MATHEMATICS

V. P. Il’in

ON THE QUESTION OF INEQUALITIES BETWEEN NORMS OF PARTIAL DERIVATIVES OF FUNCTIONS OF SEVERAL VARIABLES

(Presented by Academician I. M. Vinogradov on 12 I 1963)

1. Let \(f(x_1,\ldots,x_n)\) be a continuous function, given in some domain \(D\) of the \(n\)-dimensional Euclidean space \(E^n\) of points \(\mathbf{x}=(x_1,\ldots,x_n)\), and having continuous derivatives of arbitrary order.

Let \(n+1\) integer nonnegative vectors be given,
\[ \mathbf{r}_i=(l_1^i,\ldots,l_n^i) \]
\((i=0,1,\ldots,n;\ l_j^i\ge 0\) integers), with respect to which we require that
\[ \left| \begin{array}{cccc} 1 & l_1^0 & \ldots & l_n^0\\ 1 & l_1^1 & \ldots & l_n^1\\ \cdot & \cdot & \cdot & \cdot\\ \cdot & \cdot & \cdot & \cdot\\ 1 & l_1^n & \ldots & l_n^n \end{array} \right|\ne 0 . \tag{1} \]

Put
\[ D^{\mathbf{r}}f= \frac{\partial^{l_1}}{\partial x_1^{l_1}}\cdots \frac{\partial^{l_n}f}{\partial x_n^{l_n}}, \]
where \(\mathbf{r}=(l_1,\ldots,l_n)\), \(l_j\ge 0\) are integers.

Suppose that
\[ \|D^{\mathbf{r}_i}f\|_{L_p(D)}<\infty \qquad (i=0,1,\ldots,n), \tag{2} \]
where \(p\ge 1\).

The problem is to find the set of integer nonnegative vectors
\[ \vec{\rho}=(v_1,\ldots,v_n) \]
and the corresponding values of the parameter \(q\ge p\) for which the inequality
\[ \|D^{\vec{\rho}}f\|_{L_q(D)} \le C\sum_{i=0}^{n}\|D^{\mathbf{r}_i}f\|_{L_p(D)} \tag{3} \]
will hold, where \(C\) is a constant independent of \(f\).

Inequalities of the type of inequality (3) for the case when the vectors \(\mathbf{r}_i\) are vectors of the form
\[ \mathbf{r}_0=(0,\ldots,0),\qquad \mathbf{r}_i=(0,\ldots,l_i^i,\ldots,0) \quad (i=1,\ldots,n), \]
and the domain \(D\) coincides with the whole space \(E^n\) or is a rectangular parallelepiped with edges parallel to the coordinate axes, were first studied by S. M. Nikol’skii \((^1)\)*; for \(l_i^i=1\) \((i=1,\ldots,n)\), results of the type of inequality (3) follow from still earlier works of S. L. Sobolev \((^2)\).

We shall agree to denote by \(\Delta_R\) the rectangular parallelepiped in the space \(E^n\) of points \(\mathbf{x}=(x_1,\ldots,x_n)\), characterized by the inequalities
\[ 0<x_i<R\qquad (i=1,\ldots,n). \]

In the present note necessary and sufficient conditions are formulated which must be imposed on the vectors \(\mathbf{r}_i\) \((i=0,1,\ldots,n)\) and \(\vec{\rho}\) in order that inequality (3) hold for \(D=\Delta_R\) with some exponent \(q>p\), and a number of results of negative character are also given.

2. We shall say that inequality (3), for given \(\mathbf{r}_i\) \((i=0,1,\ldots,n)\), \(\vec{\rho}\), \(p\), \(q\), and \(D\), does not hold if, whatever constant \(C\) is chosen, there exists a function \(f\), having in \(D\) continuous derivatives of arbitrary order and satisfy-

* S. M. Nikol’skii allowed not only integer but also fractional values \(l_i^i\) \((i=1,\ldots,n)\).

satisfying conditions (2), for which inequality (3) with this constant does not hold.

We shall write \(\mathbf r_1=(l_1^1,\ldots,l_n^1)\geq \mathbf r_2=(l_1^2,\ldots,l_n^2)\) if \(l_j^1\geq l_j^2\) \((j=1,\ldots,n)\).

In what follows it will sometimes be more convenient for us to interpret the vectors \(\mathbf r_i=(l_1^i,\ldots,l_n^i)\) and \(\vec\rho=(\nu_1,\ldots,\nu_n)\) as points \(M_i(l_1^i,\ldots,l_n^i)\) and, respectively, \(N(\nu_1,\ldots,\nu_n)\) of the Euclidean space \(E^n\). The set of points \(M(l_1,\ldots,l_n)\) of the space \(E^n\) whose coordinates satisfy the conditions \(0\leq l_i\leq \nu_i\) \((i=1,\ldots,n)\) will be denoted by \(\delta(N)\).

Lemma 1. Suppose that integer nonnegative vectors \(\mathbf r_i\geq 0\) \((i=0,1,\ldots,n)\) and \(\vec\rho\geq 0\) are given. If there exists a vector \(\vec\chi=(\chi_1,\ldots,\chi_n)\) (not necessarily integer, positive) such that
\[ (\mathbf r_i,\vec\chi)=\sum_{j=1}^{n} l_j^i\chi_j=C_i \quad (i=0,1,\ldots,n), \qquad (\vec\rho,\vec\chi)=\sum_{j=1}^{n}\nu_j\chi_j=C \]
and \(C_i>C\) \((i=0,1,\ldots,n)\) or \(C_i<C\) \((i=0,1,\ldots,n)\), then for \(D=\Delta_R\) inequality (3) does not hold for any \(q\geq p\).

Corollary. If the point \(N\) lies outside the \(n\)-dimensional simplex with vertices at \(M_i\) \((i=0,1,\ldots,n)\), then for \(D=\Delta_R\) inequality (3) does not hold for any \(q\geq p\).

Lemma 2. Suppose one of the following conditions is fulfilled:

1) the points \(M_i(l_1^i,\ldots,l_n^i)\) \((i=0,1,\ldots,n)\) are contained in \(\delta(N)\);

2) the points \(M_i\) \((i=0,1,\ldots,n)\) lie outside \(\delta(N)\).

Then for \(D=\Delta_R\) inequality (3) does not hold for any \(q\geq p\).

Let now \(k\) points \(M_0,\ldots,M_{k-1}\), where \(1\leq k\leq n\), be contained in \(\delta(N)\), and \(n+1-k\) points \(M_k,\ldots,M_n\) be outside \(\delta(N)\) (the point \(M_0\), consequently, will always be considered a point of \(\delta(N)\)). Let
\[ l_1\chi_1+\cdots+l_n\chi_n=C\geq 0 \tag{4} \]
be the equation of the hyperplane passing through the points \(M_1,\ldots,M_n\) (thus, it passes through all points exterior with respect to \(\delta(N)\)). Put
\[ \nu_1\chi_1+\cdots+\nu_n\chi_n=C_1, \tag{5} \]
\[ l_1^0\chi_1+\cdots+l_n^0\chi_n=C_2. \tag{6} \]

Lemma 3. Let \(N(\nu_1,\ldots,\nu_n)\) be a point of the simplex with vertices at \(M_i\) \((i=0,1,\ldots,n)\), not belonging to the face (4) \((C_1\ne C)\).

If among the coefficients \(\chi_i\) of equation (4) there are nonpositive ones, for example, if \(\chi_i>0\) \((i=1,\ldots,k)\), \(\chi_i=0\) \((i=k+1,\ldots,m)\), \(\chi_i<0\) \((i=m+1,\ldots,n)\), then for \(D=\Delta_R\) inequality (3) is possible only for \(q=p\) and only in one of the following cases:

1) \(C_2<C_1<C\), and the point \(M_0\) lies in the hyperplane \(l_{k+1}=\nu_{k+1},\ldots,l_n=\nu_n\);

2) \(C_2>C_1>C\), and the point \(M_0\) lies in the hyperplane \(l_1=\nu_1,\ldots,l_m=\nu_m\).

Lemma 4. If the point \(N(\nu_1,\ldots,\nu_n)\) lies on the face of the simplex (with vertices at \(M_i\) \((i=0,1,\ldots,n)\)) containing the points \(M_1,\ldots,M_n\) \((C_1=C)\), then, whatever the coefficients \(\chi_i\) \((i=1,\ldots,n)\) of equation (4) may be, inequality (3) for \(D=\Delta_R\) does not hold for \(q>p\).

We note that the assertion analogous to this one, generally speaking, does not hold for points of other faces of the simplex under consideration.

Lemma 5. Suppose that for each point \(M_i\) \((i=k,\ldots,n)\) exterior with respect to \(\delta(N)\), at least one of the coordinates \(l_1,\ldots,l_s\), where \(s<n\), is greater than the corresponding coordinate of the point \(N(\nu_1,\ldots,\nu_n)\).

Then inequality (3) for \(D=\Delta_R\) can hold only for \(q=p\) and only in the case when at least one of the points \(M_0,\ldots,M_{k-1}\) contained in \(\delta(N)\) lies in the hyperplane \(l_{s+1}=\nu_{s+1},\ldots,l_n=\nu_n\).

Corollary. From Lemma 5 it follows that inequality (3) for \(q>p\) can hold only when in \(\delta(IV)\) there is one point \(M_0\), and outside \(\delta(IV)\) there are \(n\) points \(M_1,\ldots,M_n\), with only one coordinate of each point \(M_i\) \((i=1,\ldots,n)\) greater than the corresponding coordinate of the point \(N\), and for all these points these coordinates are different.

Theorem. In order that, for \(D=\Delta_R\), inequality (3) hold for some \(q>p\), it is necessary and sufficient that:

1) the coordinates of the vectors \(\mathbf r_i\) \((i=0,1,\ldots,n)\) and \(\vec\rho\) satisfy the inequalities:
\[ l_j^0 \leqslant \nu_j \quad (j=1,\ldots,n), \]
\[ l_j^i \leqslant \nu_j \quad (j=1,\ldots,n,\ j\ne i),\qquad l_i^i>\nu_i \quad \text{for } i=1,\ldots,n; \]

2) there exist numbers \(\varkappa_j>0\) \((j=1,\ldots,n)\) such that
\[ l_1^i\varkappa_1+\cdots+l_n^i\varkappa_n=A \qquad (i=1,\ldots,n); \]

3)
\[ \nu_1\varkappa_1+\cdots+\nu_n\varkappa_n=A_1<A. \]

If the indicated conditions are fulfilled, then there exists an interval of values of the parameter \(q\) for which (3) holds, determined from the relations
\[ 1\leqslant p\leqslant q\leqslant\infty,\qquad A-A_1-\left(\frac1p-\frac1q\right)\sum_{j=1}^n\varkappa_j=\varepsilon\geqslant0, \]
where, if \(\varepsilon=0\), it is assumed that \(1<p<q<\infty\).

The necessity of conditions 1)—3) follows from Lemmas 5, 3, 4, and the sufficiency is proved by the method of integral representations.

Let us note that if the vectors \(\mathbf r_i\) \((i=0,1,\ldots,n)\) satisfy the conditions of S. M. Nikol’skii, then condition 2) is fulfilled automatically, and for every point \(N(\nu_1,\ldots,\nu_n)\) of the simplex with vertices at \(M_i\) the inequalities 1) are valid.

1. Below, for the case \(n=2\), examples are given of various simplexes with vertices at \(M_i\) \((i=0,1,2)\), possessing different properties in the sense of the question considered in the article.

In Fig. 1a a case is shown when inequality (3) for \(D=\Delta_R\) does not hold for any vector \(\vec\rho\) different from \(\mathbf r_i\) \((i=0,1,2)\). In Fig. 1 b, c, d, to the points \(N_i\) \((i=1,2,3,4)\) there correspond vectors \(\vec\rho_i\) for which inequality (3) holds for \(q=p\), while to the points of the triangles \(LNM_2\) (Fig. 1c) and \(M_0M_1M_2\) (Fig. 1d), except for the points of the segments \(LM_2\) and \(M_1M_2\), there correspond inequalities with \(q>p\). These results are established by the method of integral representations. On the basis of the lemmas given above it is easy to establish that inequality (3) for \(D=\Delta_R\) holds for no other vectors \(\vec\rho\).

Fig. 1

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

Received
3 I 1963

CITED LITERATURE

1. S. M. Nikol’skii, Trudy Mat. Inst. im. V. A. Steklova AN SSSR, 38, 244 (1951).
2. S. L. Sobolev, Mat. sborn., 4 (46), 3, 471 (1938).