UDC 517.51

A. D. DZHABRAILOV

INTERPOLATION OF SPACES OF DIFFERENTIABLE FUNCTIONS DEFINED IN AN \(n\)-DIMENSIONAL DOMAIN

(Presented by Academician I. M. Vinogradov on 26 IX 1968)

1. In the paper \((^1)\) S. M. Nikol’skii proved interpolation theorems for the spaces \(S_p^{\mathbf r_i}H(E^n)\) \((i=1,\ldots,N)\) (where \(\mathbf r_i\) are positive vectors, i.e. \(r_j^i>0\)).

Subsequently, in \((^4)\), P. I. Lizorkin and S. M. Nikol’skii extended these results to the well-known spaces \(S_p^{\mathbf r_i}L(E^n)\) \((i=1,\ldots,N)\) (\(\mathbf r_i\) are positive vectors). T. I. Amanov in \((^{15})\) proved interpolation theorems for the spaces \(S_{p,\theta}^{\mathbf r_i}B(E^n)\) \((i=1,\ldots,N)\) (where \(\mathbf r_i\) are positive vectors). In \((^9)\) Grisvard proved interpolation theorems for the spaces \(S_{p_1}^{\mathbf r_1}B(E^n)\), \(S_{p_2}^{\mathbf r_2}B(E^n)\) (where \(\mathbf r_1\) and \(\mathbf r_2\) are positive vectors). In the present paper we define spaces of functions with dominating mixed derivatives, defined in an \(n\)-dimensional (not necessarily bounded) domain \(\Omega\), i.e. spaces of the type \(S_{p,\theta}^{\mathbf r}B(\Omega)\), \(S_p^{\mathbf r}W(\Omega)\). For definitions of these spaces when \(\Omega=E^n\), see \((^1,^2,^4,^8,^{10},^{12},^{14})\).

In the present paper interpolation theorems are announced for the spaces \(S_{p_i,\theta_i}^{\mathbf r_i}B(\Omega)\) \((i=1,\ldots,N)\) and \(S_{p_i}^{\mathbf r_i}W(\Omega)\) \((i=1,\ldots,N)\) (where \(\mathbf r_i\) are nonnegative vectors, i.e. \(r_j^i\geq 0\)).

For \(\theta_i=\infty\) \((i=1,\ldots,N)\) we obtain interpolation theorems for the closure of the set of smooth functions in the spaces \(S_{p_i}^{\mathbf r_i}H(\Omega)\) \((i=1,\ldots,N)\), extending the corresponding results \((^1)\) of S. M. Nikol’skii to different \(p_i\) and to the domain \(\Omega\) (without assuming strict positivity of the vectors \(\mathbf r_i\) \((i=1,\ldots,N)\).

The results of \((^1,^2,^4,^9,^{15})\) are extended to vectors \(\mathbf r_i\) \((i=1,\ldots,N)\) with nonnegative components, to different \(1\leq p_i\leq \theta_i\leq\infty\) \((i=1,\ldots,N)\), and to the domain \(\Omega\).

From the interpolation theorems that are formulated in the present paper there follows a series of new results also for the well-known spaces \(W_p^{\mathbf r}\), \(H_p^{\mathbf r}\) and \(B_p^{\mathbf r}\) (see \((^3,^5,^6)\)). In the present paper domains subject to restrictions 1) and 2) will be considered. In the present paper we shall consider domains from the class of domains with \((\mathbf h)\), i.e. from the set of domains attainable “from within” by a parallelepiped whose edges are parallel to the coordinate axes and are respectively equal to the components of the vector \(\mathbf h=(h_1,\ldots,h_n)\). Also considered are domains from the classes of domains \(\bigcap_{i=1}^n R^{(i)}\), which are defined and studied in the work of V. P. Il’in \((^5)\).

1. Let \(e_n\) be the set of natural numbers \(1,\ldots,n\), i.e. \(e_n=\{1,\ldots,n\}\). If \(\mathbf K=(k_1,\ldots,k_n)\) is some vector, then by \(\mathbf K^e=(k_1^e,\ldots,k_n^e)\) (where \(e\) is any subset of the set \(e_n\)) we shall denote the vector with components \(k_i^e=k_i\) for \(i\in e\) and \(k_i=0\) for \(i\notin e\). By \(e_k\) we denote the support of the vector \(\mathbf K\), i.e. the smallest subset \(e\) for which \(\mathbf K^e=\mathbf K\) (in other words, \(e_k\) is the set of indices of the nonzero components of the vector \(\mathbf K\); for more detail see \((^1)\)).

Definition 1. We shall say that \(f\in S_{p,\theta}^{\mathbf r}B(\Omega,\mathbf h)\), where \(1\leq p\leq \theta\leq\infty\), \(\mathbf r=(r_1,\ldots,r_n)\), \(r_i\geq 0\), \(e_r\neq\varnothing\), if \(f\in L_p(\Omega)\) and there exist

in \(\Omega\) generalized derivatives in the sense of Sobolev \((^7)\) (for all \(e \subseteq e_r\)),

\[ D^{\bar r^{\,e}} f(x) \equiv \frac{\partial^{|\bar r^{\,e}|} f(K)} {\partial x_1^{\bar r_1^{\,e}}\ldots \partial x_n^{\bar r_n^{\,e}}}, \qquad |\bar r^{\,e}|=\sum_{i=1}^n \bar r_i^{\,e} \]

(where \(\bar r_i\) is the greatest integer \(< r_i\)), satisfying the conditions

\[ \|f\|_{\mathscr L^{r^e}_{p,\theta}(\Omega,\mathbf h)} = \left( \int_{\square \mathbf h^{\,e}(0)} \!\!\cdots\!\! \int \left\| \Delta^{2\bar\omega^{\,e}}\!\left(\frac{t}{2}\right) D^{\bar r^{\,e}} f \right\|_{L_p(\Omega_{t^{\,e},2\bar\omega})}^{\theta} \prod_{i\in e}\frac{dt_i}{|t_i|^{1+\theta\alpha_i}} \right)^{1/\theta} <\infty, \]

where \(\bar\omega=(1,\ldots,1)\); \(\alpha_i=r_i-\bar r_i\); \(\Omega_{t^{\,e},2\bar\omega}^{*}=\bigcap_{i\in e}\Omega_{t_i,k_i}\); and
\(\Delta^k(t)f=\Delta_1^{k_1}(t_1)\cdots \Delta_n^{k_n}(t_n)f\);
\(\Delta_i^{k_i}(t_i)f\) is the finite difference of order \(k_i\) with respect to the variable \(x_i\) with step \(t_i\),
\(\square\mathbf h(0)=(-h_1,h_1)\times\cdots\times(-h_n,h_n)\).

We define the norm in this space as follows:

\[ \|f\|_{S^r_{p,\theta}B(\Omega,\mathbf h)} = \|f\|_{L_p(\Omega)} + \sum_{e\subseteq e_r} \|f\|_{\mathscr L^{r^e}_{p,\theta}(\Omega,\mathbf h)}. \]

The spaces \(S^r_{p,\theta}B(\Omega,\mathbf h)\) for \(\theta=\infty\) coincide with the spaces \(S^r_pH(\Omega,\mathbf h)\), defined for \(\Omega=E^n\) in \((^1)\).

Definition 2. We shall say that \(f\in S^r_pW(\Omega,\mathbf h)\) (where \(r=(r_1,\ldots,r_n)\) is a nonnegative vector with support \(e_r\ne\varnothing\)) if \(f\in L_p(\Omega)\) and for all \(e\subseteq e_r\) there exist in \(\Omega\) generalized derivatives \(D^{\tilde r^e}f\) (where \(\tilde r=(\tilde r_1,\ldots,\tilde r_n)\), \(\tilde r_i\) is the integer part of \(r_i\)), satisfying the conditions

\[ \|f\|_{L_p^{r^e}(\Omega,\mathbf h)} = \begin{cases} \|D^{r^e}f\|_{L_p(\Omega)}<\infty, & \text{if } e^*\equiv e\cdot e_r^*=\varnothing,\\[1.2em] \left( \displaystyle\int_{\square \mathbf h^{\,e^*}(0)} \!\!\cdots\!\! \int \left\|\Delta^{\vec e^{\,*}}(t)D^{\tilde r^e}f\right\|_{L_p(\Omega_{t^{\,e^*},\bar\omega}^{*})}^{p} \prod_{i\in e^*}\frac{dt_i}{|t_i|^{1+\varkappa\alpha_i}} \right)^{1/p}<\infty, & \text{if } e^*\ne\varnothing, \end{cases} \]

where \(\alpha_i=r_i-\tilde r_i\), \(e_r^*\) is the set of indices of the noninteger components of the vector \(r\).

We define the norm in these spaces in the following way:

\[ \|f\|_{S^r_pW(\Omega,\mathbf h)} = \|f\|_{L_p(\Omega)} + \sum_{e\subseteq e_r} \|f\|_{L_p^{r^e}(\Omega,\mathbf h)}. \]

The study of these spaces is the subject of the works \((^2,^4,^ {12},^{14})\).

The closure of the set of smooth functions in the corresponding norm will be denoted by
\(\dot S^r_{p,\theta}B(\Omega,\mathbf h)\), \(\dot S^r_pW(\Omega,\mathbf h)\).

3. Theorem 1. Let \(f(x)\) belong simultaneously to all spaces
\[ S^{r^i}_{p_i,\theta_i}B(\Omega,\mathbf h)\qquad (i=1,\ldots,N), \]
where \(1\le p_i\le \theta_i\le\infty\) \((i=1,\ldots,N)\), \(r^i\) are nonnegative vectors with supports \(e_{r^i}\ne\varnothing\) \((i=1,\ldots,N)\), and the domain \(\Omega\in c(\mathbf h)\) (see (5)). Suppose further that \(\lambda_i\ge0\) \((i=1,\ldots,N)\) and

\[ \sum_1^N \lambda_i=1. \]

Then, for any integer vector \(\vec v=(v_1,\ldots,v_n)\) (\(v_j\ge0\) integers) with support
\[ e_{\vec v}\subseteq \bigcup_{i=1}^N e_{r^i} \quad\text{and}\quad v_j<\sum_{i=1}^N r_j^i\lambda_i \quad (j=1,\ldots,n), \]
the function \(D^{\vec v}f\)

belongs to \(L_p(\Omega)\) \(\left(\text{where } \dfrac1p=\sum_{i=1}^N\dfrac{\lambda_i}{p_i}\right)\), and the following inequality holds

\[ {}^{*}\ \Omega_{t_i,k_i}\text{ is the set of those points }x=(x_1,\ldots,x_n)\text{ of the domain }\Omega\text{ such that the corresponding points } (x_1,\ldots,x_i+\tfrac{j}{k_i}t_i,\ldots,x_n)\ (j=0,1,\ldots,k_i) \text{ also belong to }\Omega\text{ (see (5)).} \]

\[ \left\|D^{\vec v}f\right\|_{L_p(\Omega)} \le c\prod_{i=1}^{N}\|f\|_{S_{p_i,\theta_i}^{r_i}B(\Omega,h)}^{\lambda_i} \le c\sum_{i=1}^{N}\|f\|_{S_{p_i,\theta_i}^{r_i}B(\Omega,h)}, \tag{1} \]

where \(c\) is a constant independent of \(f\).

Theorem 2. Let the conditions of Theorem 1 be satisfied and let \(\Gamma^m\) be an \(m\)-dimensional \((1\le m\le n-1)\) surface belonging to the domain \(\Omega\) (where \(\Omega\in c(h)\); see (4)) and defined by the equations:
\(x_1=x_1,\ldots,x_m=x_m,\)
\(x_{m+1}=\varphi_{m+1}(x_1,\ldots,x_m),\ldots,x_n=\varphi_n(x_1,\ldots,x_m)\), where \(\varphi_j(x_1,\ldots,x_m)\) \((j=m+1,\ldots,n)\) are continuous functions having bounded first derivatives.

Then, if \(e_m=\{m+1,\ldots,n\}\subseteq \bigcup_{i=1}^{N} e_{r_i}\) and \(\vec v=(v_1,\ldots,v_n)\) is an integer vector with components

\[ 0\le v_j<\sum_{i=1}^{N} r_j^i\lambda_i \quad (j=1,\ldots,m), \]

\[ 0\le v_j<\sum_{i=1}^{N} r_j^i\lambda_i-\frac1p \quad (j=m+1,\ldots,n), \qquad \left(\frac1p=\sum_{i=1}^{N}\frac{\lambda_i}{p_i}\right), \]

then the function \(D^{\vec v}f|_{\Gamma^m}\) belongs to \(L_p(\Omega)\), and the inequality

\[ \left\|D^{\vec v}f|_{\Gamma^m}\right\|_{L_p(\Gamma^m)} \le c\prod_{i=1}^{N}\|f\|_{S_{p_i,\theta_i}^{r_i}B(\Omega,h)}^{\lambda_i} \le c\sum_{i=1}^{N}\|f\|_{S_{p_i,\theta_i}^{r_i}B(\Omega,h)} \tag{2} \]

holds, where \(c\) is a constant independent of \(f\).

Theorem 3. If in Theorems 1 and 2 one assumes that \(\bigcup_{i=1}^{N} e_{r_i}=e_n\), \(q\ge p\) (where \(\frac1p=\sum_{i=1}^{N}\frac{\lambda_i}{p_i}\)) and

\[ \begin{aligned} \kappa_j&=\sum_{i=1}^{N} r_j^i\lambda_i-v_j-\left(\frac1p-\frac1q\right)>0 \quad (j=1,\ldots,m),\\ \kappa_j&=\sum_{i=1}^{N} r_j^i\lambda_i-v_j-\frac1p>0 \quad (j=m+1,\ldots,n), \end{aligned} \tag{3} \]

then \(D^{\vec v}f\in L_q(\Gamma^m)\), and an inequality analogous to inequalities (1) and (2) is valid, with the difference that on the left-hand side of these inequalities there will be norms in \(L_q(\Gamma^m)\) \((\Gamma^m\equiv\Omega)\).

Analogous theorems are also valid for the spaces \(S_{p_i}^{r_i}\widetilde W(\Omega,h)\).

Theorem 4. Let \(f(x)\) belong to all the spaces \(S_{p_i,\theta_i}^{r_i}\widetilde B(\Omega,h)\) \((i=1,\ldots,N)\), where \(1\le p_i\le \theta_i\le\infty\) \((i=1,\ldots,N)\), \(e_{r_i}=e_n\) \((i=1,\ldots,N)\) (i.e. \(r_j^i>0\) and \(\Omega\in\bigcap_{j=1}^{n} R^{(j)}\) (see (5)).

Suppose further that \(\lambda_i\ge0\) \((i=1,\ldots,N)\), \(\sum_{1}^{N}\lambda_i=1\) and \(\frac1p=\sum_{1}^{N}\frac{\lambda_i}{p_i}\), \(\frac1\theta=\sum_{1}^{N}\frac{\lambda_i}{\theta_i}\).

Let \(\kappa_j>0\) \((j=1,\ldots,n)\) (see (3)), and let \(\Omega^m\) be the intersection of the domain \(\Omega\) with the hyperplane \(x_{m+1}=\mathrm{const},\ldots,x_n=\mathrm{const}\).

Then \(D^{\vec v}f\in S_{q,\sigma}^{\vec\rho}\widetilde B(\Omega^m,h)\), where \(q\ge p\), \(\sigma\ge\theta\) and \(\vec\rho=(\rho_1,\ldots,\rho_m)\), \(0\le\rho_j\le\kappa_j\) \((j=1,\ldots,m)\), and the inequalities

\[ \left\|D^{\vec v}f\right\|_{S_{q,\sigma}^{\vec\rho}\widetilde B(\Omega^m,h)} \le c\prod_{i=1}^{N}\|f\|_{S_{p_i,\theta_i}^{r_i}\widetilde B(\Omega,h)}^{\lambda_i} \le c\sum_{i=1}^{N}\|f\|_{S_{p_i,\theta_i}^{r_i}\widetilde B(\Omega,h)} \]

hold, where \(c\) is a constant independent of \(f\).

Remark to Theorem 4. The theorem remains valid if the vectors \(\mathbf r^i\) are replaced by the vectors \(\mathbf l,\ (0,\ldots,r_i,\ldots,0)+\mathbf l;\ (i=1,\ldots,n)\), where \(\mathbf l\) is an arbitrary vector with nonnegative components; however, in this case it is necessary to assume that \(p_i=p,\ \theta_i=\theta\ (i=1,\ldots,n)\). The advantage of this remark is that vectors of this type may also contain zero components.

Theorem 5. If in Theorem 4 it is assumed that \(\vec\rho=(\rho_1,\ldots,\rho_m)\) and \(\rho_j<\chi_j\ (j=1,\ldots,m)\), then the assertions of this theorem remain valid also when
\[ \bigcup_{i=1}^{N} e_{r_i}=e_n,\quad e_{r_i}\ne\varnothing\quad (i=1,\ldots,N). \]

Corollary. Let \(f\in B^{\mathbf r}_{\mathbf p,\vec\theta}\) \((\mathbf p=(p_1,\ldots,p_n),\ \vec\theta=(\theta_1,\ldots,\theta_n))\),
\[ 1\le p_i\le \theta_i\le \infty\quad (i=1,\ldots,n),\quad \lambda_i>0,\quad \sum_{1}^{n}\lambda_i=1; \]
then \(f(x)\) belongs to the space \(S^{\vec\rho}_{p,\theta}B\) \((\vec\rho=(\rho_1,\ldots,\rho_n),\ \rho_i<\lambda_i r_i\ (i=1,\ldots,n))\), and the inequalities
\[ \|f\|_{S^{\vec\rho}_{p,\theta}B}\le c\prod_{i=1}^{n}\|f\|^{\lambda_i}_{B^{r_i}_{p_i,\theta_i}} \le c\|f\|_{B^{\mathbf r}_{\mathbf p,\vec\theta}} \quad \left( \frac1p=\sum_{i=1}^{n}\frac{\lambda_i}{p_i};\ \frac1\theta=\sum_{i=1}^{n}\frac{\lambda_i}{\theta_i} \right), \]
hold, where \(c\) is a constant independent of \(f\).

Theorems analogous to Theorem 5 are also valid for the spaces
\[ S^{r_i}_{p_i}W(\Omega)\quad (i=1,\ldots,n). \]

Under certain additional conditions on the domain \(\Omega\), it is proved that the sets of sufficiently smooth finite functions in \(E^n\) are dense, respectively, in the spaces \(S^{\mathbf r}_{p,\theta}B(\Omega)\) and \(S^{\mathbf r}_{p}W(\Omega)\). In particular, such domains include all bounded domains \(\Omega\) from \(\bigcap_{i=1}^{n}R^{(i)}\), the domain \(E^n\), and \(\{x;\ x_i>0,\ i\in e\subset e_n\}\). The equivalent normings of the spaces \(S^{\mathbf r}_{p,\theta}B(\Omega)\) (see (4), § 4), which occurred for \(\Omega=E^n\), remain valid also for \(\Omega\in\bigcap_{i=1}^{n}R^{(i)}\).

The indicated results are proved by the method of integral representations (see \((^{5,7,11,14})\)).

We note that some analogous results (for example, Theorem 4 and the remark to this theorem) for \(\Omega=E^n\), under a more general assumption on summability, i.e., for spaces of functions with mixed norms, are contained in a joint work of O. V. Besov and the author.

I express my sincere gratitude to Prof. L. D. Kudryavtsev for his attention to the present work.

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