UDC 517.947.42

MATHEMATICS

R. D. KULOV

ON EMBEDDING THEOREMS FOR BESOV SPACES WITH MIXED NORM

(Presented by Academician S. L. Sobolev on February 17, 1970)

Embedding theorems for the spaces \(W^l_{(p)}\) (\(l\) integer), \(H^l_{(p)}\), \(S^r_{(p)}H\), \(L^r_{(p),(\theta)}\) with mixed norm have been considered in a number of works \((^{5-15})\). In the present note embedding theorems are given for the spaces \(B^l_{(p)}(E^n)\) with mixed norm. Weighted spaces \(B^l_{(p),\alpha}(\overset{+}{E}{}^{\,n})\) are also considered. Let \(E^n\) be the \(n\)-dimensional real Euclidean space of points \(\mathbf{x}=(x_1,\ldots,x_n)\); \(\overset{+}{E}{}^{\,n}=\{\mathbf{x}:x_n>0\}\); \(E^n=E^{n_1}\times\cdots\times E^{n_k}=E^{m_1}\times\cdots\times E^{m_\tau}\); \(\overset{+}{E}{}^{\,n}=\widetilde E^{n_1}\times\cdots\times \widetilde E^{n_k}=\widetilde E^{m_1}\times\cdots\times \widetilde E^{m_\tau}\), and the decompositions into \(E^{n_i}\) and \(E^{m_j}\) (\(\widetilde E^{n_i}\) and \(\widetilde E^{m_j}\)) \((i=1,\ldots,k;\ j=1,\ldots,\tau)\) do not depend on one another; the vectors of the subspaces \(E^{n_i}\) and \(E^{m_j}\) will be denoted respectively by \(\mathbf{x}_{n_i}=(x_{i1},\ldots,x_{in_i})\) and \(\mathbf{x}_{m_j}=(x^{(j)}_1,\ldots,x^{(j)}_{m_j})\) \((i=1,\ldots,k;\ j=1,\ldots,\tau)\); \(\mathbf{p}=(p_1,\ldots,p_k)\), \(\mathbf{q}=(q_1,\ldots,q_\tau)\), \(\mathbf{l}=(l_1,\ldots,l_n)\), \(\mathbf{r}=(r_1,\ldots,r_n)\); \(\nu_i,\bar l_i,\bar r_i\) \((i=1,\ldots,n)\) are nonnegative integers, \(\alpha\geq0\).

\(\square_h^{\varkappa_{n_i}}(\mathbf{x}_{n_i})\) is an \(n_i\)-dimensional parallelepiped in \(E^{n_i}\) with vertex at the point \(\mathbf{x}_{n_i}\) and edges \(h^{\varkappa_{i1}},\ldots,h^{\varkappa_{in_i}}\) \((i=1,\ldots,k)\).

Let \(f(y)\) be a smooth finite function in \(E^n\) and

\[ \|f\|_{\mathscr L^{\,l_i}_{(p),\alpha}(\overset{+}{E}{}^{\,n})} = \left( \int_0^\infty \frac{dt}{t^{1+p_k\alpha_i}} \left\| y_n^{\alpha/p_1}\Delta_i^2(t)D_i^{\bar l_i}f(y) \right\|_{L_{(p)}(\overset{+}{E}{}^{\,n})}^{p_k} \right)^{1/p_k} <\infty \tag{1} \]

(where \(L_{(p)}(\overset{+}{E}{}^{\,n})\) is the space with mixed norm (see (8)));

\[ \|f\|_{B^l_{(p),\alpha}(\overset{+}{E}{}^{\,n})} = \|f\|_{L_{(p)}(\overset{+}{E}{}^{\,n})} + \sum_{i=1}^n \|f\|_{\mathscr L^{\,l_i}_{(p),\alpha}(\overset{+}{E}{}^{\,n})}. \tag{2} \]

Putting in (2) \(\alpha=0\) and replacing \(\overset{+}{E}{}^{\,n}\) by \(E^n\), we obtain

\[ \|f\|_{B^l_{(p)}(E^n)} = \|f\|_{L_{(p)}(E^n)} + \sum_{i=1}^n \|f\|_{\mathscr L^{\,l_i}_{(p)}(E^n)}. \tag{3} \]

By the spaces \(B^l_{(p)}(E^n)\) and \(B^l_{(p),\alpha}(\overset{+}{E}{}^{\,n})\) we shall mean the closures of the set of smooth finite functions in the norms (2) and (3).

We formulate the main results.

Theorem 1. If \(1<p_i\leq q_i<\infty\) \((i=1,\ldots,k;\ j=1,\ldots,\tau)\); \(l_i=\bar l_i+\alpha_i\), \(0<\alpha_i\leq1\) \((i=1,\ldots,n)\);

\[ 1-\sum_{i=1}^n \frac{1}{l_i}(1+\nu_i) +\sum_{i=1}^k \frac{1}{p_i}\sum_{j=1}^{n_i}\frac{1}{l_{ij}} +\sum_{i=1}^{\tau}\frac{1}{q_i}\sum_{j=1}^{m_i}\frac{1}{l^{(i)}_j} >0, \]

\(f\in B^l_{(p)}(E^n)\), then \(D_1^{\nu_1}\cdots D_n^{\nu_n}f\in L_{(q)}(E^n)\), and the inequality

\[ \left\| D_1^{\nu_1}\cdots D_n^{\nu_n}f \right\|_{L_{(q)}(E^n)} \leq C\|f\|_{B^l_{(p)}(E^n)} \tag{4} \]

holds.

Theorem 2. If \(1<p_i\le q_j<\infty\) \((i=1,\ldots,k;\ j=1,\ldots,\tau)\); \(l_i=\bar l_i+\alpha_i,\ 0<\alpha_i\le 1,\ r_i=\bar r_i+\beta_i,\ 0<\beta_i\le 1\) \((i=1,\ldots,n)\);

\[ 1-\sum_{i=1}^{n}\frac1{l_i} +\sum_{i=1}^{k}\frac1{p_i'}\sum_{j=1}^{n_i}\frac1{l_{ij}} +\sum_{i=1}^{\tau}\frac1{q_i}\sum_{j=1}^{m_j}\frac1{l_j^{(i)}} >\frac{r_s}{l_s} \qquad (s=1,\ldots,n), \]

\(f\in B_{(\mathbf p)}^{\mathbf l}(E^n)\), then \(f\in B_{(\mathbf q)}^{\mathbf r}(E^n)\) and the inequality

\[ \|f\|_{B_{(\mathbf q)}^{\mathbf r}(E^n)} \le C\|f\|_{B_{(\mathbf p)}^{\mathbf l}(E^n)} \tag{5} \]

holds.

Theorem 3. If \(1<p_i\le q_j<\infty,\quad 0\le \alpha<p_1/\max_i p_i'\) \((i=1,\ldots,k;\ j=1,\ldots,\tau)\); \(l_i=\bar l_i+\alpha_i,\ 0<\alpha_i\le 1\) \((i=1,\ldots,n)\);

\[ 1-\sum_{i=1}^{n}\frac1{l_i}(1+\nu_i)-\frac{\alpha}{p_1l_n} +\sum_{i=1}^{k}\frac1{p_i'}\sum_{j=1}^{\tilde n_i}\frac1{l_{ij}} +\sum_{i=1}^{\tau}\frac1{q_i}\sum_{j=1}^{\tilde m_i}\frac1{l_j^{(i)}}>0, \]

\(f\in B_{(\mathbf p),\alpha}^{\mathbf l}(\overset{+}{E}{}^n)\), then \(D_1^{\nu_1}\cdots D_n^{\nu_n}f\in L_{(\mathbf q)}(\overset{+}{E}{}^n)\) and the inequality

\[ \|D_1^{\nu_1}\cdots D_n^{\nu_n}f\|_{L_{(\mathbf q)}(\overset{+}{E}{}^n)} \le C\|f\|_{B_{(\mathbf p),\alpha}^{\mathbf l}(\overset{+}{E}{}^n)} \tag{6} \]

holds.

Theorem 4. If \(1<p_i\le q_j<\infty,\quad 0\le \alpha<p_1/\max_i p_i'\) \((i=1,\ldots,k;\ j=1,\ldots,\tau)\); \(l_i=\bar l_i+\alpha_i,\quad 0<\alpha_i\le 1,\quad r_i=\bar r_i+\beta_i,\ 0<\beta_i\le 1\) \((i=1,\ldots,n)\);

\[ 1-\sum_{i=1}^{n}\frac1{l_i}-\frac{\alpha}{p_1l_n} +\sum_{i=1}^{k}\frac1{p_i'}\sum_{j=1}^{\tilde n_i}\frac1{l_{ij}} +\sum_{i=1}^{\tau}\frac1{q_i}\sum_{j=1}^{\tilde m_i}\frac1{l_j^{(i)}} >\frac{r_s}{l_s} \qquad (s=1,\ldots,n); \]

\(f\in B_{(\mathbf p),\alpha}^{\mathbf l}(\overset{+}{E}{}^n)\), then \(f\in B_{(\mathbf q)}^{\mathbf r}(\overset{+}{E}{}^n)\) and the inequality

\[ \|f\|_{B_{(\mathbf q)}^{\mathbf r}(\overset{+}{E}{}^n)} \le C\|f\|_{B_{(\mathbf p),\alpha}^{\mathbf l}(\overset{+}{E}{}^n)} \tag{7} \]

holds.

In inequalities (4)—(7) the constant \(C\) does not depend on \(f\).

Let us indicate the scheme of proof, for example, of Theorem 2. Consider the norm
\(\|f\|_{\mathscr L^{r_s}_{(\mathbf q)}(E^n)}\): its estimate, in view of Theorem 1, reduces to estimating the expression

\[ N= \left( \int_{0}^{h^{\chi_s}} \frac{dz}{z^{1+q_\tau\beta_s}} \|\Delta_s^2(z)D_s^{\bar r_s}f(x)\|_{L_{(\mathbf q)}(E^n)}^{q_\tau} \right)^{1/q_\tau}, \qquad \text{where }\chi_s=\frac1{l_s}\quad (s=1,\ldots,n). \]

Using the integral identity of V. P. Il’in \((^3)\), and making the necessary transformations and estimates, we obtain

\[ \begin{aligned} N \le{}& C\left[ \int_{0}^{h^{\chi_s}} \frac{dz}{z^{1+(\beta_s-2)q_\tau}} h^{(-\delta-\bar r_s\chi_s-2\chi_s)} \|f\|_{L_{(\mathbf p)}(E^n)}^{q_\tau} \right]^{1/q_\tau} \\ &+ C\sum_{i=1}^{n} \left[ \int_{0}^{h^{\chi_s}} \frac{dz}{z^{1+\beta_s q_\tau}} \times \left\| \int_{0}^{z^{1/\chi_s}} \frac{dv}{v^{1+\lambda_i-\varepsilon+\gamma_i\chi_i+r_s\chi_s}} \int_{0}^{v^{\chi_i}} t^{\gamma_i-\left(\frac1{p_k}+\alpha_i\right)}\,dt \int_x^{x+v^k} \Delta_i^2\!\left(\frac{t}{2}\right)D_i^{\bar l_i}f(y)\,dy \right\|_{L_{(\mathbf q)}(E^n)}^{q_\tau} \right]^{1/q_\tau} \\ &+ C\sum_{i=1}^{n} \left[ \int_{0}^{h^{\chi_s}} \frac{dz}{z^{1+(\beta_s-2)q_\tau}} \left\| \int_{z^{1/\chi_s}}^{h} \frac{dv}{v^{1+\lambda_i-\varepsilon+\bar r_s\chi_s+2\chi_s+\gamma_i\chi_i}} \int_{0}^{v^{\chi_i}} t^{\gamma_i-\left(1/p_k+\alpha_i\right)}\,dt \right.\right.\\ &\qquad\qquad\left.\left. \times \int_x^{x+v^k} \Delta_i^2\!\left(\frac{t}{2}\right)D_i^{\bar l_i}f(y)\,dy \right\|_{L_{(\mathbf q)}(E^n)}^{q_\tau} \right]^{1/q_\tau} = N_1+N_2+N_3, \end{aligned} \]

where

\[ \delta=\sum_{i=1}^{n}\chi_i-\sum_{i=1}^{k}\frac{1}{p_k}\sum_{j=1}^{n_i}\chi_{ij} -\sum_{i=1}^{\tau}\frac{1}{q_i}\sum_{j=1}^{m_i}\chi_j^{(i)}, \]

\[ 0<\gamma_i\leq \frac{1}{p_k}+\alpha_i,\qquad \lambda_i=\frac{\chi_i}{p_k}+\sum_{j=1}^{n}\chi_j-\delta,\qquad \varepsilon=1-\delta. \]

Let us estimate one of the expressions, for example \(N_2\) (\(N_1, N_3\) are estimated analogously). For this we choose \(\gamma_i\) so that \(\varepsilon-r_s\chi_s>\gamma_i\chi_i\) \((i=1,\ldots,n;\ s=1,\ldots,n)\); then, after elementary estimates, we shall have

\[ N_2\leq ch^{\varepsilon-r_s\chi_s}\sum_{i=1}^{n} \left[ \int_{0}^{h^{\chi_s}} \frac{dz}{1+\dfrac{q_\tau}{\chi_s}(\gamma_i\chi_i-\mu)} \left\| \left( \int_{0}^{z^{1/\chi_s}} \frac{dv}{ 1+p_k\displaystyle\sum_{i=1}^{\tau}\frac{1}{q_i} \sum_{j=1}^{m_i}\chi_j^{(i)}+\mu p_k } \right.\right.\right. \]

\[ \left.\left.\left. \times \int_{0}^{v^{\chi_i}} t^{\chi_i p_k-(1+p_k\alpha_i)}\,dt\, \left\|\Delta_i^2\!\left(\frac{t}{2}\right)\bar D_i^{\,l_i}f(y)\right\|_{L^{(p)} \left[ \Box_v x_{n_1}(x_{n_1}),\ldots,\Box_v x_{n_k}(x_{n_k}) \right]}^{p_k} \right)^{1/p_k} \right\|_{L^{(q)}(E^n)}^{q_\tau} \right]^{1/q_\tau} \leq \]

(where \(\mu\) is an arbitrary positive number); using the generalized Minkowski inequality and lemma (1) of A. Kh. Gudiev \(^{5}\) the required number of times, we shall have

\[ \leq ch^{\varepsilon-r_s\chi_s}\sum_{i=1}^{n} \left[ \int_{0}^{h^{\chi_s}} \frac{dz}{1+\dfrac{q_\tau}{\chi_s}(\gamma_i\chi_i-\mu)} \left( \int_{0}^{z^{1/\chi_s}} \frac{dv}{v^{1+\mu p_k}} \int_{0}^{v^{\chi_i}} t^{\chi_i p_k-(1+p_k\alpha_i)}\,dt \right.\right. \]

\[ \left.\left. \times \left\|\Delta_i^2\!\left(\frac{t}{2}\right)\bar D_i^{\,l_i}f(y)\right\|_{L^{(p)}(E^n)}^{p_k} \right)^{q_\tau} \right]^{1/q_\tau} \leq \]

changing the order of integration; choosing \(\mu\) so that \(\gamma_i\chi_i-\mu>0\), we obtain

\[ \leq ch^{\varepsilon-r_s\chi_s}\sum_{i=1}^{n}\|f\|_{\mathcal L_{(p)}^{\,l_i}(E^n)}. \]

From the estimates obtained, for \(h=1\) we obtain inequality (5).

In conclusion I express my deep gratitude to Acad. S. L. Sobolev for his attention to this work, and also to A. Kh. Gudiev for posing the question and for valuable advice.

Institute of Mathematics
Siberian Branch of the Academy of Sciences of the USSR
Novosibirsk

Received
9 II 1970

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