UDC 517.947.42+513.881

MATHEMATICS

A. D. BERIEV

EMBEDDING THEOREMS FOR WEIGHTED FUNCTION SPACES

(Presented by Academician S. L. Sobolev, 13 III 1968)

In the present paper we consider embedding theorems for the weighted function spaces \(W_{p,\alpha}^{r}(\overset{+}{E}{}^{\,n})\), \(L_{p,\alpha}^{r}(\overset{+}{E}{}^{\,n})\) into a space with mixed norm. \(E^n\) is the set of points \(\{x_1,x_2,\ldots,x_n\}\) of the \(n\)-dimensional Euclidean space \(E^n\) for which \(x_n>0\).

Let \(f(\bar{x})\) be a smooth function defined in the space \(E^n\). The numbers \(\alpha\), \(r_i\) \((i=1,\ldots,n)\), and \(p\) are real and satisfy the conditions \(\alpha>-1\), \(0<r_i<1\), \(1\le p\le \infty\). We shall say that \(f\in L_{p,\alpha}^{r_i}(\overset{+}{E}{}^{\,n})\) if

\[ \|f\|_{L_{p,\alpha}^{r_i}(\overset{+}{E}{}^{\,n})} = \left( \int_{0}^{\infty}\frac{dt}{t^{1+pr_i}} \int_{\overset{+}{E}{}^{\,n}} x_n^\alpha\,|\Delta_i(t)f(\mathbf{x})|\,d\mathbf{x} \right)^{1/p} <\infty, \]

\[ \|f\|_{L_{p,\alpha}^{r}(\overset{+}{E}{}^{\,n})} = \sum_{i=1}^{n} \|f\|_{L_{p,\alpha}^{r_i}(\overset{+}{E}{}^{\,n})}, \]

\[ \|f\|_{W_{p,\alpha}^{r}(\overset{+}{E}{}^{\,n})} = \sum_{i=1}^{n} \|f\|_{L_{p,\alpha}^{r_i}(\overset{+}{E}{}^{\,n})} + \|f\|_{L_p(\overset{+}{E}{}^{\,n})}. \]

The closures of smooth finite functions in the corresponding norms will be denoted by \(W_{p,\alpha}^{r}(\overset{+}{E}{}^{\,n})\) and \(L_{p,\alpha}^{r}(\overset{+}{E}{}^{\,n})\). We shall say that \(f(\mathbf{x})\in L^{\rho_s}(p,p_1,p_2,\ldots,p_m)(E^m)\) if

\[ \|f\|_{L^{\rho_s}_{(p,p_1,p_2,\ldots,p_m)}(E^m)} = \left( \int_{0}^{\infty}\frac{dt}{t^{1+\rho_s p}} \|\Delta_s(t)f(\mathbf{x})\|_{L(p_1,p_2,\ldots,p_m)(E^m)}^{p} \right)^{1/p} <\infty, \]

\[ \|f\|_{\vec{L}^{\,\rho}_{(p,p_1,p_2,\ldots,p_m)}(E^m)} = \sum_{s=1}^{m} \|f\|_{L^{\rho_s}_{(p,p_1,p_2,\ldots,p_m)}(E^m)}, \]

\[ \|f\|_{\vec{W}^{\,\rho}_{(p,p_1,p_2,\ldots,p_m)}(E^m)} = \|f\|_{L(p_1,\ldots,p_m)(E^m)} + \|f\|_{\vec{L}^{\,\rho}_{(p,p_1,\ldots,p_m)}(E^m)}, \]

where \(0<\rho_s<1\) \((s=1,\ldots,m)\),

\[ \|f\|_{L(p_1,p_2,\ldots,p_m)(E^m)} = \left( \int_{-\infty}^{\infty} \left( \int_{-\infty}^{\infty} \cdots \left( \int_{-\infty}^{\infty} |f|^{p_1}\,dx_1 \right)^{p_2/p_1} \cdots \right)^{p_m/p_{m-1}} dx_m \right)^{1/p_m} <\infty. \]

We formulate the results obtained in the form of embedding theorems.

Theorem 1. If \(f(\bar{x})\in W_{p,\alpha}^{r}(\overset{+}{E}{}^{\,n})\) and the conditions

\[ 1<p\le p_1\le \cdots \le p_m<\infty,\quad 0<m\le n-1,\quad \alpha>-1, \]

\[ 0<r_i<1\quad (i=1,\ldots,n), \]

are satisfied,

\[ \varepsilon=1+\sum_{i=1}^{m}\frac{1}{p_i r_i}-\frac{1}{p}\sum_{i=1}^{n}\frac{1}{r_i}-\frac{\alpha}{p r_n}>0, \]

then, for \(x_n=0\), \(f\in L_{(p_1,p_2,\ldots,p_m)}(E^m)\), and the inequality

\[ \|f\|_{L_{(p_1,p_2,\ldots,p_m)}(E^m)} \leqslant c\|f\|_{W^{\mathbf r}_{p,\alpha}(E^n_+)} \]

holds.

Theorem 2. If \(f(x)\in L^{\mathbf r}_{p,\alpha}(E^n_+)\) and the conditions

\[ 1<p\leqslant p_1\leqslant\cdots\leqslant p_m<\infty,\quad 0<m\leqslant n-1,\quad \alpha>-1, \]

\[ 0<r_i<1\quad (i=1,\ldots,n), \]

\[ \varepsilon=1+\sum_{i=1}^{m}\frac{1}{p_i r_i}-\frac{1}{p}\sum_{i=1}^{n}\frac{1}{r_i}-\frac{\alpha}{p r_n}>0, \]

are satisfied, then, for \(x_n=0\), \(f\in L_{(p_1,p_2,\ldots,p_m)}(E^m)\), and the inequality

\[ \|f\|_{L_{(p_1,p_2,\ldots,p_m)}(E^m)} \leqslant c\|f\|_{L^{\mathbf r}_{p,\alpha}(E^n_+)} \]

holds.

Theorem 3. If \(f(x)\in W^{\mathbf r}_{p,\alpha}(E^n_+)\) and the conditions

\[ 1<p\leqslant p_1\leqslant\cdots\leqslant p_m<\infty,\quad 0<m\leqslant n-1,\quad \alpha>-1, \]

\[ 0<r_i<1\quad (i=1,\ldots,n), \]

\[ 0<\rho_s\leqslant \varepsilon r_s\quad (s=1,\ldots,m),\qquad \varepsilon=1+\sum_{i=1}^{m}\frac{1}{p_i r_i}-\frac{1}{p}\sum_{i=1}^{n}\frac{1}{r_i}-\frac{\alpha}{p r_n}>0, \]

are satisfied, then, for \(x_n=0\), \(f\in L^{\vec\rho}_{(p,p_1,\ldots,p_m)}(E^m)\), and the inequality

\[ \|f\|_{L^{\vec\rho}_{(p,p_1,\ldots,p_m)}(E^m)} \leqslant c\|f\|_{W^{\mathbf r}_{p,\alpha}(E^n_+)} \]

holds.

Theorem 4. If \(f(x)\in W^{\mathbf r}_{p,\alpha}(E^n_+)\) and the conditions

\[ 1<p\leqslant p_1\leqslant\cdots\leqslant p_m<\infty,\quad 0<m\leqslant n-1,\quad \alpha>-1, \]

\[ 0<r_i<1\quad (i=1,\ldots,n), \]

\[ 0<\rho_s\leqslant \varepsilon r_s\quad (s=1,\ldots,m),\qquad \varepsilon=1+\sum_{i=1}^{m}\frac{1}{r_i p_i}-\frac{1}{p}\sum_{i=1}^{n}\frac{1}{r_i}-\frac{\alpha}{p r_n}>0, \]

are satisfied, then, for \(x_n=0\), \(f\in W^{\vec\rho}_{(p,p_1,\ldots,p_m)}(E^m)\), and the inequality

\[ \|f\|_{W^{\vec\rho}_{(p,p_1,\ldots,p_m)}(E^m)} \leqslant c\|f\|_{W^{\mathbf r}_{p,\alpha}(E^n_+)} \]

holds.

In proving these theorems, the integral representation of V. P. Il’in \((^2)\) is used.

We shall outline the proof of one of the stated theorems, for example, Theorem 3.

We have

\[ \|f\|_{L^{\rho_s}_{(p,p_1,\ldots,p_m)}(E^m)} = \left( \int_{0}^{\infty} \frac{dt}{t^{1-\rho_s p}} \|\Delta_s(t)f(x)\|^{p}_{L_{(p_1,p_2,\ldots,p_m)}(E^m)} \right)^{1/p}; \tag{1} \]

we represent the function \(f(x)\) in the form

\[ f(x)=\sum_{i=1}^{?} f_i, \tag{2} \]

where

\[ f_1(\bar x,h)=\frac{c}{h^\omega}\int_0^{h^{\sum_1^n\sigma_j}} f(\bar x+\bar y)\Pi(\bar y,h)\,d\bar y, \]

\[ f_2(\bar x,t,h)=-c\sum_{i=1}^n\int_0^{t^{1/\sigma_s}}\frac{dv}{v^{1+\omega}} \int_0^{v^{\sum_1^n\sigma_j}} d\bar y \int_0^{v^{\sigma_i-y_i}}\Delta_i(\tau)f(\bar x+\bar y)R_i(\bar y,\tau,v)\,d\tau, \]

\[ f_3(x,t,h)=-c\sum_{i=1}^n\int_{t^{1/\sigma_s}}^h\frac{dv}{v^{1+\omega}} \int_0^{v^{\sum_1^n\sigma_j}} d\bar y \int_0^{v^{\sigma_i-y_i}}\Delta_i(\tau)f(\bar x+\bar y)R_i(\bar y,\tau,v)\,d\tau, \]

where \(\sigma_j=1/r_j\).

Taking equality (2) into account, we have

\[ \left\|\Delta_s(t)f(x)\right\|_{L_{(p_1,p_2,\ldots,p_m)}(E^m)} \leq \sum_{i=1}^3 \left\|\Delta_s(t)f_i\right\|_{L_{(p_1,p_2,\ldots,p_m)}(E^m)} . \tag{3} \]

The following estimates hold:

\[ \left\|\Delta_s(t)f_1(\bar x h)\right\|_{L_{(p)}(E^m)} \leq cth^{ -\frac1p\sum_1^n\sigma_j+\sum_1^m\frac{\sigma_j}{p_j}-\sigma_s } \left\|f\right\|_{L_p^+(E^n)}; \tag{4} \]

\[ \left\|\Delta_s(t)f_2(\bar x,t,h)\right\|_{L_{(p)}(E^m)} \leq c\sum_{i=1}^n \left\| \int_0^{t^{1/\sigma_s}} \frac{dv}{v^{1+\lambda_i-\varepsilon+\gamma\sigma_i-\alpha/pr_n}} \times \right. \]

\[ \left. \times \int_0^{v^{\sigma_i}}\tau^\gamma\,d\tau \int_x^{x+v^{\sum_1^n\sigma_j}} \frac{|\Delta_i(\tau)f(\bar y)|}{\tau^{1/p+r_i}}\,d\bar y \right\|_{L_{(p)}(E^m)}; \tag{5} \]

\[ \left\| \Delta_s(t)f_3(\bar x,t,h) \right\|_{L_{(p)}(E^m)} \leq ct\sum_{i=1}^n \left\| \int_{t^{1/\sigma_s}}^h \frac{dv}{v^{1+\lambda_i-\varepsilon+\gamma\sigma_i+\sigma_s-\alpha/pr_n}} \times \right. \]

\[ \left. \times \int_0^{v^{\sigma_i}}\tau^\gamma\,d\tau \int_x^{x+v^{\sum_1^n\sigma_j}} \frac{|\Delta_i(\tau)f(\bar y)|}{\tau^{1/p+r_i}}\,d\bar y \right\|_{L_{(p)}(E^m)} . \tag{6} \]

We have

\[ \left( \int_0^\infty \frac{dt}{t^{1+\rho_s p}} \left\|\Delta_s(t)f(\bar x)\right\|_{L_{(p)}(E^m)}^p \right)^{1/p} \leq \]

\[ \leq \left( \int_0^{h^{\sigma_s}} \frac{dt}{t^{1+\rho_s p}} \left\|\Delta_s(t)f(\bar x)\right\|_{L_{(p)}(E^m)}^p \right)^{1/p} + \left( \int_{h^{\sigma_s}}^\infty \frac{dt}{t^{1+\rho_s p}} \left\|\Delta_s(t)f(\bar x)\right\|_{L_{(p)}(E^m)}^p \right)^{1/p} \leq \]

\[ \leq \sum_{i=1}^3 \left( \int_0^{h^{\sigma_s}} \frac{dt}{t^{1+\rho_s p}} \left\|\Delta_s(t)f_i\right\|_{L_{(p)}(E^m)}^p \right)^{1/p} + \left( \int_{h^{\sigma_s}}^\infty \frac{dt}{t^{1+\rho_s p}} \left\|\Delta_s(t)f(\bar x)\right\|_{L_{(p)}(E^m)}^p \right)^{1/p} = \]

\[ = \sum_{i=1}^3 A_i+B . \tag{7} \]

If we take into account (4), (5), and (6), then after a series of estimates we obtain

\[ A_1= \left( \int_{0}^{h^{\sigma_s}} \frac{dt}{t^{1+\rho_s p}}\, \|\Delta_s(t)f_1\|_{L(p)(E^m)}^p \right)^{1/p} \ll c h^{-(\delta+\rho_s\sigma_s)} \|f\|_{L_p(\stackrel{+}{E^n})}, \]

\[ A_2,\ A_3 \ll c h^{\varepsilon-\rho_s\sigma_s} \|f\|_{L_{p,\alpha}^{\,r}(\stackrel{+}{E^n})}. \]

Consequently,

\[ \sum_{i=1}^{3} A_i \ll c\left( h^{-(\delta+\rho_s\sigma_s)} \|f\|_{L_p(\stackrel{+}{E^n})} + h^{\varepsilon-\rho_s\sigma_s} \|f\|_{L_{p,\alpha}^{\,r}(\stackrel{+}{E^n})} \right). \tag{8} \]

It is not difficult to show that \(B\) is no greater than the right-hand side of inequality (8); therefore

\[ \|f\|_{L_{\vec p}(p,p_1,\ldots,p_m)(E^m)} \ll c\left( h^{-(\delta+\rho_s\sigma_s)} \|f\|_{L_p(\stackrel{+}{E^n})} + h^{\varepsilon-\rho_s\sigma_s} \|f\|_{L_{p,\alpha}^{\,r}(\stackrel{+}{E^n})} \right). \]

Hence, putting \(h=1\), we obtain the required result.

Corollary. If \(\varepsilon-\rho_s\sigma_s=0\), then as \(h\to\infty\)

\[ \|f\|_{L_{\vec p}(p,p_1,\ldots,p_m)(E^m)} \ll c\|f\|_{L_{p,\alpha}^{\,r}(\stackrel{+}{E^n})}. \]

Received
24 I 1968

CITED LITERATURE

\({}^{1}\) S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics, Leningrad, 1950. \({}^{2}\) V. P. Il’in, V. A. Solonnikov, Trudy Mat. Inst. im. V. A. Steklova AN SSSR, 66, 205 (1962). \({}^{3}\) L. D. Kudryavtsev, Trudy Mat. Inst. im. V. A. Steklova AN SSSR, 55 (1959). \({}^{4}\) A. D. Dzhabrailov, Author’s abstract of Candidate’s dissertation, 1966.