UDC 513.88:513.83

MATHEMATICS

A. D. BERIEV

EMBEDDING THEOREMS FOR THE SPACES \(W_{p_0\bar p}^{(\bar l)}\), \(B_{p_0\bar\theta}^{(\bar l)}\)

(Presented by Academician S. L. Sobolev on 18 V 1965)

In the present note embedding theorems are proved for the spaces

\[ W_{p_0\bar p}^{(\bar l)}(E^n),\quad B_{p_0\bar\theta}^{(\bar l)}(E^n), \]

generalizing the results of V. P. Il’in \((^4)\) and A. Kh. Gudiyev \((^5)\); \(l_i\) \((i=1,\ldots,s)\) are arbitrary positive numbers.

Let \(E^\mu, E^{n_j}, E^{n(i)}\) be, respectively, \(\mu\)-, \(n_j\)-, and \(n^{(i)}\)-dimensional Euclidean spaces of points \(\bar t(t_1,t_2,\ldots,t_\mu)\), \(\bar x^j(x_1^j,\ldots,x_{n_j}^j)\), \(\bar y^{(i)}(y_1^{(i)},\ldots,y_{n(i)}^{(i)})\);

\[ n=\sum_{i=1}^{s} n^{(i)}=\sum_{j=1}^{\tau} n_j;\quad E^{n(i)}\cap E^{n_j}=E^{m_j^{(i)}};\quad r=r(\bar x,\bar y,\bar t)=\left[\sum_{i=1}^{s} r_i^{2/\chi_i}+|\bar t|^{2/\chi_0}\right]^{1/2}, \]

where

\[ r_i=\bigl[|\bar y^{(i)}-\bar x^{(i)}|\bigr]^{1/2} =\left[\sum_{k=1}^{n(i)}(y_k^{(i)}-x_k^{(i)})^2\right]^{1/2}; \quad \chi_i>0\ (i=0,1,\ldots,s); \quad D_{0;s} \]

\[ =Ш_{h^{\chi_0}}^\mu(\bar 0)\times\prod_{i=1}^{s} Ш_{h^{\chi_i}}^{n(i)}(\bar x^{(i)}); \quad Ш_{h^{\chi_0}}^\mu(\bar 0)\subset E^\mu \]
is the \(\mu\)-dimensional ball of radius \(h^{\chi_0}\) with center at \((\bar 0)\); \(Ш_{h^{\chi_i}}^{n(i)}(\bar x^{(i)})\subset E^{n(i)}\) is the \(n^{(i)}\)-dimensional ball of radius \(h^{\chi_i}\) with center at \((\bar x^{(i)})\); \((\bar l)=(l_1,\ldots,l_s)\); \(\bar p=(p_1,\ldots,p_s)\); \(\theta=(\theta_1,\ldots,\theta_n)\); \(s\le n\); \((\rho_{0;s})=(\rho_0,\rho_1,\ldots,\rho_s)\); \((\bar q)=(q_1,\ldots,q_\tau)\),

\[ A_{(D_{0;s})}^{(\rho_{0;s})}\bigl[|f|^{\rho_s}\bigr] \equiv \|f\|_{L_{(\rho_{0;s})}(D_{0;s})}. \]

Theorem 1. If

\[ F(\bar y,\bar t)\in L_{(\rho_{0;s})}(E^{n+\mu}), \]

\[ 1\le \rho_i\le q_j<\infty\quad (i=0,1,\ldots,s;\ j=1,\ldots,\tau);\quad \beta>0, \]

\[ \gamma>-\mu\rho_s/\rho_s'\rho_0, \]

\[ \lambda=\frac{\rho_s}{\rho_s'}\left(\sum_{i=1}^{s}\frac{n^{(i)}\chi_i}{\rho_i}+\frac{\mu\chi_0}{\rho_0}\right) +\sum_{j=1}^{\tau}\frac{1}{q_j}\sum_{i=1}^{s}m_j^{(i)}\chi_i, \]

then

\[ \left\| \left( A_{(D_{0;s})}^{(\rho_{0;s})} \left[ \frac{|\bar t|^\gamma F(\bar y,\bar t)} {\left(\sqrt{r^2+H^2}\right)^{\lambda+\gamma\chi_0+\alpha}} \right] \right)^{\rho_s} \right\|_{L_{(\bar q)}(E^n)} \le \]

\[ \leq \begin{cases} c_1 h^\beta \|F\|_{L(\bar\rho_0;s)(E^{n+\mu})}, & \text{if } \alpha=-\beta,\ H=0,\\[4pt] c_2 H^{-\beta}\|F\|_{L(\bar\rho_0;s)(E^{n+\mu})}, & \text{if } \alpha=\beta,\ 0<H\le h,\\[6pt] c_3\left\| \dfrac{|\bar t|^\gamma F(\bar y,\bar t)} {\left(\sqrt{|\bar t|^{2/\chi_0}+H^2}\right)^{\gamma\chi_0}} \right\|_{L(\bar\rho_0;s)(E^n\times \Pi^\mu_{h\chi_0}(\bar 0))}, & \text{if } \alpha=0,\ \gamma\ge0,\ 0<H\le h, \end{cases} \]
where \(c_1,c_2,c_3\) are constants independent of \(H,h,F\).

Theorem 2. If
\[ F(\bar y,\bar t)\in L_p(E^{n+\mu}),\qquad 1<p<q_j<\infty\quad (j=1,\ldots,\tau), \]
\[ \lambda=\frac{\mu\chi_0}{p'}+ \sum_{j=1}^{\tau}\left(\frac1{p'}+\frac1{q_j}\right) \sum_{i=1}^{s} m_j^{(i)}\chi_i, \]
then
\[ \left\| \int_{E^\mu}\! d\bar t\int_{E^n}\frac{F(\bar y,\bar t)}{r^\lambda}\,d\bar y \right\|_{L(\bar q)(E^n)} \le c\,\|F\|_{L_p(E^{n+\mu})}, \tag{1} \]
where \(c\) does not depend on \(F\).

Proof. On the basis of the theorem of Benedek and Panzone \((^3)\), the proof of the inequality
\[ \int_{E_x^n}\int_{E_y^n}\int_{E_t^\mu} \frac{\varphi(\bar x)F(\bar y,\bar t)}{r^\lambda}\, d\bar t\,d\bar y\,d\bar x \le c_1\|F\|_{L_p(E^{n+\mu})}\|\varphi\|_{L(\bar q')(E^n)} \tag{2} \]
is equivalent to the proof of inequality (1).

The following estimate holds:
\[ \int_{E_t^\mu}\frac{|F(\bar y,\bar t)|}{r^\lambda}\,d\bar t \le c_2\, \bar r^{-\sum_{j=1}^{\tau}(1/p'+1/q_j)\sum_{i=1}^{s}m_j^{(i)}\chi_i} \, \|F\|_{L_p(E^\mu)}, \tag{3} \]
where
\[ \bar r^2=\sum_{i=1}^{s} r_i^{2/\chi_i} > c_3\sum_{j=1}^{\tau}\sum_{i=1}^{s} r_{j,i}^{2/\chi_i}, \qquad r_{j,i}= \sum_{k=\sum_{\eta=1}^{j-1}m_\eta^{(i)}+1}^{\sum_{\eta=1}^{j}m_\eta^{(i)}} \bigl(x_k^{(i)}-y_k^{(i)}\bigr)^2. \]

If inequality (3) is taken into account, then
\[ \int_{E_x^n}\int_{E_y^n}\int_{E_t^\mu} \frac{\varphi(\bar x)F(\bar y,\bar t)}{r^\lambda}\, d\bar t\,d\bar y\,d\bar x \le \]
\[ \le c_4 \int_{E_x^{m_\tau^{(1)}}} \int_{E_y^{m_\tau^{(1)}}} \cdots \int_{E_x^{m_\tau^{(s)}}} \int_{E_y^{m_\tau^{(s)}}} \int_{E_x^{m_1^{(1)}}} \int_{E_y^{m_1^{(1)}}} \cdots \]
\[ \cdots \int_{E_x^{m_1^{(s)}}} \int_{E_y^{m_1^{(s)}}} \frac{ \varphi(\bar x)\|F\|_{L_p(E^\mu)} }{ \left(\sum_{j=1}^{\tau}\sum_{i=1}^{s} r_{j,i}^{2/\chi_i}\right)^{ \frac12\left[ \sum_{j=1}^{\tau}(1/p'+1/q_j)\sum_{i=1}^{s}m_j^{(i)}\chi_i \right]} } \,d\bar y_1^{(s)}\,d\bar x_1^{(s)}\cdots \]
\[ \cdots d\bar y_1^{(1)}\,d\bar x_1^{(1)} \cdots d\bar y_\tau^{(s)}\,d\bar x_\tau^{(s)} \cdots d\bar y_\tau^{(1)}\,d\bar x_\tau^{(1)}. \tag{4} \]

If Sobolev’s theorem \((^1)\) is applied \(s-\tau\) times to the right-hand side of inequality (4), we obtain the required result.

On the basis of Theorems 1 and 2, with the aid of the integral inequalities of V. P. Il’in (⁴), the following embedding theorems are proved for \(E^n\).

Theorem 3. If \(f \in W^{(\bar l)}_{p_0\bar p}(E^n)\), \(\nu_j^{(i)}\) \((i=1,\ldots,s;\ j=1,\ldots,n(i))\) are nonnegative integers satisfying the conditions

\[ \nu \sum_{i=1}^{s}\nu^{(i)},\qquad \nu^{(i)}=\sum_{j=1}^{n(i)}\nu_j^{(i)},\qquad 1\le p_i\le q_j<\infty \quad (i=0,1,\ldots,s;\ j=1,\ldots,\tau), \]

\[ \varepsilon = 1-\sum_{i=1}^{s}\frac{n^{(i)}}{l_i p_i} -\sum_{i=1}^{s}\chi_i\nu^{(i)} +\sum_{j=1}^{\tau}\frac{1}{q_j}\sum_{i=1}^{s}m_j^{(i)}\chi_i \ge 0,\qquad h>0, \]

then:

\[ 1.\quad \left\|D_x^\nu f(\bar x)\right\|_{L_{\bar q}(E^n)} \le c_1\|f\|_{W^{(\bar l)}_{p_0\bar p}(E^n)}, \qquad \text{if } \varepsilon>0. \]

\[ 2.\quad \left\|D_x^\nu f(\bar x)\right\|_{L_{\bar q}(E^n)} \le c_2\|f\|_{L^{(\bar l)}_{\bar p}(E^n)}, \qquad \text{if } \varepsilon=0,\ 1<p_i<q_j<\infty \]

\[ (i=1,\ldots,s;\ j=1,\ldots,\tau). \]

Theorem 4. If

\[ f\in B^{(\bar l)}_{p_0\bar p,\bar\theta}(E^n),\qquad 1\le p_i\le \theta_i<\infty\ (i=1,\ldots,n),\qquad 1\le p_i\le q_j<\infty \]

\[ (i=0,1,\ldots,n;\ j=1,\ldots,\tau), \]

\[ \varepsilon = 1-\sum_{i=1}^{n}\frac{1}{l_i p_i} -\sum_{i=1}^{n}\nu^{(i)}\chi_i +\sum_{j=1}^{\tau}\frac{1}{q_j}\sum_{i=1}^{n}\chi_i \ge 0, \]

then:

\[ 1.\quad \left\|D_x^\nu f(\bar x)\right\|_{L_{\bar q}(E^n)} \le c_1\|f\|_{B^{(\bar l)}_{p_0\bar p,\bar\theta}(E^n)}, \qquad \text{if } \varepsilon>0. \]

\[ 2.\quad \left\|D_x^\nu f(\bar x)\right\|_{L_{\bar q}(E^n)} \le c_2\|f\|_{\mathscr L^{(\bar l)}_{\bar p}(E^n)}, \qquad \text{if } \varepsilon=0,\ p_i=\theta_i,\ 1<p_i<q_j<\infty \]

\[ (i=1,\ldots,n;\ j=1,\ldots,\tau). \]

We indicate the path of the proof of Theorem 4. For the proof we use inequality (29.2) of V. P. Il’in (⁴) for the domain

\[ D_{1;s}= \prod_{i=1}^{s}\Pi_{h\chi_i}^{\,n(i)}(\bar x^{(i)}). \]

Applying Minkowski’s inequality to (29.2), we shall have

\[ \begin{aligned} \left\|D_x^\nu f(\bar x)\right\|_{L_{\bar q}(E^n)} &\le C_3 h^{-\sum_{i=1}^{n}\chi_i(1+\nu^{(i)})} \left\| \int_{D_{1;s}} |f(\bar y)|\,d\bar y \right\|_{L_{\bar q}(E^n)} \\ &\quad+ \sum_{i=1}^{n} \left\| \int_{I_{h\chi_i}(0)} dt \int_{D_{1;s}} \frac{\left|\Delta_i^2(t/2)D_{y^{(i)}}^{l_i}f(\bar y)\right|}{r^{\lambda_i}} \,d\bar y \right\|_{L_{\bar q}(E^n)} . \end{aligned} \tag{5} \]

The estimate holds

\[ h^{-\sum_{i=1}^{n}\chi_i(1+\nu^{(i)})} \left\| \int_{D_{1;s}} |f|\,dy \right\|_{L_{\bar q}(E^n)} \le c_4 h^{-\delta}\|f\|_{L_{p_0}(E^n)}, \tag{6} \]

where

\[ \delta=1-\varepsilon+\sum_{i=1}^{n}\frac{1}{l_i}\left(\frac{1}{p_0}-\frac{1}{p_i}\right). \]

If \(\varepsilon>0\), then, on the basis of Theorem 1, for \(\chi_0=\chi_i\), \(\mu=1\), \(\beta=\varepsilon\), \(\rho_j=p_i\) \((j=0,1,\ldots,s)\),

\[ \bar{\lambda}=\lambda=\frac{1}{p_i}\left(\sum_{i=1}^{n}\chi_i+\chi_i\right)+\sum_{j=1}^{\tau}\frac{1}{q_j}\sum_{i=1}^{n}\chi_i \]

after simple transformations we obtain the estimate

\[ \left\| \int_{h}^{\chi_i(\bar{0})} dt \int_{D_{1;s}} \frac{F(\bar{y},\bar{t})\,|\bar{t}|^{\gamma}}{r^{\bar{\lambda}+\gamma\chi_i-\varepsilon}}\,d\bar{y} \right\|_{L_{\bar{(q)}}(E^n)} \leq c_5 h^{\varepsilon}\, \|f\|_{\mathscr{L}_{p_i,y^{(i)}}^{\,l_i}(E^n)}. \tag{7} \]

If \(\varepsilon=0\), then, on the basis of Theorem 2, we obtain the estimate

\[ \left\| \int_{h}^{\chi_i(\bar{0})} dt \int_{D_{1;s}} \frac{F(\bar{y},\bar{t})}{r^{\bar{\lambda}}}\,d\bar{y} \right\|_{L_{\bar{(q)}}(E^n)} \leq c_6\, \|f\|_{\mathscr{L}_{p_i,y^{(i)}}^{\,l_i}(E^n)}. \tag{8} \]

From inequalities (5), (6), (7), and (8) we obtain

\[ D_{\bar{x}}^{\,\nu} f(\bar{x})\big\|_{L_{\bar{(q)}}(E^n)} \leq \begin{cases} c_7\left(h^{-\delta}\|f\|_{L_{p_0}(E^n)} +h^{\varepsilon}\|f\|_{\mathscr{L}_{\frac{\bar{p}}{\bar{\theta}}}^{\,\bar{l}}(E^n)}\right), & \text{if } \varepsilon>0,\\[6pt] c_8\left(h^{-\delta}\|f\|_{L_{p_0}(E^n)} +\|f\|_{\mathscr{L}_{\bar{p}}^{\,\bar{l}}(E^n)}\right), & \text{if } \varepsilon=0,\ \theta_i=p_i,\\[6pt] 1<p_i<q_j<\infty\quad (i=1,\ldots,n;\ j=1,\ldots,\tau). \end{cases} \tag{9} \]

\[ \tag{10} \]

If for the right-hand side of (9) one finds the minimum of the function for the corresponding \(h\), and in (10) lets \(h\to\infty\), then we obtain what was required.

North Ossetian State
Medical Institute

Received
13 V 1965

REFERENCES

1. S. L. Sobolev, Some applications of functional analysis in mathematical physics, L., 1950.
2. S. L. Sobolev, Matem. sborn., 4 (46), No. 3, 471 (1938).
3. A. Benedek, R. Panzone, Duke Math. J., 28, No. 3 (1961).
4. V. P. Il’in, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 66 (1962).
5. A. Kh. Gudiev, DAN, 160, No. 2 (1965).
6. A. Kh. Gudiev, DAN, 149, No. 2 (1963).