MATHEMATICS

V. P. PALAMODOV

ON CONDITIONS FOR CORRECT SOLVABILITY IN THE LARGE OF A CERTAIN CLASS OF EQUATIONS WITH CONSTANT COEFFICIENTS

(Presented by Academician P. S. Aleksandrov on 21 I 1960)

In the present note equations of the form

[
p\left(i\frac{\partial}{\partial x}\right)u=w,
\tag{1}
]

are considered, where (p(s)=p(s_1,\ldots,s_n)) is a polynomial in (n) complex variables (s_j=\sigma_j+i\tau_j,\ 1\leq j\leq n), which does not vanish on the real manifold, i.e., for (\tau_1=\cdots=\tau_n=0) (or a matrix whose entries are polynomials with (\det\ne 0)).

The problem is posed as follows: to find, in terms of smoothness and growth at infinity, the classes of uniqueness and the classes of right-hand sides (w) for which there exists a solution of the equation satisfying one or another restriction imposed on its growth. For (n>1) we introduce the number (\gamma)—the genus of the equation (system), equal to the least upper bound of those (\gamma) such that, for some (c), in the domain

[
T(c,\gamma)={\sigma+i\tau:\ |\tau|\leq c|\sigma|^\gamma};
]

[
|\sigma|=|\sigma_1|+\cdots+|\sigma_n|;\quad
|\tau|=|\tau_1|+\cdots+|\tau_n|,
]

for sufficiently large (|\sigma|) the polynomial (p(s)) continues not to vanish*. For (n=1) one may henceforth take (\gamma=1). In general it is not known whether such a domain (T(c,\gamma)) exists with (\gamma) equal to the genus of the equation; therefore in what follows by (\gamma) we shall mean any number less than the genus.

In the case (\gamma>0), the results of the present work are an extension of certain results obtained in ((^3)).

The reasoning is based on the following estimate:

[
\left|D^q \frac{1}{p(\sigma)}\right|
\leq A_\rho B_\rho^{|q|}\, |q|^{|q|}\, |\sigma+i|^{-\gamma|q|+m};
\qquad
D^q=\frac{\partial^q}{\partial s_1^{q_1}\cdots \partial s_n^{q_n}};
]

[
|q|=|q_1|+\cdots+|q_n|
]

((m) is the order of (p(s))), in view of the fact that finding the classes of uniqueness and the classes of existence reduces to studying the operator of multiplication by the function (\dfrac{1}{p(\sigma)}) in various spaces.

Using this estimate and the technique of spaces of type (S) (the definition and properties of these spaces are set forth in ((^1)), Chap. 4), we find classes

* Formally speaking, for the value (\gamma) the quantity (-\infty) is also admissible; the subsequent results remain valid also in this case. But, using the methods of Seidenberg–Tarski ((^2)), it can be shown that always (\gamma>-\infty).

uniqueness. In the case (\gamma \geqslant 0), the solution of equation (1) will be unique only in the space ((S_{1,A}^{\beta})'), for (\beta>0) and some (A); in particular, it is unique in the class of functions satisfying the inequality

[
|u(x)| \leqslant C \exp \left[\sum a_j |x_j|\right], \qquad (x=(x_1,\ldots,x_n)),
]

where it turns out that one may put

[
a_j=\inf_{p(s)=0}|\tau_j|-\varepsilon,
]

the lower bound being taken over all roots of the polynomial (p(s)); but if one puts

[
a_j=\inf_{p(s)=0}|\tau|+\varepsilon,
]

then uniqueness will be violated.

In the case (\gamma<0), as a uniqueness class one may take any space ((S_{\alpha}^{\beta})'), if (\alpha+\gamma\beta \geqslant 1) and (\alpha>1); in particular, uniqueness will hold in the class of functions satisfying the inequality

[
|u(x)| \leqslant C \exp [a|x|^{1-\varepsilon}].
]

for any positive (a) and (\varepsilon). At the same time, uniqueness will already be violated in the class of functions satisfying this inequality with (\varepsilon=0) and arbitrarily small (a).

To find existence classes we shall need the following spaces:

[
E_{\pm\alpha,A}=\left{\chi(x): \bar{\chi}(x)\exp\left[\pm \frac{1}{A}|x|^{1/\alpha}\right]\in L_2\right}\qquad (A>0);
]

[
H_{(\pm k)}={\chi(x): \chi(x)|x+i|^{\pm k}\in L_2}\qquad (k>0).
]

Theorem 1. Let (\gamma \geqslant 0). Then, if the function (w) belongs to the spaces: 1) (E_{\alpha,A}), (\alpha \geqslant 1); 2) (E_{-\alpha,A}), (\alpha \geqslant 1); 3) (H_{(-k)}); 4) (H_{(k)}) for some (A), there exists a solution of equation (1) belonging respectively to the spaces: 1) (E_{\alpha,A_1}); 2) (E_{-\alpha,A_1}); 3) (H_{(-k)}); 4) (H_{(k)}), for some (A_1).

Thus, in the case (\gamma \geqslant 0), for the existence of a solution it is sufficient that the right-hand side not grow too rapidly. In the case (\gamma<0) the situation changes, namely, the faster the growth of the right-hand side, the greater the smoothness that must be required of it in order to ensure the existence of a solution among ordinary functions; for example, if (\gamma=-1), then for the existence of a solution with arbitrary right-hand side (w\in H_{(-k)}) it is necessary that the function (w) be (k) times differentiable.

However, in a certain sense the converse is also true: if the function (w) can be expanded in a series (w=\sum_{k=1}^{\infty} w_k) such that the inequality

[
\sum_{k=1}^{\infty}\sum_{j=0}^{k} C_k^{j} B^{|j|}|j|^{|j|}a_{k-|j|}\left|(D+1)^{[-\gamma(j+m)]+1}\frac{w_k}{|x+i|^k}\right|_{L_2}<\infty,
\tag{2}
]

holds, where ([\delta]) is the integer part of (\delta), and

[
a_k=\max_x \frac{|x+i|^k}{e(x)},
]

then there exists a solution of equation (1) such that (e^{-1}(x)u(x)\in L_2).

In order to write condition (2) in a more explicit form for part of the functions (w), we introduce spaces of type (E):

[
E_{\alpha,A}^{\beta,B}=\left{\chi(x):\ |D^q\chi(x)|\leqslant C B^{|q|}\cdot |q|^{|q|/\beta}\exp\left[\frac{1}{A}|x|^{1/\alpha}\right]\right}
]

The following theorem is proved by using inequality (2).

Theorem 2. Let (\gamma<0). Then, if the function (w) belongs to the spaces: 1) (E_{\alpha,A}^{\beta,B}) for (\alpha+\gamma\beta\geq 1,\ \alpha>1) and some (A) and (B); 2) (S_{\alpha,A}^{\beta,B}) for (\alpha+\gamma\beta\geq 1,\ \alpha>1) and some (A) and (B); 3) (H_{(-k)}) together with all its derivatives up to order ([-\gamma(k+m)+m+1]); 4) (H_{(k)}) with the same number of derivatives, then there exists a solution of equation (1) belonging respectively to the spaces: 1) (E_{\alpha,A_1}^{\beta,B_1}) for some (B_1) and (A_1); 2) (S_{\alpha,A_1}^{\beta,B_1}) for some (B_1) and (A_1); 3) (H_{(-k)}); 4) (H_{(k)}).

It is interesting to note the symmetry in the existence classes exhibited by Theorems 1 and 2.

Since all the classes in which we have ensured the existence of a solution are contained in the corresponding uniqueness classes, they are at the same time classes of well-posedness.

We now observe that, excluding the first two classes of Theorem 2, we have ensured the existence of a solution only in the generalized sense; however, in order that it be a solution in the ordinary sense, it is enough to require that not only the function (w), but also all its derivatives needed for the operator (p\left(i\frac{\partial}{\partial x}\right)) to be applicable to it, possess the properties required by these theorems.

We note that the definition of the genus does not require that the polynomial (p(s)) have no zeros; it is sufficient that it have no zeros outside a bounded domain. We indicate a method for finding the genus of an equation from the values of the polynomial (p(s)) on a real manifold.

Theorem 3. The genus of equation (1) is equal to the number

[
-\sup \lim_{|\sigma|\to\infty}
\left{
\frac{\ln \dfrac{|\operatorname{grad} p(\sigma)|}{|p(\sigma)|}}
{\ln |\sigma|}
\right};
\qquad
|\operatorname{grad} p(\sigma)|
=
\left[
\sum_{j=1}^{n} |p'_{s_j}(\sigma)|^2
\right]^{1/2}.
\tag{3}
]

In the case when the operator (p\left(i\frac{\partial}{\partial x}\right)) is hypoelliptic in the sense of Hörmander ({}^{4}), formula (3) makes it possible to determine the class of Gevrey to which all solutions of the corresponding homogeneous equation belong (see ({}^{5})).

In conclusion the author expresses gratitude to G. E. Shilov for his great attention and valuable critical remarks.

Moscow State University
named after M. V. Lomonosov

Received
8 I 1960

References

({}^{1}) I. M. Gelfand, G. E. Shilov, Spaces of basic and generalized functions, Moscow, 1958.
({}^{2}) A. Seidenberg, Ann. of Math., Ser. 2, 60, 365 (1954).
({}^{3}) V. P. Palamodov, DAN, 129, No. 4 (1959).
({}^{4}) L. Hörmander, On the theory of general differential operators in partial derivatives, IL, 1959.
({}^{5}) G. E. Shilov, UMN, 14, issue 5 (89) (1959).