Full Text
UDC 513.881
IMBEDDING THEOREMS FOR MIXED SPACES
A. Kh. Gudiev
The paper studies properties of differentiable functions of several variables. The main content of the work is the derivation of imbedding theorems for new functional spaces
\(WB_{p_0,p_1,\ldots,p_s}^{l_1,\ldots,l_s}(k; s; D)\),
\(BW_{p_0,p_1,\ldots,p_s}^{l_1,\ldots,l_s}(k; s; D)\)—mixed spaces.
We note that the mixed spaces
\(WB_{p_0,p_1,\ldots,p_s}^{l_1,\ldots,l_s}(k; s; D)\), under certain conditions on \(p_0,p_1,\ldots,p_s,l_1,\ldots,l_s\), contain the spaces
\(W_{p_0,p_1,\ldots,p_s}^{l_1,\ldots,l_s}(D)\),
\(B_{p_0,p_1,\ldots,p_s}^{l_1,\ldots,l_s}(D)\), and in other cases are intermediate between them.
At the end of the paper a definition of general mixed spaces is given.
Let \(s\) be a natural number \(\le n\); let \(n_1,n_2,\ldots,n_s\) be natural numbers whose sum is \(n\). Let \(E^{n_i}\) be the \(n_i\)-dimensional Euclidean space of points
\(\bar{x}^{(i)}(x_1^{(i)},x_2^{(i)},\ldots,x_{n_i}^{(i)})\) \((i=1,2,\ldots,s)\);
\(E^n=E^{n_1}\times E^{n_2}\times\cdots\times E^{n_s}\) is the space of points
\(\bar{x}=(\bar{x}^{(1)},\bar{x}^{(2)},\ldots,\bar{x}^{(s)})=(x_1^{(1)},\ldots,x_{n_1}^{(1)},\ldots,x_1^{(s)},\ldots,x_{n_s}^{(s)})=(x_1,x_2,\ldots,x_n)\).
If \(n_\sigma=1\) \((\sigma\le s)\), then we shall write
\[ \bar{x}^{(\sigma)}=x_{n_1+n_2+\cdots+n_\sigma+1}=\tilde{x}_\sigma . \]
By \(D^{(i)}(x)\) we shall denote the set of points \(\bar{y}\in D\) for which
\(\bar{y}^{(j)}=\bar{x}^{(j)}\) \((j=1,2,\ldots,i-1,i+1,\ldots,s)\), and by \(\Pi_{H_i}^{(i)}(\bar{x})\) we shall denote the set of points \(\bar{y}\in E^n\) for which
\(\bar{y}^{(j)}=\bar{x}^{(j)}\) \((j=1,2,\ldots,i-1,i+1,\ldots,s)\) and
\(|\bar{y}^{(i)}-\bar{x}^{(i)}|<H_i\). Put
\[ D_{H_i}^{(i)}(\bar{x})=D\cap \Pi_{H_i}^{(i)}(\bar{x}),\qquad D_{\bar H}(\bar{x})=D\cap \prod_{i=1}^{s}\Pi_{H^{\chi_i}}^{(i)}(\bar{x}^{(i)}), \]
where \(\chi_i>0\) \((i=1,2,\ldots,s)\);
\(\bar{H}=(H^{\chi_1},H^{\chi_2},\ldots,H^{\chi_s})\),
\(|\bar{x}^{(i)}-\bar{y}^{(i)}|\) is the distance between the points \(\bar{x}^{(i)}\) and \(\bar{y}^{(i)}\) in \(E^{n_i}\); \(|x-y|\) is the distance between \(x\) and \(y\) in \(E^n\).
Let \(\bar{t}\) be a vector in \(E^n\), and let \(k\) be a natural number. By \((D)_{\bar{t},k}\) we denote the intersection of the domains \(D_j\) obtained by shifting the domain \(D\) respectively by the vectors \(-\dfrac{j}{k}\bar{t}\) \((j=0,1,\ldots,k)\). The sets
\((D(x,H))_{\bar{t},k}\) and \((D_i(x,H^{(i)}))_{\bar{t}^{(i)},k}\) are defined analogously. For \(k=1\) we shall write
\((D)_{\bar{t}}\), \((D(x,H))_{\bar{t}}\), \((D^{(i)}(x,H^{(i)}))_{\bar{t}^{(i)}}\).
We shall say that the domain \(D\) belongs to the class \(C^{(i)}(H_i,\omega_i)\),
if for each point \(\bar x \in D\) there exists an \(n_i\)-dimensional spherical sector \(V^{(i)}_{H_i}(\bar x)\) of radius \(H_i\), with solid angle \(\omega_i\) and with vertex at \(\bar x\), which is contained in \(D^{(i)}(\bar x)\).
We shall further say that \(D \in C(\bar H^{\,s}, \bar\omega^{\,s})\) \((\bar H^{\,s}=\bar H=(H_1,\ldots,H_s),\ \omega^s=(\omega_1,\ldots,\omega_s))\), if for each point \(\bar x \in D\) there exists a hypercylinder
\[
\Pi_{\bar H}(\bar x)=\prod_{i=1}^{s} V_{H_i}(\bar x^{(i)})
\]
of constant dimensions (with fixed \(H_i\) and \(\omega_i\)), contained in \(D\). In particular, when \(s=n\), i.e., when \(n_i=1\) \((i=1,2,\ldots,n)\), the hypercylinder will be an \(n\)-dimensional rectangle with vertex at \(\bar x\) and with edges parallel to the coordinate axes. In this case we shall write \(D \in C(\bar H^{\,n})\) \((\bar H^{\,n}=(H_1,\ldots,H_n)=\bar H)\).
Let \(\nu_j^{(i)}\) \((i=1,2,\ldots,s;\ j=1,2,\ldots,n_i)\) be nonnegative integers,
\[
\nu^{(i)}=\sum_{j=1}^{n_i}\nu_j^{(i)},\qquad \nu=\sum_{i=1}^{s}\nu^{(i)},
\]
and let \(f\) be a smooth function given in \(E^n\),
\[
D_{x_j^{(i)}}^{\nu_j^{(i)}} f(\bar x)=
\frac{\partial^{\nu_j^{(i)}} f(\bar x)}
{\bigl(\partial x_j^{(i)}\bigr)^{\nu_j^{(i)}}},\qquad
D_{\bar x^{(i)}}^{\nu^{(i)}} f=
\left(\prod_{j=1}^{n_i} D_{x_j^{(i)}}^{\nu_j^{(i)}}\right)f,
\]
\[
D_{\bar x}^{\nu} f=
\left(\prod_{i=1}^{s}\prod_{j=1}^{n_i} D_{x_j^{(i)}}^{\nu_j^{(i)}}\right)f.
\]
Let \(S^m\) be a finite or infinite \(m\)-dimensional surface in the space \(E^n\) \((1\le m\le n-1)\) of points \(\bar x=(x_1,\ldots,x_n)\), given by the equations
\[
x_i=
\begin{cases}
x_i, & \text{if } 1\le i\le m,\\
\varphi_{m+i}(x_1,\ldots,x_m), & \text{if } m<i\le n,
\end{cases}
\tag{1}
\]
where the functions \(\varphi_k(x_1,\ldots,x_m)\) are defined in some domain \(\Omega^m\) of the \(m\)-dimensional space \(E^m\) of points \(\bar x^{\,m}=(x_1,\ldots,x_m)\).
We shall say that \(S^m\) is a surface of class \(\widetilde C^{(1)}\) in \(E^n\), if it is given by equations of the form (1), and the functions \(\varphi_k(x_1,\ldots,x_m)=\varphi_k(\bar x^{\,m})\) are continuous and have continuous bounded partial derivatives of first order in \(\Omega^m\).
We shall say that a function \(f\in L^{l_i}_{p_i,\bar y^{(i)}}(D;H)\), if it has all generalized derivatives, in the Sobolev sense, of order \(l_i\) with respect to the variables \(y_1^{(i)},\ldots,y_{n_i}^{(i)}\) in the domain \(D\), satisfying the conditions:
\[
\|f\|_{L^{(l_i)}_{p_i,\bar y^{(i)}}(D)}
=
\sum_{j_1,\ldots,j_{l_i}=1}^{n_i}
\left[
\int_D
\left|
\frac{\partial^{l_i} f(\bar y)}
{\partial y_{j_1}^{(i)}\cdots \partial y_{j_{l_i}}^{(i)}}
\right|^{p_i}
\,d\bar y
\right]^{\frac1{p_i}}
<\infty,
\]
if \(l_i\) is an integer, or
\[
\|f\|_{L^{(l_i)}_{p_i,\bar y^{(i)}}(D;H)}
=
\sum_{j_1,\ldots,j_{l_i}=1}^{n_i}
\left[
\int_{\Pi_{\bar H}^{\,n_i}(0)}
\frac{d\bar t^{(i)}}{|\bar t|^{\,n_i+\alpha_i p_i}}
\times
\right.
\]
\[ \times \left\|\Delta\bigl(\bar t^{(i)}\bigr) \frac{\partial^{\bar l_i} f(\bar y)} {\partial y_{j_1}^{(i)} \cdots \partial y_{\bar j_{\bar l_i}}^{(i)}} \right\|_{L_{p_i}(D_{\bar t}^{(i)})}^{p_i} \right]^{1/p_i}, \tag{2} \]
if \(l_i\) is noninteger and \(l_i=\bar l_i+\alpha_i\).
We shall say below that a function \(f\) belongs to the space
\(W_{p_0,p_i,\bar y^{(i)}}^{(l_i)}(D;H)\) if \(f\in L_{p_0}(D)\),
\(f\in \mathscr L_{p_i,\bar y^{(i)}}^{(l_i)}(D;H)\); in this case we set
\[ \|f\|_{W_{p_0,p_i,\bar y^{(i)}}^{(l_i)}(D;H)} = \|f\|_{L_{p_0}(D)} + \|f\|_{\mathscr L_{p_i,\bar y^{(i)}}^{(l_i)}(D;H)} . \]
If \(p_0=p_i\), we shall write
\(f\in W_{p_i,\bar y^{(i)}}^{(l_i)}(D;H)\).
We shall say that a function \(f\in \mathscr L_{p_i,y_i}^{(l_i)}(D;H)\)
\((l_i=\bar l_i+\alpha_i;\ 0<\alpha_i\le 1)\), if it has a generalized derivative, in the sense of Sobolev, of order \(\bar l_i\) with respect to \(y_i\) in the domain \(D\), satisfying the condition
\[ \|f\|_{\mathscr L_{p_i,y_i}^{(l_i)}(D;H)} = \left[ \int_{J_H(0)} \frac{dt}{|t|^{1+\alpha_i Q_i}} \left\| \Delta_i^2\left(\frac{t}{2}\right) \frac{\partial^{\bar l_i} f(\bar y)}{\partial y_i^{\bar l_i}} \right\|_{L_{p_i}(D_{t\bar e_i})}^{Q_i} \right]^{1/Q_i} <\infty, \tag{3} \]
where \(\bar e_i\) is the unit vector directed along the \(y_i\)-axis.
We shall say that \(f\in B_{p_0,p_i,y_i}^{(l_i)}(D;H)\), if
\(f\in L_{p_0}(D)\), \(f\in \mathscr L_{p_i,y_i}^{(l_i)}\), and set
\[ \|f\|_{B_{p_0,p_i,y_i}^{(l_i)}(D;H)} = \|f\|_{L_{p_0}(D)} + \|f\|_{\mathscr L_{p_i,y_i}^{(l_i)}(D;H)} . \]
If in (2) and (3) the integration over \(\Pi_H^{n_i}(0)\), \(J_H(0)\) is replaced, respectively, by \(E^{n_i}\), \(E^1\), then we shall write
\(W_{p_0,p_i,\bar y^{(i)}}^{(l_i)}(D)\),
\(B_{p_0,p_i,y_i}^{(l_i)}(D)\).
Let \(D\) be a domain of the space \(E^n\) of points
\[ \bar y=(\bar y^{(1)},\ldots,\bar y^{(s)})=(y_1,y_2,\ldots,y_n), \]
\[ \bar y^{(i)}=(y_1^{(i)},\ldots,y_{n_i}^{(i)}),\quad i=1,2,\ldots,s;\qquad \sum_1^s n_i=n, \]
where
\[ n_i= \begin{cases} >1 & \text{for } 1\le i\le k,\\ =1 & \text{for } k<i\le s. \end{cases} \]
Definition. If
\[ f\in \begin{cases} W_{p_0,p_i,\bar y^{(i)}}^{(l_i)}(D;H) & \text{for } i=1,2,\ldots,k,\\ B_{p_0,p_i,y_i}^{(l_i)}(D;H) & \text{for } i=k+1,\ldots,s, \end{cases} \]
then we shall say that
\(f\in WB_{p_0,p_1,\ldots,p_s}^{\,l_1,\ldots,l_k,l_{k+1},\ldots,l_s}{}_{\,Q_{k+1},\ldots,Q_s}(k;s;D;H)\).
We define the norm in
\(WB_{p_0,p_1,\ldots,p_s}^{\,l_1,\ldots,l_k,l_{k+1},\ldots,l_s}{}_{\,Q_{k+1},\ldots,Q_s}(k;s;D;H)\)
by the equality
\[ \left\| f \right\|_{WB^{l_1,\ldots,l_{k+1},\ldots,l_s}_{p_0,p_1,\ldots,p_{k+1},\ldots,p_s;\,Q_{k+1},\ldots,Q_s}(k;\ s;\ D;\ H)} = \]
\[ = \left\| f \right\|_{L_{p_0}(D)} + \sum_{i=1}^{k}\left\| f \right\|_{L^{(l_i)}_{p_i,y^{(i)}}(D;\ H)} + \sum_{i=k+1}^{s}\left\| f \right\|_{\mathscr L^{(l_i)}_{p_i,y_1;\,Q_i}(D;\ H)} . \]
We shall call the spaces
\[ WB^{l_1,\ldots,l_{k+1},\ldots,l_s}_{p_0,p_1,\ldots,p_{k+1},\ldots,p_s;\,Q_{k+1},\ldots,Q_s}(k;\ s;\ D;\ H) \]
mixed spaces.
From the definition of the spaces
\[ WB^{l_1,\ldots,l_{k+1},\ldots,l_s}_{p_0,p_1,\ldots,p_{k+1},\ldots,p_s;\,Q_{k+1},\ldots,Q_s}(k;\ s;\ D;\ H) \]
it is clear that
\[ WB^{l_1,\ldots,l_{k+1},\ldots,l_s}_{p_0,p_1,\ldots,p_{k+1},\ldots,p_s;\,Q_{k+1},\ldots,Q_s}(k;\ s;\ D;\ H) = \begin{cases} B^{l_1,\ldots,l_s}_{p_0,p_1,\ldots,p_s;\,Q_1,\ldots,Q_s}(D;\ H), & \text{for } k=0,\\[4pt] W^{l_1,\ldots,l_s}_{p_0,p_1,\ldots,p_s}(D;\ H), & \text{for } k=s. \end{cases} \]
It is now clear how to define the spaces
\[ BW^{l_1,\ldots,l_k,l_{k+1},\ldots,l_s}_{p_0,p_1,\ldots,p_k,p_{k+1},\ldots,p_s;\,Q_1,\ldots,Q_k}(k;\ s;\ D;\ H). \]
If the coordinate axes are fixed, then, generally speaking,
\[ WB^{l_1,\ldots,l_s}_{p_0,p_1,\ldots,p_s}(k;\ s;\ D;\ H) \ne BW^{l_1,\ldots,l_s}_{p_0,p_1,\ldots,p_s}(k;\ s;\ D;\ H). \]
If \(l_i\) \((i=1,2,\ldots,s)\) are nonintegers and \(s=n\), then
\[ W^{l_1,\ldots,l_n}_{p_0,p_1,\ldots,p_n}(k;\ n;\ D;\ H) = BW^{l_1,\ldots,l_n}_{p_0,p_1,\ldots,p_n}(k;\ n;\ D;\ H). \]
Let us establish relations between the classes of functions
\[ WB^{l_1,\ldots,l_s}_{p_0,p_1,\ldots,p_s}(k;\ s;\ D;\ H) \]
and the known classes of functions
\[ W^{l_1,\ldots,l_s}_{p_0,p_1,\ldots,p_s}(D;\ H) \quad\text{and}\quad B^{l_1,\ldots,l_s}_{p_0,p_1,\ldots,p_s}(D;\ H). \]
For simplicity, consider the case \(n=2;\ s=2\). Suppose that \(l_1\) and \(l_2\) are integers, for, as noted above, when \(l_1\) and \(l_2\) are nonintegers,
\[ WB^{l_1,l_2}_{p_1,p_2}(1;\ 2;\ D;\ H) = W^{l_1,l_2}_{p_1,p_2}(D;\ H) = B^{l_1,l_2}_{p_1,p_2}(D;\ H). \]
Thus, consider the spaces
\[ WB^{l_1,l_2}_{p_1,p_2}(1;\ 2;\ D;\ H) \]
and compare them with the spaces
\[ W^{l_1,l_2}_{p_1,p_2},\qquad B^{l_1,l_2}_{p_1,p_2}. \]
1) \(1<p_1<2<p_2\). In this case
\[ W^{l_1,l_2}_{p_1,p_2}\subset WB^{l_1,l_2}_{p_1,p_2}, \qquad B^{l_1,l_2}_{p_1,p_2}\subset WB^{l_1,l_2}_{p_1,p_2}. \]
Indeed, let
\[ \text{a) }\quad f\in W^{l_1,l_2}_{p_1,p_2} = W^{l_1}_{p_1,x_1}\cap W^{l_2}_{p_2,x_2}; \]
since \(p_2>2\), we have \(W^{l_2}_{p_2,x_2}\subset B^{l_2}_{p_2,x_2}\), and therefore
\[ f\in W^{l_1}_{p_1,x_1}\cap B^{l_2}_{p_2,x_2} = WB^{l_1,l_2}_{p_1,p_2}. \]
b) If
\[ f\in B^{l_1,l_2}_{p_1,p_2} = B^{l_1}_{p_1,x_1}\cap B^{l_2}_{p_2,x_2}, \]
then, since \(p_1<2\), we have \(B^{l_1}_{p_1,x_1}\subset W^{l_1}_{p_1,x_1}\), and consequently
\[ f\in W^{l_1}_{p_1,x_1}\cap B^{l_2}_{p_2,x_2} = WB^{l_1,l_2}_{p_1,p_2}. \]
2) \(1<p_1,p_2<2\). In this case
\[ B^{l_1,l_2}_{p_1,p_2}\subset WB^{l_1,l_2}_{p_1,p_2}\subset W^{l_1,l_2}_{p_1,p_2}. \]
Indeed, let
\[ f\in B^{l_1,l_2}_{p_1,p_2} = B^{l_1}_{p_1,x_1}\cap B^{l_2}_{p_2,x_2}; \]
since \(p_1<2\), we have
\[ B^{l_1}_{p_1,x_1}\subset W^{l_1}_{p_1,x_1}, \]
and therefore
\[ f\in B^{l_1}_{p_1,x_1}\cap B^{l_2}_{p_2,x_2} \subset W^{l_1}_{p_1,x_1}\cap B^{l_2}_{p_2,x_2} = WB^{l_1,l_2}_{p_1,p_2}. \]
Since \(p_2<2\), we have \(B^{l_2}_{p_2,x_2}\subset W^{l_2}_{p_2,x_2}\), and therefore
\[ f\in W^{l_1}_{p_1,x_1}\cap B^{l_2}_{p_2,x_2} \subset W^{l_1}_{p_1,x_1}\cap W^{l_2}_{p_2,x_2} = W^{l_1,l_2}_{p_1,p_2}. \]
Similarly one can establish the validity of the following assertions:
3) \(2<p_1,\ p_2\). In this case
\[ W_{p_1,p_2}^{l_1,l_2}\subset WB_{p_1,p_2}^{l_1,l_2}\subset B_{p_1,p_2}^{l_1,l_2}. \]
4) \(1<p_2<2<p_1\). In this case
\[ WB_{p_1,p_2}^{l_1,l_2}\subset W_{p_1,p_2}^{l_1,l_2},\qquad WB_{p_1,p_2}^{l_1,l_2}\subset B_{p_1,p_2}^{l_1,l_2}. \]
Let natural numbers \(n_i\) and positive numbers \(l_i,\ p_i\) \((i=1,2,\ldots,s)\) be given. Define the system of numbers \(\chi_i\) by setting
\[ \chi_i=\frac{1}{l_i}\left[\,1+\frac{1}{p_i}\sum_{j=1}^{s}\frac{n_j}{l_j} -\sum_{j=1}^{s}\frac{n_j}{l_jp_j}\,\right]. \tag{4} \]
Thus, to each space \(W_{p_0,p_1,\ldots,p_s}^{l_1,\ldots,l_s}\), \(B_{p_0,p_1,\ldots,p_s}^{l_1,\ldots,l_s}\), \(WB_{p_0,p_1,\ldots,p_s}^{l_1,\ldots,l_s}\) of functions defined in a domain \(D\) of the space \(E^n\), one may put in correspondence a system of numbers \(\chi_i\), defined by formulas (4).
Theorem 1. Suppose
\[ D\in C(\bar H^{\,s},\bar\omega^{\,s}),\qquad f\in WB_{p_0,p_1,\ldots,p_k,Q_{k+1},\ldots,Q_s}^{l_1,\ldots,l_k,l_{k+1},\ldots,l_s}(k;\ s;\ D), \]
\[ 1\le p_i\le Q_i\le\infty\quad (i=k+1,\ldots,s). \]
Natural numbers \(m_i\), nonnegative integers \(\nu_j^{(i)}\), and numbers \(q,\ \chi_j\) satisfy the conditions
\[ 1\le m_i\le n_i\quad (i=1,2,\ldots,\tau);\qquad \tau\le k;\qquad \sum_{i=1}^{k}m_i+\sigma=m\le n; \]
\[ \sum_{j=1}^{n_i}\nu_j^{(i)}=\nu^{(i)};\qquad \sum_{i=1}^{s}\nu^{(i)}=\nu;\qquad \chi_i>0; \]
\[ 1-\sum_{i=1}^{s}\frac{n_i}{l_ip_i} -\sum_{i=1}^{s}\nu^{(i)}\chi_i +\frac{1}{q}\sum_{i=1}^{k}m_i\chi_i +\frac{1}{q}\sum_{i=k+1}^{k+\sigma}\chi_i =\varepsilon_m\ge0, \]
\(S^m\) is an \(m\)-dimensional surface of class \(\widetilde C^{(1)}\), contained in \(D\) and given by the equations
\[ x_j^{(i)}=x_j^{(i)}\qquad (i=1,2,\ldots,\tau;\quad j=1,2,\ldots,m_i), \]
\[ \bar x^{(i)}=\widetilde x_i=x_{n-s+i}\qquad (i=k+1,\ldots,k+\sigma), \]
\[ x_j^{(i)}=\varphi_j^{(i)} \left(x_1^{(1)},\ldots,x_{m_1}^{(1)},\ldots, x_1^{(\tau)},\ldots,x_{m_\tau}^{(\tau)}; \widetilde x_{k+1},\ldots,\widetilde x_{k+\sigma}\right) \]
\[ \left( \begin{array}{ll} i=1,2,\ldots,\tau; & j=m_i+1,\ldots,n_i\\ i=\tau+1,\ldots,k; & j=1,2,\ldots,n_i \end{array} \right), \]
\[ \bar x^{(i)}=x_{n-s+i}=\varphi_i \left(x_1^{(1)},\ldots,x_{m_1}^{(1)},\ldots, \bar x_1^{(\tau)},\ldots,\bar x_{m_\tau}^{(\tau)}; \widetilde x_{k+1},\ldots,\widetilde x_{k+\sigma}\right) \]
\[ (i=k+\sigma+1,\ldots,s), \]
then
\[ \text{a) }\ \|D_x^\nu f\|_{L_q^{\tau}(S^m)} \le C_1\|f\|_{L_{p_0}(D)}h^{-\delta_m}+ \]
Imbedding Theorems for Mixed Spaces
\[ + C_2\left(\sum_{i=1}^{k}\|f\|_{L_{p_i}^{l_i}(D)}+ \sum_{i=k+1}^{s}\|f\|_{\mathscr L_{Q_i}^{l_i}(D)}\right)h^{\varepsilon_m}, \tag{5} \]
if \(\varepsilon_m>0\) and \(1\leqslant p_i\leqslant q<\infty\) \((i=0,1,2,\ldots,s)\), and
\[ b)\quad \|D_x^\nu f\|_{L_q(S^m)}\leqslant C_3\|f\|_{L_{p_0}(D)}h^{-\delta_m}+ \]
\[ + C_4\left(\sum_{i=1}^{k}\|f\|_{L_{p_i}^{l_i}(D)}+ \sum_{i=k+1}^{s}\|f\|_{\mathscr L_{p_i}^{l_i}(D)}\right), \tag{6} \]
if \(\varepsilon_m=0;\ 1<p_i<q<\infty,\ i=1,2,\ldots,s;\ Q_i=p_i,\ i=k+1,\ldots,s,\) where \(h\) is an arbitrary positive number satisfying the inequalities
\[ 0<h\leqslant H^{1/\chi_i}\quad (i=1,2,\ldots,s), \]
\[ \delta_m=1-\varepsilon_m+\sum_{i=1}^{s}\frac{n_i}{l_i} \left(\frac{1}{p_0}-\frac{1}{p_i}\right). \]
Remark. With a corresponding choice of \(h\) we shall have
\[ \|D_x^\nu f\|_{L_q(S^m)}\leqslant C\|f\|_{L_{p_0}(D)}^{\frac{\varepsilon_m}{\varepsilon_m+\delta_m}} \|f\|_{WB_{p_0,p_1,\ldots,p_{k+1},\ldots,p_s;\,Q_{k+1},\ldots,Q_s}^{l_1,\ldots,l_k,l_{k+1},\ldots,l_s}(k;\,s;\,D)}^{ \frac{\delta_m}{\varepsilon_m+\delta_m}} . \]
The following estimate holds, which follows directly from V. P. Il’in’s integral estimates:
\[ |D_x^\nu f|\leqslant C_0h^{-\sum_{i=1}^{s}\chi_i(\nu^{(i)}+n_i)} \int_{D_h(\bar x)} |f(\bar y)|\,d\bar y+ \]
\[ +\sum_{i=1}^{k}\sum_{j_1,\ldots,j_{\bar l_i-\nu^{(i)}}}^{n_i} C_i\times \]
\[ \times \left\{ \begin{array}{ll} \displaystyle \int_{D_h(\bar x)} \frac{\left|D_{\bar y^{(i)};\,j_1,\ldots,j_{\bar l_i-\nu^{(i)}}}^{\bar l_i} f(\bar y)\right|\,d\bar y} {\left(\sqrt{\sum_{j=1}^{s}|\bar y^{(j)}-\bar x^{(j)}|^{2/\chi_j}}\right)^{\lambda_i^{(0)}}}, & \text{if } k_i=0,\\[3ex] \displaystyle \int_{\mathrm{ш}_{h^{\chi_i(0)}}^{\bar n_i}} d\bar t \int_{\left(D_h(\bar x)\right)_{\bar t}^{(i)}} \frac{|\Delta(\bar t^{(i)})|\, \left|D_{\bar y^{(i)};\,j_1,\ldots,j_{l_i-\nu(i)}}^{l_i} f(\bar y)\right|\,d\bar y} {\left(\sqrt{\sum|\bar y^{(i)}-\bar x^{(i)}|^{2/\chi_j}+|\bar t|^{2/\chi_0}}\right)^{\lambda_i}}, & \text{if } k_i=1 \end{array} \right\} + \]
\[ +\sum_{i=k+1}^{s} C_i \int_{J_{h^{\chi_i(0)}}} d\bar t \int_{\left(D_h(\bar x)\right)_{t_i,2}} \frac{\left|\Delta_i^2\!\left(\frac{t}{2}\right)D_{y_i}^{l_i}f(\bar y)\right|\,d\bar y} {\left(\sqrt{\sum|\bar y^{(i)}-\bar x^{(i)}|^{2/\chi_j}+|\bar t|^{2/\chi_0}}\right)^{\lambda_i}} = \]
\[ = C_0J_1+\sum_{i=1}^{k}\sum_{j_1,\ldots,j_{l_i-\nu^{(i)}}=1}^{n_i} J_{i;j_1,\ldots,j_{l_i-\nu^{(i)}}}^{(k_i)} +\sum_{i=k+1}^{s}J_i, \]
where
\[
\lambda_i^{(0)}=\sum_{j=1}^{s}\chi_i(\nu^{(i)}+n_i)-l_i\chi_i,\qquad
\lambda_i=\lambda_i^{(0)}+n_i\chi_i,\qquad
\nu^{(i)}<l_i+k_i,
\]
and \(C_0, C_i\) are constants independent of \(h\).
Using this inequality and Minkowski’s inequality, we shall have:
\[
\begin{aligned}
\|D_x^\nu f\|_{L_q(S^m)}
&\le C_0\|J_1\|_{L_q(\Omega^m)}\\
&\quad+\sum_{i=1}^{k}\sum_{j_1,\ldots,j_{l_i-\nu^{(i)}}=1}^{n_i}
\left\|J_{i;j_1,\ldots,j_{l_i-\nu^{(i)}}}^{(k_i)}\right\|_{L_q(\Omega^m)}
+\sum_{i=k+1}^{s}\|J_i\|_{L_q(\Omega^m)}\\
&=C_0\bar J_1+\sum_{i=1}^{k}\sum_{j_1,\ldots,j_{l_i-\nu^{(i)}}=1}^{n_i}
\bar J_{i;j_1,\ldots,j_{l_i-\nu^{(i)}}}^{(k_i)}
+\sum_{i=k+1}^{s}\bar J_i,
\end{aligned}
\tag{7}
\]
where \(\Omega^m\) is an \(m\)-dimensional domain of the space \(E^m\) of points
\[
\bar x^m=(x_1^{(1)},\ldots,x_{m_1}^{(1)},\ldots,x_1^{(\tau)},\ldots,x_{m_\tau}^{(\tau)},\tilde x_{k+1},\ldots,\tilde x_{k+\sigma}),
\]
which is mapped into \(S^m\) by the mapping (1).
The following estimates hold:
\[
\text{I) }\quad \bar J_1\le Ch^{-\delta_m}\|f\|_{L_{p_0}(D)},
\tag{8}
\]
where
\[
\delta_m=\frac1{p_0}\sum_{i=1}^{s}n_i\chi_i-\frac1q\sum_{i=1}^{\tau}m_i\chi_i-\frac1q\sum_{i=k+1}^{k+\sigma}\chi_i
+\sum_{i=1}^{\tau}\nu^{(i)}\chi_i+\sum_{i=k+1}^{k+\sigma}\nu_i\chi_i.
\]
II) For \(i=1,2,\ldots,k\) and \(\varepsilon_m>0\),
\[
\bar J_{i;j_1,\ldots,j_{l_i-\nu^{(i)}}}^{(k_i)}
\le C_{k_i}h^{\varepsilon_m}\|f\|_{L_{p_i}^{l_i}(D)}.
\tag{9}
\]
III) For \(i=k+1,\ldots,s\) and \(\varepsilon_m>0\),
\[
\bar J_i\le Ch^{\varepsilon_m}\|f\|_{\mathscr L_{p_i;y_i}^{\,l_i}(D)}.
\tag{10}
\]
From (7)—(10) follows (5).
The case b) is proved analogously.
Using the results of V. P. Il’in, we can formulate the following theorems for mixed spaces.
Theorem 2. Let
\[
D\subset \bigcap_{i=1}^{n}R^{(i)}(\bar H^n,H),\qquad
f\in B_{p_0,p_1,\ldots,p_n}^{l_1,\ldots,l_n}{}_{Q_1,\ldots,Q_n}(D),
\]
where \(1\le p_i\le Q_i\le\infty\) and the numbers \(\chi_i>0\) \((i=1,2,\ldots,n)\). Suppose the natural number \(m\), the nonnegative integers \(\nu_i\) \((i=1,2,\ldots,n)\), and the numbers \(q,\sigma,\rho_i\) \((i=1,2,\ldots,n)\) satisfy the conditions
\[
1\le m\le n,\qquad 1\le p_i\le q<\infty\quad (i=0,1,\ldots,n),\qquad \sigma>Q_i,
\]
\[
\varepsilon_m=1-\sum_{i=1}^{n}\frac1{l_ip_i}-\sum_{i=1}^{n}\nu_i\chi_i+\frac1q\sum_{i=1}^{m}\chi_i>0;\qquad
\rho_j\chi_j<\varepsilon_m\quad (j=1,2,\ldots,m),
\]
\(D^{m}\) is the section of the domain \(D\) by the hyperplane \(x_{m+1}=\mathrm{const},\ldots,x_n=\mathrm{const}\) \((D^{n}\equiv D)\), then
\[ \|D_{\bar x}^{\nu} f\|_{W B_{q,\ldots,q}^{\rho_1,\ldots,\rho_m}(k;m;D;H)} \le \]
\[ \le C_1(h)\|f\|_{L_{p_0}(D)}+C_2(h)\|f\|_{\mathcal L_{p_1,\ldots,p_n}^{l_1,\ldots,l_n}(D)}, \]
where \(0<h<H_i^{1/\chi_i}\) \((i=1,2,\ldots,n)\), and the \(C_i\) are constants not depending on \(f\).
Theorem 3. Let \(D\subset \bigcap_{i=1}^{s} R^{(i)}(\bar H^{s},\bar \omega^{s},2,H)\),
\[ m_i= \begin{cases} >1, & \text{for } 1\le i\le \sigma,\\ =1, & \text{for } \sigma<i\le k, \end{cases} \]
\(f\in W_{p_0,p_1,\ldots,p_s}^{l_1,\ldots,l_s}(D)\), and suppose that the numbers \(\chi_i>0\) \((i=1,2,\ldots,s)\), the natural numbers \(m_i\) \((i=1,2,\ldots,k)\), the nonnegative integers \(\nu_j^{(i)}\) \((i=1,2,\ldots,s;\ j=1,2,\ldots,n_i)\), and the numbers \(q,\rho_i\) \((i=1,2,\ldots,k)\) satisfy the inequalities
\[ 1\le m_i\le n_i \quad (i=1,2,\ldots,k),\qquad k\le s,\qquad \sum_{i=1}^{k} m_i=m\le n, \]
\[ \sum_{j=1}^{n_i}\nu_j^{(i)}=\nu^{(i)}\quad (i=1,2,\ldots,s),\qquad \sum_{i=1}^{s}\nu^{(i)}=\nu, \]
\[ 1\le p_i\le q<\infty \quad (i=0,1,\ldots,s),\qquad \rho_i\chi_i=\varepsilon_m \quad (i=1,2,\ldots,k), \]
\[ \varepsilon_m = 1-\sum_{i=1}^{s}\frac{n_i}{l_i p_i} -\sum_{i=1}^{s}\nu^{(i)}\chi_i +\frac{1}{q}\sum_{i=1}^{k}m_i\chi_i >0; \]
\(D^{m}\) is the \(m\)-dimensional section of the domain \(D\), consisting of points \(\bar x^{(1)},x^{(1)},\ldots,\bar x^{(k)},x^{(k)},\bar x^{(k+1)},\ldots,\bar x^{(s)}\) with fixed \(\bar{\bar x}^{(1)},\ldots,\bar{\bar x}^{(k)},\bar x^{(k+1)},\ldots,\bar x^{(s)}\), where
\[ \bar x^{(i)}=(x_1^{(i)},\ldots,x_{m_i}^{(i)}) \qquad (i=1,2,\ldots,k), \]
\[ \bar{\bar x}^{(i)}=(x_{m_i+1}^{(i)},\ldots,x_{n_i}^{(i)}), \]
\[ \bar x^{(i)}=(x_1^{(i)},\ldots,x_{n_i}^{(i)}) \qquad (i=k+1,\ldots,s), \]
then
\[ \|D_x^{\nu}f\|_{W B_{q,\ldots,q}^{\rho_1,\ldots,\rho_m}(\sigma;k;D^m;H)} \le \]
\[ \le C\left[ \left(\sum_{i=1}^{k} h^{-\delta_i}\right)\|f\|_{L_{p_0}(D)} + \|f\|_{\mathcal L_{p_1,\ldots,p_n}^{l_1,\ldots,l_n}(D)} \right] \left(1+h^{\varepsilon_m}\right), \]
\[ \delta_i = 1-(\varepsilon_m-\rho_i\chi_i) + \sum_{j=1}^{s}\frac{n_j}{l_j}\left(\frac{1}{\rho_0}-\frac{1}{\rho_j}\right), \qquad 0<h\le H^{1/\chi_i}, \]
\(C\) is a constant not depending on the sets standing next to it.
In conclusion we give the definition of the general mixed spaces
\(X_1X_2\ldots X_\tau(k_1;k_2;\ldots;k_{\tau-1};s;D)\).
Denote by \(X_\sigma^{(i)}(D)\) \((i=1,2,\ldots,s;\ \sigma=1,\ldots,\tau \leq s)\) a normed space of functions \(f(\bar y)\) defined in \(D\) and possessing some property in the direction of the vector \(\bar y^{(i)}\).
Let \(\|f\|_{X_\sigma^{(i)}}\) be the norm of the function \(f\) in \(X_\sigma^{(i)}\).
Definition. We shall say that \(f \in X_\sigma^{(1,2,\ldots,s)}\) if \(f \in X_\sigma^{(i)}\) for all \(i=1,2,\ldots,s\), or \(f \in \bigcap_{i=1}^{s} X_\sigma^{(i)}\). We define the norm in \(X_\sigma^{(1,2,\ldots,s)}\) by the equality
\[ \|f\|_{X_\sigma^{(1,2,\ldots,s)}}= \sum_{i=1}^{s}\|f\|_{X_\sigma^{(i)}}. \]
In particular, if \(X_\sigma^{(i)}=W_{p_i,\bar y^{(i)}}^{l_i}\), then \(X_\sigma^{(1,2,\ldots,s)}=W_{p_1,\ldots,p_s}^{l_1,\ldots,l_s}\); if \(X_\sigma^{(i)}=H_{p,\bar y_i}^{r_i}\), then \(X_\sigma^{(1,2,\ldots,s)}=H_{p_1,\ldots,p_s}^{r_1,\ldots,r_s}\).
Definition. We shall say that a function \(f\), defined in \(D\), belongs to the mixed space
\(X_1X_2\cdots X_\tau^{(1,2,\ldots,s)}(k_1,k_2,\ldots,k_{\tau-1};s;D)\), if
\[ f\in \bigcap_{\sigma=1}^{\tau} \left( \bigcap_{i=k_{\sigma-1}+1}^{k_\sigma} X_\sigma^{(i)} \right), \]
where \(k_1\leq k_2\leq \cdots \leq k_{\tau-1}\leq s;\ \tau\leq s;\ k_0=1\). We define the norm of
\(X_1X_2\cdots X_\tau^{(1,2,\ldots,s)}(k_1;k_2;\ldots;k_{\tau-1};s;D)\) by the equality
\[ \|f\|_{X_1X_2\cdots X_\tau^{(1,2,\ldots,s)}(k_1,k_2,\ldots,k_{\tau-1};s;D)} = \sum_{\sigma=1}^{\tau} \sum_{i=k_{\sigma-1}+1}^{k_\sigma} \|f\|_{X_\sigma^{(i)}}. \]
From the definition of the mixed space it follows that
\[ X_1X_2\cdots X_\tau^{(1,2,\ldots,s)}(k_1;k_2;\ldots;k_{\tau-1};s;D) = X_\sigma^{(1,2,\ldots,s)}(D) \]
for \(k_1=\cdots=k_{\sigma-1}=0,\quad k_\sigma=k_{\sigma+1}=\cdots=s\).
References
- V. P. Il’in, Trudy MIAN, vol. LXVI, 1962, pp. 227–363.
- O. V. Besov, Trudy MIAN, vol. IX, 1961, pp. 42–81.
- I. A. Kipriyanov, DAN SSSR, 147, No. 3, 540–543, 1962.
Received by the editors
May 10, 1966
Institute of Mathematics, Siberian Branch, Academy of Sciences of the USSR