Reports of the Academy of Sciences of the USSR

1958, Volume 123, No. 1

MATHEMATICS

M. S. AGRANOVICH

GENERAL SOLUTIONS OF DIFFERENTIAL-DIFFERENCE EQUATIONS WITH CONSTANT COEFFICIENTS

(Presented by Academician S. L. Sobolev on 25 VI 1958)

Consider the equation

[
\sum_{k=0}^{l} P_k(\partial/\partial x)u(x-h_k)=f(x).
\tag{1}
]

Here (x=(x_1,\ldots,x_n)) is a point of the (n)-dimensional real space (\mathfrak R);
(\partial/\partial x=(\partial/\partial x_1,\ldots,\partial/\partial x_n)); (P_k) are polynomials with constant complex coefficients; (h_k=(h_{k1},\ldots,h_{kn})) are fixed points in (\mathfrak R), with (h_0=0) and (h_k\ne h_j) for (k\ne j). Let (A) and (B) be bounded regions in (\mathfrak R), and assume that all (x+h_k\in B) if (x\in A). By (K_A) and (K_B) we denote the spaces of infinitely differentiable complex-valued functions (\varphi(x)), equal to zero respectively outside (A) and outside (B), with the usual topology defined by the norms

[
|\varphi|\mu=\sup|D^k\varphi(x)|,\qquad \mu=0,1,\ldots,
]

where (|k|=\sum_1^n k_\nu) and

[
D^k=\partial^{|k|}/\partial x_1^{k_1}\cdots \partial x_n^{k_n}.
]

By (K'_A) and (K'_B) we denote the spaces of linear continuous functionals over (K_A) and (K_B), respectively. We shall seek the general form of the solution (u(x)), assuming that (u\in K'_B), (f\in K'_A), and (\varphi\in K_A). In this case equation (1) is understood in the following sense:

[
\left(u(x),\sum \overline{P}_k(-\partial/\partial x)\varphi(x+h_k)\right)=(f(x),\varphi(x)),
]

where the bar denotes replacement of the coefficients in (P_k) by their conjugate numbers. The main results of the note are readily transferred to systems of (m) analogous equations with (m) unknowns.

Let us perform the Fourier transform (F) (cf. (1), Ch. II and (2), Ch. III). The spaces (K_A) and (K_B) then pass into spaces (Z_A) and (Z_B) of entire functions defined in the (n)-dimensional complex space (\mathfrak C),

[
\psi(s)=\psi(\sigma+i\tau)=F[\varphi]=\int \varphi(x)\exp(x,is)\,dx,
]

where ((x,s)=\sum_1^n x_\nu s_\nu), with a topology in which neighborhoods of zero are the images, under the mapping (F), of neighborhoods of zero of the spaces (K_A) and (K_B). The spaces (K'_A) and (K'_B) pass into the spaces (Z'_A) and (Z'_B) of linear continuous functionals over (Z_A) and (Z_B), acting according to the formula

[
(F[u],F[\varphi])=(2\pi)^n(u,\varphi).
]

Let (v=F[u]), (g=F[v]),
[
Q(s)=\sum P_k(-is)\exp(h_k,is)
]
and
[
\overline{Q}(s)=\sum \overline{P}_k(is)\exp(h_k,-is).
]
Under the Fourier transform, equation (1) becomes the equation
[
Q(s)v=g,
]
i.e.
[
(v,\overline{Q}(s)\psi(s))=(g,\psi(s)).
\tag{2}
]

Every functional (u) from (K_A') or (K_B') can be represented in the form (cf. (3), Ch. III, §§ 6–9 and (2), Ch. II, § 4) of a finite sum
[
(u,\varphi)=\sum_{|r|\le N}\int D^r\varphi(x)\,d\mu_r(x),
\tag{3}
]
where (d\mu_r(x)) are arbitrary measures ((3), Ch. 1, § 1), which it is convenient for us to regard as concentrated in the closure (\overline{C}) of a fixed bounded domain (C) containing all (x+h_k) for (x\in B). We emphasize that if in equation (1) (l=0), i.e. if it is a partial differential equation, then in what follows one may take (A=B=C). From (3) it follows that every functional (v) from (Z_A') or (Z_B') has the form (cf. (2), Ch. III, § 2)
[
(v,\psi)=\int_{\mathfrak{R}} v(\sigma)\psi(\sigma)\,d\sigma,
\tag{4}
]
where
[
v(s)=\sum_{|r|\le N}s^r\int_C \exp(x,-is)\,d\mu_r(x).
\tag{5}
]
Here, for (\psi\in Z_A), the entire function (v(s)) is determined up to the addition of any function (\omega(s)) of the form (5) for which
[
\int_{\mathfrak{R}}\omega(\sigma)\psi(\sigma)\,d\sigma=0
]
for all (\psi\in Z_A). We shall call such an (\omega(s)) a zero function.

For every polynomial (P(s)) one can construct in (\mathbb{C}) a contour (H), called a Hörmander staircase for (P), possessing the following properties (see (4) and (2), Ch. II, § 3): 1) (|P(s)|\ge \mathrm{const}>0) on (H); 2) (H) lies in some “strip” (|\tau_j|<M) ((j=1,\ldots,n)); 3) if a function (V(s)) is analytic in this strip and decreases sufficiently rapidly in it at infinity, then
[
\int_{\mathfrak{R}}V(\sigma)\,d\sigma=\int_H V(s)\,ds.
]
Let us verify that an analogous contour (H) can be constructed also for any function (Q(s)), with a slight modification of 2).

I. Let (l>0) and let (P_k(s)\not\equiv \mathrm{const}) for some (k). Performing in (1), if necessary, a nondegenerate real transformation
[
y_k=\sum \alpha_{kj}x_j+\beta_k,
]
one can arrange that: 1) all (P_k(-is)) have the form
[
c_k s_1^{m_k}+\text{terms of order lower than }m_k\text{ in }s_1,
]
where (c_k=\mathrm{const}\ne0); 2) (h_{j1}\ne h_{k1}>0). Let (h_{l1}) be the largest of the numbers (h_{k1}). We now define (H) by the equalities
[
\tau_2=\cdots=\tau_n=0,\qquad \tau_1=T(\sigma_1,\ldots,\sigma_n).
]
Here (T) is a function whose values lie between
[
-M\ln(|\sigma|+e)
]
and
[
-M\ln(|\sigma|+e)-N
]
(where (M>0) and (N>0) are sufficiently large), such that
[
\left|c_l\right|\exp(-h_{l1}\tau_1)-
\left|\sum_{k=0}^{l-1}P_k(-is)\exp\bigl((h_k,i\sigma)-h_{k1}\tau_1\bigr)\right|\ge 1
]
and
[
|P_l(-is)|\ge |c_l|
]
on (H).

II. Let (l>0), (P_k\equiv \mathrm{const}) for all (k). Then no change of coordinates is needed, and for (H) one may take the product of (n) straight lines (\tau_k=\mathrm{const}), where all constants, except perhaps one, are equal to zero.

III. Let (l=0). Then (Q(s)) is a polynomial.

We pass to the main results. Let (H) be a fixed Hörmander ladder for the function (Q(s)), and let (g(s)) be a fixed function corresponding to the functional (g\in Z'_A) by formula (4). One can verify that the functional (v\in Z'_B) then and only then satisfies equation (2) over (Z_A), when

[
(v,\psi)=\int_H \frac{g(s)+w(s)}{Q(s)}\psi(s)\,ds
\qquad (\psi\in Z'_B),
\tag{6}
]

where (w(s)) is an arbitrary zero function.

Denote by (\Delta) the complement of (A) in (\overline C). Let the regions (A) and (C) be such that (\Delta) is regular ((3), Ch. III, § 9). It can be proved that then every zero function has the form (5), where the integrals must be taken not over (\overline C), but over (\Delta). Conversely, every function of this form is zero. Thus, one obtains a description of all (v\in Z'_B) satisfying equation (2) over (Z_A), and thereby of all (u\in K'_B) satisfying equation (1) over (K_A). In particular, for (f=0)

[
(u,\varphi)=\sum_{|r|\le N}\int_\Delta
\left{\int_H \frac{s^r}{Q(s)}
\int \varphi(x)\exp(x-y,is)\,dx\,ds\right}d\mu_r(y).
\tag{7}
]

For each (y\in\Delta) the functional standing in braces may be regarded as a solution over the space (K) of all finite infinitely differentiable functions ((1^{-3})) of equation (1), in which the right-hand side is a derivative of some order of the (\delta)-function at the point (x=y), multiplied by a constant.

Let us consider in more detail the case when (H) lies in the strip (|\tau_j|\le M) (see II and III above). Let (y\in\mathfrak A) and (\varepsilon=(\varepsilon_1,\ldots,\varepsilon_n)), where each (\varepsilon_k) is equal to (+1) or (-1). Denote by (L(y,\varepsilon)) the “coordinate angle with vertex at the point (y),” i.e. the set of points (x) of (\mathfrak A) whose coordinates satisfy the inequalities ((x_k-y_k)\varepsilon_k\ge 0) for all (k). Suppose that the set (\Delta) can be divided into a finite number of pairwise nonintersecting parts (\Delta_1,\ldots,\Delta_q) and that to each part (\Delta_j) one can assign a set (\varepsilon^j=(\varepsilon^j_1,\ldots,\varepsilon^j_n)) of (+1)’s and (-1)’s in such a way that the complement of (\mathfrak A_j) to the set (\displaystyle\bigcup_{y\in\Delta_j} L(y,\varepsilon^j)) contains the region (A).

Let (m_k) be the highest order of differentiation with respect to (x_k) in equation (1). Fix arbitrarily: integers (l_j\ge m_j+2) ((j=1,\ldots,n)); a Hörmander ladder (H) for the function (Q(s)); real numbers (b_{jk}) sufficiently large in modulus ((>M), if (H) lies in the strip (|\tau_j|\le M,\ j=1,\ldots,n)), ((j=1,\ldots,q;\ k=1,\ldots,n)), such that (b_{jk}\varepsilon^j_k>0). Consider the function

[
U_{j;\mu}(x)=\int_{\Delta_j}\int_H
\frac{\exp(y-x,is)\,ds}
{Q(s)(ib_{j1}-s_1)^{l_1}\cdots(ib_{jn}-s_n)^{l_n}}
\,d\mu(y).
\tag{8}
]

This function is continuously differentiable everywhere at least up to order ((l_1-2,\ldots,l_n-2)) and is an ordinary solution of equation (1) with (f=0) for (x\in\mathfrak A_j). From formula (7) it follows that any functional (u\in K'_B) satisfying equation (1) with (f=0) over (K_A) can be represented in the form of a finite sum of derivatives, in the sense of generalized functions, of ordinary solutions of the form (8) (cf. (3)):

[
(u,\varphi)=\sum_{|r|\le N'}\sum_{j=1}^{q}
\int \overline{U_{j;\mu_r}(x)}\,D^r\varphi(x)\,dx.
\tag{9}
]

Hence, for equation (1) with (f=0), one obtains a new proof of L. Schwartz’s theorem (\left({}^{3}\right)), Ch. VI, Theorem 29, stating that if all sufficiently smooth solutions are infinitely differentiable (analytic), then all solutions are infinitely differentiable (analytic).

The separability condition on the set (\Delta) given above singles out those domains (A) for which the representation of the solution in the form (8)—(9) can be obtained without resorting to a change of coordinates or to a decomposition of the domain (A) itself into parts. This condition is satisfied, in particular, by any convex bounded domain; moreover, for such a domain the set (\Delta) can always be taken to be regular. To split (\Delta) in this case into parts (\Delta_j), we draw supporting hyperplanes parallel to the coordinate hyperplanes; let (c_1,\ldots,c_{2n}) be the vertices of the circumscribed parallelepiped thus obtained. The parts (\Delta_j) may be formed so that, for each (y\in\Delta_j), the segment joining (y) and (c_j) does not intersect (A). In the case where (A) is a convex polyhedron, it is simplest to split (\Delta) into parts (\Delta_j) by drawing hyperplanes through all its ((n-1))-dimensional faces. Then each (\mathfrak A_j) will contain the half-space (\mathfrak A_j^) bounded by one of the hyperplanes drawn; (\bigcap \mathfrak A_j^ = A). From (7) it is clear that every functional (u\in K'_B) satisfying equation (1) with (f=0) on (K_A) is a finite sum of functionals (u_j) on (K) satisfying the equation on functions (\varphi) from (K) that are equal to (0) outside (\mathfrak A_j^*).

If the (\tau_j) are not bounded on (H) (see I) and if (A) is the (n)-dimensional parallelepiped (|x_j|