V. P. IL'IN

SOME INEQUALITIES BETWEEN NORMS OF PARTIAL DERIVATIVES OF FUNCTIONS OF MANY VARIABLES

(Presented by Academician I. M. Vinogradov on 19 III 1963)

Let \(f(x_1,\ldots,x_n)\) be a continuous function given in a domain \(\Omega\) of the \(n\)-dimensional Euclidean space \(E^n\) of points \(\mathbf{x}=(x_1,\ldots,x_n)\), having continuous derivatives of arbitrary order. Let \(n+1\) integral nonnegative vectors be given,
\[ \mathbf{r}_i=(l_1^i,\ldots,l_n^i)\quad (i=0,1,\ldots,n)\quad (l_j^i\geqslant 0\ \text{integers}). \]
The problem is to find integral nonnegative vectors \(\vec{\rho}=(\nu_1,\ldots,\nu_n)\) and numbers \(q\geqslant p\geqslant 1\) for which the inequality
\[ \|D^{\vec{\rho}}f\|_{L_q(\Omega)} \leqslant C \sum_{i=0}^{n}\|D^{\mathbf{r}_i}f\|_{L_p(\Omega)} \tag{1} \]
holds, where \(C\) is a constant independent of \(f\),
\[ D^{\mathbf{r}} f = \frac{\partial^{l_1}}{\partial x_1^{l_1}}\cdots \frac{\partial^{l_n} f}{\partial x_n^{l_n}} \quad (\mathbf{r}=(l_1,\ldots,l_n)). \]

The results of the present note are a development of the results of note \((^1)\) on this question. Some of the inequalities given below may be regarded as a generalization, in a certain direction, of the known inequalities of S. L. Sobolev \((^2)\) and S. M. Nikol'skii \((^3)\); some new inequalities are also given.

I. We shall say that a domain \(D\) of the space \(E^n\) belongs to the class \(C(\mathcal H^k)\), where \(1\leqslant k\leqslant n\), if for every point \(\mathbf{x}\in D\) there exists a \(k\)-dimensional cube with vertex at \(\mathbf{x}\) and with edges of length \(\mathcal H\), parallel to the coordinate axes \(x_{n-k+1},\ldots,x_n\), contained in \(\Omega\). We shall further say that \(S^m\), an \(m\)-dimensional surface of the space \(E^n\), is a surface of class \(C^{(1)}\) if it is given by the equations \(x_1=x_1,\ldots,x_m=x_m\), \(x_{m+1}=\varphi_{m+1}(x_1,\ldots,x_m),\ldots,x_n=\varphi_n(x_1,\ldots,x_m)\), where the functions \(\varphi_i(x_1,\ldots,x_m)\) are defined in some domain \(\Omega^m\) of the space \(E^m\) of points \(\mathbf{x}^m=(x_1,\ldots,x_m)\) and have continuous bounded partial derivatives of first order in \(\Omega^m\).

II. Let natural numbers \(s\leqslant n\) and \(n_i\) \((i=1,\ldots,s)\) be given, for which \(n_1+\cdots+n_s=n\).

Consider the following table of numbers, consisting of \(s+1\) rows (the rule for forming the numbers is clear from the table):
\[ \begin{array}{c|l} i & \\ \hline 0 & 1\\ 1 & 1,\ n_1\\ 2 & 1,\ n_1,\ n_2,\ n_1n_2\\ 3 & 1,\ n_1,\ n_2,\ n_3,\ n_1n_2,\ n_1n_3,\ n_2n_3,\ n_1n_2n_3\\ \cdot & \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\\ \cdot & \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\\ s & 1,\ n_1,\ n_2,\ldots,n_s,\ n_1n_2,\ldots,n_1n_2\ldots n_s \end{array} \tag{2} \]

By \(\alpha_i\) we denote an arbitrary number taken from the \(i\)-th row; \(\alpha_i\) is either equal to \(1\) (the first number of the row), or is a number of the form \(\alpha_i=n_{i_1}\cdots n_{i_{k_i}}\), where \(k_i\) denotes the number of factors in \(\alpha_i\), and the indices \(i_1,\ldots,i_{k_i}\) indicate exactly which factors enter into \(\alpha_i\) (\(i_1,\ldots,i_{k_i}\) are certain natural numbers not exceeding \(i\)). To the number \(\alpha_i\) we associate \(\alpha_i\) vectors, for which we introduce the notation: \(\mathbf r_{i;0}\), if \(\alpha_i\) is equal to the first number of the \(i\)-th row (\(\alpha_i=1\)); \(\mathbf r_{i;j_{i_1},\ldots,j_{i_{k_i}}}\), \((j_{i_t}=1,\ldots,n_{i_t};\ t=1,\ldots,k_i)\), if \(\alpha_i=n_{i_1}\cdots n_{i_{k_i}}\).

The coordinates of an arbitrary vector \(\mathbf r\) will be denoted by \(r_{\lambda,\mu}\) \((\lambda=1,\ldots,s;\ \mu=1,\ldots,n_\lambda)\), i.e.
\[ \mathbf r=(l_{1,1},\ldots,l_{1,n_1};\ldots;l_{s,1},\ldots,l_{s,n_s}); \]
in particular, the coordinates of \(\mathbf r_{i;j_{i_1},\ldots,j_{i_{k_i}}}\) are denoted by
\[ l_{\lambda,\mu}^{\,i;j_{i_1},\ldots,j_{i_{k_i}}}. \]

We now state a somewhat more general theorem, from which, in particular, sufficient conditions for the validity of inequality (1) for \(q=p\) will follow.

Theorem 1. Let natural numbers \(s\le n\) and \(n_i\) \((i=1,\ldots,s)\) be given, for which
\[ n_1+\cdots+n_s=n, \]
and the table (2) corresponding to them, as well as numbers \(\alpha_i\) \((i=0,1,\ldots,s)\),
\[ \sum_{i=0}^s \alpha_i=N. \]
Suppose, further, that \(N\) integral nonnegative vectors \(\mathbf r_{i;j_{i_1},\ldots,j_{i_{k_i}}}\) (or \(\mathbf r_{i;0}\)), corresponding to the numbers \(\alpha_i\), and an integral nonnegative vector
\[ \vec\rho=(v_{1,1},\ldots,v_{s,n_s}) \]
are given, whose coordinates satisfy the conditions:

1.
\[ \mathbf r_{i;0}:\quad l_{\lambda,\mu}^{\,i;0}=v_{\lambda,\mu} \quad (\lambda=1,\ldots,i,\ i+2,\ldots,s;\ \mu=1,\ldots,n_\lambda), \]
\[ l_{i+1,\mu}^{\,i;0}\le v_{i+1,\mu} \quad (\mu=1,\ldots,n_{i+1}); \]
\[ \mathbf r_{i;j_{i_1},\ldots,j_{i_{k_i}}}:\quad l_{\lambda,\mu}^{\,i;j_{i_1},\ldots,j_{i_{k_i}}}=v_{\lambda,\mu} \quad \text{for } \lambda=1,\ldots,s,\ \text{but } \lambda\ne i_1,\ldots,\lambda\ne i_{k_i}, \]
\[ \lambda\ne i+1,\quad \mu=1,\ldots,n_\lambda; \]
\[ l_{i_t,j_{i_t}}^{\,i;j_{i_1},\ldots,j_{i_{k_i}}}>v_{i_t,j_{i_t}} \quad (t=1,\ldots,k_i); \]
\[ l_{i_t,\mu}^{\,i;j_{i_1},\ldots,j_{i_{k_i}}}\le v_{i_t,\mu} \quad (\mu=1,\ldots,n_{i_t},\ \mu\ne j_{i_t};\ t=1,\ldots,k_i); \]
\[ l_{i+1,\mu}^{\,i;j_{i_1},\ldots,j_{i_{k_i}}}\le v_{i+1,\mu} \quad (\mu=1,\ldots,n_{i+1}). \]

1. For all \(\lambda=i_1,\ldots,i_{k_i}\) \((i=1,\ldots,s)\) and \(\mu=1,\ldots,n_\lambda\) there exist numbers \(x_{\lambda,\mu}>0\) such that
   \[ \sum_{\mu=1}^{n_{i_t}} v_{i_t,\mu}x_{i_t,\mu} < \sum_{\mu=1}^{n_{i_t}} l_{i_t,\mu}^{\,i;j_{i_1},\ldots,j_{i_{k_i}}}x_{i_t,\mu} \quad (t=1,\ldots,k_i) \]
   for all vectors \(\mathbf r_{i;j_{i_1},\ldots,j_{i_{k_i}}}\)
   \[ (j_{i_t}=1,\ldots,n_{i_t},\ t=1,\ldots,k_i,\ i=1,\ldots,s). \]

Assume that
\[ D^{\mathbf r_i}f\in L_p(\Omega)\quad (i=1,\ldots,N)\quad (p\ge1), \]
where \(\mathbf r_i\) are the newly renumbered given \(N\) vectors, \(\Omega\in C(\mathcal H^n)\). Then the inequality
\[ \|D^{\vec\rho}f\|_{L_p(\Omega)} \le C\sum_{i=1}^N \|D^{\mathbf r_i}f\|_{L_p(\Omega)}, \tag{3} \]
holds, where \(C\) is a constant independent of \(f\).

From inequality (3) there follows inequality (1) in the case when

\[ N=\sum_{i=0}^{s}\alpha_i\leq n+1. \]

In particular, for \(n_1=\cdots=n_{s-1}=1,\ n_s=n+1-s\),

\[ \sum_{i=0}^{s}\alpha_i \]

will not exceed \(n+1\) for any choice of \(\alpha_i\). To this case there corresponds the following simpler theorem.

Theorem \(1'\). Let a natural number \(s\leq n\) and integer nonnegative vectors \({\bf r}_i\) \((i=0,1,\ldots,n)\) and \(\vec\rho=(v_1,\ldots,v_n)\) be given, whose coordinates satisfy the conditions:

1. \({\bf r}_i\) \((i=0,1,\ldots,s-2)\):
   \[ l_j^i\geq v_j\quad (j=1,\ldots,i),\qquad l_{i+1}^i\leq v_{i+1},\qquad l_j^i=v_j\quad (j=i+2,\ldots,n); \]

\[ {\bf r}_{s-1}:\quad l_j^i\geq v_j\quad (j=1,\ldots,s-1),\qquad l_j^i\leq v_j\quad (j=s,\ldots,n); \tag{A} \]

\({\bf r}_i\) \((i=s,\ldots,n)\): for each fixed \(j=1,\ldots,s-1\), either
\[ l_j^i>v_j\quad (i=s,\ldots,n), \]
or
\[ l_j^i=v_j\quad (i=s,\ldots,n), \]
and either a)
\[ l_i^i>v_i\quad (i=s,\ldots,n),\qquad l_j^i\leq v_i\quad (j=s,\ldots,n;\ j\ne i;\ i=s,\ldots,n), \]
or b)
\[ l_j^i=v_j\quad (i=s,\ldots,n;\ j=s,\ldots,n). \]

1. If the vectors \({\bf r}_i\) \((i=s,\ldots,n)\) satisfy conditions a), then let there exist numbers \(\chi_i>0\) \((i=s,\ldots,n)\) such that
   \[ \sum_{j=s}^{n}v_j\chi_j<\sum_{j=s}^{n}l_j^i\chi_j \quad (i=s,\ldots,n). \]

Let, furthermore, \(D^{{\bf r}_i}f\in L_p(\Omega)\) \((p\geq1)\), \(\Omega\in C(\mathscr H^n)\). Then

\[ \bigl\|D^{\vec\rho}f\bigr\|_{L_p(\Omega)} \leq C\sum_{i=0}^{n}\bigl\|D^{{\bf r}_i}f\bigr\|_{L_p(\Omega)}, \tag{4} \]

where \(C\) is a constant independent of \(f\).

For a concrete choice of the parameters in Theorem \(1'\) one obtains various inequalities. For example, for \(s=n\), \(l_i>v_i,\ 0\leq k_i\leq v_i\) \((i=1,\ldots,n)\), we obtain the inequality

\[ \bigl\|D^{v_1+\cdots+v_n}f\bigr\|_{L_p(\Omega)} \leq \]

\[ \leq C\left( \bigl\|D^{l_1+\cdots+l_n}f\bigr\|_{L_p(\Omega)} + \sum_{i=1}^{n} \bigl\|D^{v_1+\cdots+v_{i-1}+k_i+v_{i+2}+\cdots+v_n}f\bigr\|_{L_p(\Omega)} \right). \]

III. From the results of work [1] it follows that inequality (1) for an arbitrary domain of class \(C(\mathscr H^n)\) for \(q>p\) is possible only when the coordinates of the vectors \({\bf r}_i\) \((i=0,1,\ldots,n)\) and \(\vec\rho\) satisfy the conditions:

\[ l_j^0\leq v_j\quad (j=1,\ldots,n)\qquad \text{for } i=0, \]
\[ l_j^i\leq v_j\quad (j=1,\ldots,i-1,i+1,\ldots,n),\qquad l_i^i>v_i\quad \text{for } i=1,\ldots,n, \tag{Б} \]

which are obtained from conditions (A) for \(s=1\). Under these conditions the usual embedding theorems hold.

Theorem 2. Let integer nonnegative vectors \({\bf r}_i\) \((i=0,1,\ldots,n)\) and \(\vec\rho\) be given, for which conditions (Б) are valid, and, in addition, let there exist numbers \(\chi_j>0\) \((j=1,\ldots,n)\) such that

\[ F_1=\sum_{j=1}^{n}v_j\chi_j<\sum_{j=1}^{n}l_j^i\chi_j=F \quad (i=1,\ldots,n). \tag{5} \]

Let, further:

1. A natural number \(m\) and numbers \(p\) and \(q\) are given, satisfying the inequalities:
   \[ 1 \leq m \leq n,\quad 1 \leq p \leq q \leq \infty,\quad F-F_1-\frac1p\sum_{j=1}^n\varkappa_j+\frac1q\sum_{j=1}^m\varkappa_j=\varepsilon_m\geq 0. \]

2. \(D^{\mathbf r_i}f\in L_p(\Omega)\) \((i=0,1,\ldots,n)\), \(\Omega\in C(\overline{\mathfrak H^n})\).

3. \(S^m\) is an \(m\)-dimensional surface of class \(C^{(1)}\), contained in \(\overline\Omega\) \((S^n\equiv \Omega)\).

Then, under one of the following conditions: a) \(\varepsilon_m>0\), b) \(\varepsilon_m=0,\ 1<p<q<\infty\), the inequality
\[ \bigl\|D^{\vec\rho}f\bigr\|_{L_q(S^m)} \leq C_1h^{-\delta_m}\bigl\|D^{\mathbf r_0}f\bigr\|_{L_p(\Omega)} +C_2h^{\varepsilon_m}\sum_{i=1}^n\bigl\|D^{\mathbf r_i}f\bigr\|_{L_p(\Omega)} \]
holds, where
\[ \delta_m=F-\varepsilon_m-\sum_{j=1}^n l_j^0\varkappa_j,\quad 0<h\leq \mathfrak H^{1/\varkappa_j}\ (j=1,\ldots,n), \]
and \(C_1\) and \(C_2\) are constants independent of \(f\) and \(h\).

Remark. If the vectors \(\mathbf r_i\) and \(\vec\rho\) satisfy conditions (B), but among the coefficients \(\varkappa_i\) satisfying (5) there are also negative ones, then inequality (1) is possible only for \(q=p\) and only in the case when \(k+1\) vectors \(\mathbf r_i\) \((1\leq k<n)\) and the vector \(\vec\rho\) satisfy conditions 1–2 of the following theorem.

Theorem 3. Let the integer nonnegative vectors \(\mathbf r_i\) \((i=0,n-k+1,\ldots,n)\) and \(\vec\rho\) satisfy the conditions:

1.
\[ l_j^0=v_j\ (j=1,\ldots,n-k),\quad l_j^0\leq v_j\ (j=n-k+1,\ldots,n); \]
\[ l_j^i=v_j\ (j=1,\ldots,n-k),\quad l_i^i>v_i, \]
\[ l_j^i\leq v_j\ (j=n-k+1,\ldots,n;\ j\ne i)\quad \text{for } i=n-k+1,\ldots,n. \]

1. There exist numbers \(\varkappa_j>0\) \((j=n-k+1,\ldots,n)\) such that
   \[ F_1=\sum_{j=n-k+1}^n v_j\varkappa_j < \sum_{j=n-k+1}^n l_j^i\varkappa_j=F \quad (i=n-k+1,\ldots,n). \]

Let, further:

3.
\[ m\geq n-k,\quad p\geq 1,\quad F-F_1-\frac1p\sum_{j=n-m+1}^n\varkappa_j>0. \]

1. \(D^{\mathbf r_i}f\in L_p(\Omega)\) \((i=0,n-k+1,\ldots,n)\), \(\Omega\in C(\overline{\mathfrak H^k})\).

2. \(S^m\) is an \(m\)-dimensional surface of class \(C^{(1)}\), contained in \(\overline\Omega\) \((S^n\equiv\Omega)\).

Then
\[ \bigl\|D^{\vec\rho}f\bigr\|_{L_p(S^m)} \leq C\left( \bigl\|D^{\mathbf r_0}f\bigr\|_{L_p(\Omega)} + \sum_{i=n-k+1}^n \bigl\|D^{\mathbf r_i}f\bigr\|_{L_p(\Omega)} \right), \]
where \(C\) does not depend on \(f\).

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

Received
19 III 1963

REFERENCES

\({}^{1}\) V. P. Il’in, DAN, 150, No. 5 (1963).
\({}^{2}\) S. L. Sobolev, Matem. sborn., 4 (46), 3, 471 (1938).
\({}^{3}\) S. M. Nikol’skii, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 38, 244 (1951).