MATHEMATICS

A. D. DZHABRAILOV

ON DIRECT AND INVERSE EMBEDDING THEOREMS FOR WEIGHTED SPACES

(Presented by Academician I. M. Vinogradov, 19 III 1965)

1. Let \(n_1,\ldots,n_s\) be natural numbers whose sum is equal to \(n\); \(E^{n_i}\) \((i=1,\ldots,s)\) are \(n_i\)-dimensional Euclidean spaces of points

\[ \mathbf y^{(i)}=(y_1^{(i)},\ldots,y_{n_i}^{(i)});\qquad E_s^n=\prod_{i=1}^s E^{n_i} \]

are spaces of points

\[ \mathbf y=(\mathbf y^{(1)},\ldots,\mathbf y^{(s)}) =(y_1^{(1)},\ldots,y_{n_i}^{(1)};\ldots;y_1^{(s)},\ldots,y_{n_s}^{(s)}) =(y_1,\ldots,y_n); \]

if \(n_i=1\), then \(\mathbf y^{(i)}=y_i\).

Let

\[ E^{+n_i}=\{\mathbf y^{(i)},\, y_{j_i}^{(i)}>0\ (j_i=1,\ldots,n_i)\},\qquad E_s^{+n}=\prod_{i=1}^s E^{+n_i}. \]

For \(s=n\), instead of \(E_n^n\) and \(E_n^{+n}\) we shall write, respectively, \(E^n\) and \(E^{+n}\).

1. Let \(f(\mathbf y)\) be an arbitrary smooth function defined in the space \(E_s^{+n}\). Suppose \(\mathbf r=(\mathbf r^{(1)},\ldots,\mathbf r^{(s)})\), where \(\mathbf r^{(i)}=(r_1^{(i)},\ldots,r_{n_i}^{(i)})\), is a vector with positive integer components, \(1\le p\le\infty\); \(\vec\alpha=(\alpha_1,\ldots,\alpha_n)\) is a vector whose components satisfy the conditions \(\alpha_j>-1\) \((j=1,\ldots,n)\). We introduce the following norms:

\[ \|f\|_{L_p(E_s^{+n},\vec\alpha)} = \left( \int_{E_s^{+n}} \prod_{j=1}^n y_j^{\alpha_j}\, |f(\mathbf y)|^p\, d\mathbf y \right)^{1/p}, \]

\[ \|f\|_{L_p^{(\mathbf r)}(E_s^{+n},\vec\alpha)} = \sum_{i=1}^s \left\| D_{(i)}^{|\mathbf r^{(i)}|} f \right\|_{L_p(E_s^{+n},\vec\alpha)}, \]

where

\[ D_{(i)}^{|\mathbf r^{(i)}|} f(\mathbf y) = \frac{\partial^{|\mathbf r^{(i)}|} f(\mathbf y)} {(\partial y_1^{(i)})^{r_1^{(i)}}\cdots(\partial y_{n_i}^{(i)})^{r_{n_i}^{(i)}}}, \qquad |\mathbf r^{(i)}|=\sum_{j_i=1}^{n_i} r_{j_i}^{(i)}, \]

\[ \|f\|_{W_p^{(\mathbf r)}(E_s^{+n},\vec\alpha)} = \|f\|_{L_p(E_s^{+n},\vec\alpha)} + \|f\|_{L_p^{(\mathbf r)}(E_s^{+n},\vec\alpha)}. \]

Definition. The spaces \(L_p(E_s^{+n},\vec\alpha)\), \(L_p^{(\mathbf r)}(E_s^{+n},\vec\alpha)\), and \(B_p^{(\mathbf r)}(E_s^{+n},\vec\alpha)\) shall be called the closures of the set of smooth functions finite in \(E^n\), respectively in the norms

\[ \|f\|_{L_p(E_s^{+n},\vec\alpha)},\qquad \|f\|_{L_p^{(\mathbf r)}(E_s^{+n},\vec\alpha)} \]

and

\[ \|f\|_{B_p^{(\mathbf r)}(E_s^{+n},\vec\alpha)}. \]

1. Let \(\varphi(\mathbf y)\) be a smooth function defined in \(E^n\). Suppose that numbers \(\rho_j>0\) \((j=1,\ldots,n)\) are given. For each \(j\) set \(\rho_j=\bar\rho_j+\beta_j\), where \(\bar\rho_j\) is the greatest integer less than \(\rho_j\), so that \(0<\beta_j\le1\). Introduce

norms

\[ \|\varphi\|_{\mathscr L_{p,y_j}^{\rho_j}(E^{+n},\vec\alpha)} = \left( \int_0^\infty \|\Delta_j^{1+[\beta_j]}(t)D_j^{\bar\rho_j}\varphi\|_{L_p(E^{+n},\vec\alpha)} \,\frac{dt}{t^{1+p\beta_j}} \right)^{1/p}, \]

where \(\Delta_j^{1+[\beta_j]}(t)\varphi\) is the finite difference of order \(1+[\beta_j]\) with respect to the variable \(y_j\) with step \(t\); \([\beta_j]\) is the integer part of \(\beta_j\), \(D_j^{\bar\rho_j}\varphi(y)=\partial^{\rho_j}\varphi(y)/\partial y_i^{\bar\rho_j}\),

\[ \|\varphi\|_{\mathscr L_p^{(\vec\rho)}(E^{+n},\vec\alpha)} = \sum_{i=1}^n \|\varphi\|_{\mathscr L_{p,y_i}^{\rho_i}(E^{+n},\vec\alpha)}, \]

\[ \|\varphi\|_{B_p^{(\vec\rho)}(E^{+n},\vec\alpha)} = \|\varphi\|_{L_p(E^{+n},\vec\alpha)} + \|\varphi\|_{\mathscr L_p^{(\vec\rho)}(E^{+n},\vec\alpha)} . \]

Definition. By the spaces \(\mathscr L_p^{(\vec\rho)}(E^{+n},\vec\alpha)\) and \(B_p^{(\vec\rho)}(E^{+n},\vec\alpha)\) we shall mean the closure of the set of smooth finite functions in \(E^n\), respectively in the norms \(\|\varphi\|_{\mathscr L_p^{(\vec\rho)}(E^{+n},\vec\alpha)}\) and \(\|\varphi\|_{B_p^{(\vec\rho)}(E^{+n},\vec\alpha)}\).

For \(\alpha_i=0\) \((i=1,\ldots,n)\) these spaces are denoted respectively by \(\mathscr L_p^{(\vec\rho)}(E^{+n})\) and \(B_p^{(\vec\rho)}(E^{+n})\).

1. Some embedding theorems have been obtained for the spaces defined above; these are a development of the corresponding results of works \((1\text{–}11)\).

Theorem 1. Suppose:

1) \(p>1,\quad \mathbf r=(\mathbf r^{(1)},\ldots,\mathbf r^{(s)})=(r_1^{(1)},\ldots,r_{n_1}^{(1)};\ldots; r_1^{(s)},\ldots,\)
\[ \ldots,r_{n_s}^{(s)})=(r_1,\ldots,r_n) \]
is a vector with integer positive components;

2) \(\vec\nu=(\vec\nu^{(1)},\ldots,\vec\nu^{(s)})=(\nu_1^{(1)},\ldots,\nu_{n_1}^{(1)};\ldots;\nu_1^{(s)},\ldots,\nu_{n_s}^{(s)})=(\nu_1,\ldots,\)
\[ \ldots,\nu_n) \]
is a vector with integer nonnegative components; 3) \(m\) is a natural number such that \(0<m\le n\); 4) \(\vec\alpha=(\alpha_1,\ldots,\alpha_n)\), \(\vec\gamma=(\gamma_1,\ldots,\gamma_m)\) are vectors whose components satisfy the conditions
\[ \alpha_j\ge \gamma_j\ge 0 \quad (j=1,\ldots,m),\qquad \alpha_\eta>-1 \quad (\eta=m+1,\ldots,n); \]
5)
\[ \varepsilon= \min_{i\in\{1+,\ldots,s\}}\{n_i\} -\frac1p\sum_{j=m+1}^n\frac1{r_j} -\sum_{j=1}^n\frac{\nu_j}{r_j} -\frac1p\sum_{j=1}^n\frac{\alpha_j}{r_j} +\frac1p\sum_{j=1}^m\frac{\gamma_j}{r_j} >0; \]
6) \(f(y)\in W_p^{(\mathbf r)}(E_s^{+n},\alpha)\). Then

\[ \Psi\in L_p(E^{+m},\vec\gamma), \quad \text{where } \Psi = D_{(1)}^{|\vec\nu^{(1)}|}\cdots D_{(s)}^{|\vec\nu^{(s)}|} f(y_1,\ldots,y_m,0,\ldots,0), \]

and the inequality

\[ \|\Psi\|_{L_p(E^{+m},\vec\gamma)} \le c\,\|f\|_{W_p^{(\mathbf r)}(E_s^{+n},\vec\alpha)} \]

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

Theorem 2. Under the hypotheses of Theorem 1, if all \(n_i\) \((i=1,\ldots,s)\) are equal to one another, then the inequality

\[ \|\Psi\|_{L_p(E^{+m},\vec\gamma)} \le c\,\|f\|_{L_p(E_s^{+n},\vec\alpha)}^{\varepsilon/(\varepsilon+\delta)} \|f\|_{L_p^{(\mathbf r)}(E_s^{+n},\vec\alpha)}^{\delta/(\varepsilon+\delta)} \]

holds, where

\[ \delta= \frac1p\sum_{j=m+1}^n\frac1{r_j} +\sum_{j=1}^n\frac{\nu_j}{r_j} +\frac1p\sum_{j=1}^n\frac{\alpha_j}{r_j} -\frac1p\sum_{j=1}^m\frac{\gamma_j}{r_j} >0. \]

Theorem 3. Let the conditions 1)–6) of Theorem 1 be satisfied and, in addition, let one of the following conditions be satisfied:

I. \(0<\rho_j\leqslant \varepsilon r_j\) \((j=1,\ldots,m)\), if \(m<n\). II. \(0<\rho_j<\varepsilon r_j\) \((j=1,\ldots,m)\), if \(m\leqslant n\). Then \(\Psi\in \mathscr L_p^{(\vec\rho)}(E^{+m},\vec\gamma)\), and the inequality
\[ \|\Psi\|_{\mathscr L_p^{(\vec\rho)}(E^{+m},\vec\gamma)} \leqslant c\|f\|_{W_p^{(r)}(E_s^{+n},\vec\alpha)} \]
holds; \(c\) is a constant independent of \(f\).

From Theorems 1 and 3 the following assertion follows:

Theorem 4. Under the conditions of Theorem 3
\[ \Psi\in B_p^{(\vec\rho)}(E^{+m},\vec\gamma), \]
and the inequality
\[ \|\Psi\|_{B_p^{(\vec\rho)}(E^{+m},\vec\gamma)} \leqslant c\|f\|_{W_p^{(r)}(E_s^{+n},\vec\alpha)} \]
holds, where \(c\) is a constant independent of \(f\).

Theorem 5. Suppose that all the conditions of Theorem 1 are satisfied, \(m<n\); moreover, \(\rho_j=\varepsilon r_j\) \((j=1,\ldots,m)\), and all \(n_i\) \((i=1,\ldots,s)\) are equal to one another. Then the inequality
\[ \|\Psi\|_{\mathscr L_p^{(\vec\rho)}(E^{+m},\vec\gamma)} \leqslant c\|f\|_{L_p^{(r)}(E_s^{+n},\vec\alpha)} \]
holds.

Theorem 6. Suppose that all the conditions of Theorem 1 hold and that \(\rho_j\) \((j=1,\ldots,m)\) are positive integers such that \(\rho_j<\varepsilon r_j\) \((j=1,\ldots,m)\). Then
\[ \Psi\in L_p^{(\vec\rho)}(E^{+m},\vec\gamma) \]
and the inequality
\[ \|\Psi\|_{L_p^{(\vec\rho)}(E^{+m},\vec\gamma)} \leqslant c\|f\|_{W_p^{(r)}(E_s^{+n},\vec\alpha)} \]
holds, where \(c\) is a constant independent of \(f\).

From Theorems 1 and 6 the following assertion follows:

Theorem 7. Under the conditions of Theorem 6
\[ \Psi\in W_p^{(\vec\rho)}(E^{+m},\vec\gamma), \]
and the inequality
\[ \|\Psi\|_{W_p^{(\vec\rho)}(E^{+m},\vec\gamma)} \leqslant c\|f\|_{W_p^{(r)}(E_s^{+n},\vec\alpha)} \]
holds; where \(c\) is a constant independent of \(f\).

1. Some theorems of inverse character have also been obtained.

Theorem 8. Let \(p\geqslant 1\); \(\nu\) be a natural number; \(r_i\) \((i=1,\ldots,n)\) be positive integers; \(\alpha_j\geqslant 0\) \((j=1,\ldots,n-1)\), \(\alpha_n>-1\). It is assumed that
\[ \varepsilon=1-\nu/r_n-(1+\alpha_n)/pr_n>0,\quad \rho_j=\varepsilon r_j\quad (j=1,\ldots,n-1). \]
Then: 1) if on the hyperplane \(y_n=0\) a function
\[ \varphi_1(y_1,\ldots,y_{n-1})\in \mathscr L_p^{(\vec\rho)}(E^{+\,n-1},\vec\alpha^{(n-1)}) \]
is given, then in the space \(E^{+n}\) one can construct a function
\[ f_1(y_1,\ldots,y_n)\in L_p^{(r)}(E^{+n},\vec\alpha) \]
such that
\[ D_n^\nu f_1(y)\big|_{y_n=0}=\varphi_1(y_1,\ldots,y_{n-1}), \]
\[ \|f_1\|_{L_p^{(r)}(E^{+n},\vec\alpha)} \leqslant c\|\varphi_1\|_{\mathscr L_p^{(\vec\rho)}(E^{+\,n-1},\vec\alpha^{(n-1)})}; \]

2) if on the hyperplane \(y_n=0\) a function \(\varphi_2(y_1,\ldots,y_{n-1}) \in B_p^{(\vec{\rho})}(E^{+n-1},\vec{\alpha}^{(n-1)})\) is given, then in \(\Omega\) one can construct a function \(f_2(y_1,\ldots,y_n)\in W_p^{(r)}(\Omega,\vec{\alpha})\) such that

\[ D_n^{\nu} f_2(y)\big|_{y_n=0}=\varphi_2(y_1,\ldots,y_{n-1}), \]

\[ \|f_2\|_{W_p^{(r)}(\Omega,\vec{\alpha})}\le c\|\varphi_2\|_{B_p^{(\vec{\rho})}(E^{+n-1},\vec{\alpha}^{(n-1)})}, \]

where \(c\) is a constant independent of \(\varphi_1\) and \(\varphi_2\),

\[ \Omega=\{y\in E^{+n},\, y_1<\infty\},\qquad \vec{\alpha}^{(n-1)}=(\alpha_1,\ldots,\alpha_{n-1}). \]

Theorem 9. Let \(p\ge 1\), \(0<m<n\), \(\nu_j\) \((j=m+1,\ldots,n)\) be nonnegative integers, and \(a_n>-1\). Suppose that

\[ \varepsilon^*=1-\sum_{j=m+1}^{n}\frac{\nu_j}{r_j} -\frac{1}{p}\sum_{j=m+1}^{n}\frac{1}{r_j} -\frac{\alpha_n}{p r_n}>0, \]

\[ \rho_j=\varepsilon^* r_j\quad (j=1,\ldots,m). \]

Then: 1) if on the hyperplane \(y_{m+1}=\cdots=y_n=0\) a function
\(\varphi_1(y_1,\ldots,y_m)\in \mathscr{L}_p^{(\vec{\rho})}(E^m)\) is given, then in the space \(E_n^{+*}\) one can construct a function
\(f_1(y_1,\ldots,y_n)\in L_p^{(r)}(E_n^{+*},a_n)\) such that

\[ D_{m+1}^{\nu_{m+1}}\cdots D_n^{\nu_n} f_1(y)\big|_{y_{m+1}=0,\ldots,y_n=0} =\varphi_1(y_1,\ldots,y_m), \]

\[ \|f_1\|_{L_p^{(r)}(E_n^{+*},a_n)} \le c\|\varphi_1\|_{\mathscr{L}_p^{(\vec{\rho})}(E^m)}; \]

2) if on the hyperplane \(y_{m+1}=\cdots=y_n=0\) a function
\(\varphi_2(y_1,\ldots,y_m)\in B_p^{(\vec{\rho})}(E^m)\) is given, then in \(\Omega^*\) one can construct a function
\(f_2(y_1,\ldots,y_n)\in W_p^{(r)}(\Omega^*,a_n)\) such that

\[ D_{m+1}^{\nu_{m+1}}\cdots D_n^{\nu_n} f_2(y)\big|_{y_{m+1}=0,\ldots,y_n=0} =\varphi_2(y_1,\ldots,y_m), \]

\[ \|f_2\|_{W_p^{(r)}(\Omega^*,a_n)} \le c\|\varphi_2\|_{B_p^{(\vec{\rho})}(E^m)}, \]

where \(c\) is a constant independent of \(\varphi_1,\varphi_2\),

\[ E_n^{+*}=\{y\in E^n,\, y_n>0\},\qquad \Omega^*=\{y\in E_n^{+*},\ |y_i|<\infty \]

\[ (i=m+1,\ldots,n)\}. \]

In conclusion the author takes the opportunity to express gratitude to Prof. L. D. Kudryavtsev for posing the problems and for valuable advice.

Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR

Received
11 III 1965

CITED LITERATURE

1. L. D. Kudryavtsev, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 55 (1959).
2. L. D. Kudryavtsev, Nauch. dokl. vyssh. shkoly, fiz.-matem. nauki, No. 3 (1959).
3. L. D. Kudryavtsev, DAN, 153, No. 3 (1963).
4. V. P. Il’in, V. A. Solonnikov, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 66, 205 (1962).
5. V. P. Il’in, ibid., 64, 61 (1962).
6. O. V. Besov, ibid., 60, 42 (1961).
7. S. V. Uspenskii, ibid., 60, 282 (1961).
8. P. I. Lizorkin, DAN, 132, No. 3 (1960).
9. G. N. Yakovlev, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 60, 325 (1961).
10. E. Gagliardo, Rend. Seminario matem. Univ. Padova, 1957, p. 27.
11. A. D. Dzhabrailov, DAN, 159, No. 2 (1964).