\), \(p>1\). Let natural numbers \(\nu_i\) \((i=1,\ldots,n)\) and \(m\) \((0<m\le n)\), as well as numbers \(q\) \((p\le q\le \infty)\), \(\rho_s\) \((s=1,\ldots,m)\), be given, and suppose that the conditions (A), (B) of Theorem 1 are satisfied. Then, for any fixed \(x_{m+1},\ldots,x_n\), the function \(f^{(\nu)}(x)\in S_q^{(\rho)}B(E^m)\), and the inequality
\[ \bigl\| f^{(\nu)}(x)\bigr\|_{S_q^{(\rho)}B(E^m)} \le C \bigl\| f(x)\bigr\|_{S_p^{(r)}B(E^n)} . \tag{9} \]
holds.
We now introduce weighted spaces with fractional indices and formulate for them some embedding and extension theorems.
Let \(f(x)\) be a smooth function given in the space \(\bar E^{+n}=\{x=(x_1,\ldots,x_n), x_n\ge 0\}\); \(\Delta_i^2(t)f(x)\) is the finite difference of order 2 with respect to the variable \(x_i\) with step \(t\). Let \(1\le p<\infty\), \(0<r_i<1\) \((i=1,\ldots,n)\). Introduce the following norms:
\[ \bigl\|f(x)\bigr\|_{\mathscr L_{p,x_i,\alpha}^{\,r_i}(\bar E^{+n})} = \left( \int_0^\infty \frac{dt}{t^{1+pr_i}} \int_{\bar E^{+n}} x_n^\alpha \bigl|\Delta_i^2(t)f(x)\bigr|^p\,d\bar E^{+n} \right)^{1/p}, \]
\[ \bigl\|f(x)\bigr\|_{\mathscr L_{p,\alpha}^{(r_1,\ldots,r_n)}(\bar E^{+n})} = \sum_{i=1}^n \bigl\|f(x)\bigr\|_{\mathscr L_{p,x_i,\alpha}^{\,r_i}(\bar E^{+n})}, \]
\[ \bigl\|f(x)\bigr\|_{B_{p,\alpha}^{(r_1,\ldots,r_n)}(\bar E^{+n})} = \bigl\|f(x)\bigr\|_{L_p(\bar E^{+n})} + \bigl\|f(x)\bigr\|_{\mathscr L_{p,\alpha}^{(r_1,\ldots,r_n)}(\bar E^{+n})}. \]
Definition 5. By the spaces \(\mathscr L_{p,\alpha}^{(r_1,\ldots,r_n)}(\bar E^{+n})\) and \(B_{p,\alpha}^{(r_1,\ldots,r_n)}(\bar E^{+n})\) we shall mean the closures of the set of smooth finite functions, respectively, in the norms \(\|f\|_{\mathscr L_{p,\alpha}^{(r_1,\ldots,r_n)}(\bar E^{+n})}\), \(\|f\|_{B_{p,\alpha}^{(r_1,\ldots,r_n)}(\bar E^{+n})}\).
Theorem 5. Let \(p>1\), \(0<r_i<1\) \((i=1,\ldots,n)\), and let \(m\) be a natural number such that \(0<m<n\) and
\[ \varepsilon = 1-\frac1p\sum_{j=m+1}^n \frac1{r_j} -\frac{\alpha}{pr_n} >0. \]
Then, if \(f(x)\in B_{p,\alpha}^{(r_1,\ldots,r_n)}(\bar E^{+n})\), then for any fixed \(x_{m+1},\ldots,x_{n-1}\) and \(x_n=0\) the function \(f(x)\in L_p(E^m)\), and the inequality
\[ \|f\|_{L_p(E^m)} \le C \|f\|_{B_{p,\alpha}^{(r_1,\ldots,r_n)}(\bar E^{+n})}. \tag{10} \]
holds.
Theorem 6. Let \(p>1\), \(0<r_i<1\) \((i=1,\ldots,n)\), and let \(m\) be a natural number such that \(0<m<n\), and suppose the conditions
\[ \varepsilon = 1-\frac1p\sum_{m+1}^n \frac1{r_i} -\frac{\alpha}{pr_n} >0, \qquad 0<\rho_s\le \varepsilon r_s \quad (s=1,\ldots,m). \]
are satisfied. Then, if the function \(f\in B_{p,\alpha}^{(r_1,\ldots,r_n)}(\bar E^{+n})\), then for any fixed \(x_{m+1},\ldots,x_{n-1}\) and \(x_n=0\) the function
\[ f(x)\in \mathscr L_p^{(\rho_1,\ldots,\rho_m)}(E^m), \]
and, moreover, the inequality
\[ \|f\|_{\mathscr L_p^{(\rho_1,\ldots,\rho_m)}(E^m)} \le C \|f\|_{B_{p,\alpha}^{(r_1,\ldots,r_n)}(\bar E^{+n})}, \tag{11} \]
holds; and if \(\rho_s=\varepsilon r_s\) \((s=1,\ldots,m)\), then
\[ \|f\|_{\mathscr L_p^{(\rho_1,\ldots,\rho_m)}(E^m)} \le C \|f\|_{\mathscr L_{p,\alpha}^{(r_1,\ldots,r_n)}(\bar E^{+n})}. \tag{12} \]
As a consequence of Theorems 5 and 6 we obtain that, under the conditions of Theorem 6, if
\(f \in B_{p,\alpha}^{(r_1,\ldots,r_n)}(\bar E^n_+)\), then for any fixed \(x_{m+1},\ldots,x_{n-1}\) and \(x_n=0\) it also belongs to the space \(B_p^{(\rho_1,\ldots,\rho_m)}(E^m)\), and the inequality
\[ \|f\|_{B_p^{(\rho_1,\ldots,\rho_m)}(E^m)} \leq C\|f\|_{B_{p,\alpha}^{(r_1,\ldots,r_n)}(\bar E^n_+)} \tag{13} \]
holds.
Theorem 7. Let \(p>1\), \(0<r_i<1\) \((i=1,\ldots,n)\), let \(m\) be a natural number such that \(0<m<n\), \(\alpha>-1\), and
\[ \varepsilon = 1-\frac1p\sum_{j=m+1}^n\frac1{r_j} -\frac{\alpha}{pr_n} >0. \]
Then, if on the hyperplane \(x_{m+1}=\cdots=x_n=0\) a function
\(\varphi(x_1,\ldots,x_m)\in \mathscr L_p^{(\rho_1,\ldots,\rho_m)}(E^m)\) is given, with \(\rho_s=\varepsilon r_s\) \((s=1,\ldots,m)\), there exists a function
\(f(x_1,\ldots,x_n)\in \mathscr L_{p,\alpha}^{(r_1,\ldots,r_n)}(\bar E^n_+)\) such that
\[ f\big|_{E^m}=\varphi,\qquad \|f\|_{\mathscr L_{p,\alpha}^{(r_1,\ldots,r_n)}(\bar E^n_+)} \leq C\|\varphi\|_{\mathscr L_p^{(\rho_1,\ldots,\rho_m)}(E^m)} . \tag{14} \]
Theorem 8. Let \(p>1\), \(0<r_i<1\) \((i=1,\ldots,n)\), let \(m\) be a natural number such that \(0<m<n\), \(-1<\alpha\), and
\[ \varepsilon = 1-\frac1p\sum_{i=m+1}^n\frac1{r_i} -\frac{\alpha}{pr_n} >0. \]
Then, if on the hyperplane \(x_{m+1}=\cdots=x_n=0\) a function
\(\varphi(x_1,\ldots,x_m)\in B_p^{(\rho_1,\ldots,\rho_m)}(E^m)\) is given, with \(\rho_s=\varepsilon r_s\) \((s=1,\ldots,m)\), there exists a function
\(f(x_1,\ldots,x_n)\in B_{p,\alpha}^{(r_1,\ldots,r_n)}(\bar E^n_+\setminus\infty)\) such that
\[ f\big|_{E^m}=\varphi,\qquad \|f\|_{B_{p,\alpha}^{(r_1,\ldots,r_n)}(\bar E^n_+\setminus\infty)} \leq C\|\varphi\|_{B_p^{(\rho_1,\ldots,\rho_m)}(E^m)}, \tag{15} \]
where
\(\bar E^n_+\setminus\infty=\{x\in \bar E^n_+;\ x_i<\infty,\ i=m+1,\ldots,n\}\).
Inequality (12) was obtained earlier by L. D. Kudryavtsev for \(r_i=1\) \((i=1,\ldots,n)\), \(m=n-1\), but under a more general assumption, i.e. there the \(p\)-summability of the function itself is not required. Inequality (14) was also proved earlier by L. D. Kudryavtsev for \(r_i=1\) \((i=1,\ldots,n)\), \(m=n-1\). A result close to inequality (9) was obtained simultaneously and independently of the author, by another method, by T. I. Amanov.
The author expresses sincere gratitude to his adviser Prof. L. D. Kudryavtsev for posing the problem and for valuable advice.
Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR
Received
25 V 1964
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