V. P. PALAMODOV

THE GENERAL FORM OF SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS

(Presented by Academician P. S. Aleksandrov, 1 XII 1961)

As is known, every solution of an ordinary linear equation with constant coefficients and with zero right-hand side is a linear combination of the corresponding exponentials (exponential polynomials). For partial differential equations with constant coefficients

\[ p(D)u=0,\qquad D=i\frac{\partial}{\partial x_1},\ldots,i\frac{\partial}{\partial x_n}. \tag{1} \]

this fact was generalized by L. Ehrenpreis in \((^1)\) and independently by the author in \((^4)\) as follows: every solution of equation (1) is the Fourier transform of a measure concentrated on the complex variety of roots of the polynomial \(p(s)\).*

In \((^4)\) a formulation was given and a proof of this theorem was outlined for the cases: 1) the solution \(u\) is an entire function; 2) the solution \(u\) is an ordinary or generalized function growing at infinity no faster than \(\exp(A|x|^\alpha)\) for certain \(A>0\) and \(\alpha>0\). In the present note we indicate the general form of the solutions of (1) in the broadest spaces \(S_0^{\beta'}\) and \(\mathscr E_0^\beta\) (see \((^{2,4})\)) of ordinary and generalized functions.

We begin with the definition of the functional spaces needed below. Let \(K\subset R^n\) be some compact set. By \(S_0^\beta(K)\), \(1<\beta\leq\infty\), we denote the space of all finite functions whose supports belong to \(K\), infinitely differentiable for \(\beta=\infty\), and belonging to the Gevrey class of order \(\beta\) for \(1<\beta<\infty\), with the natural topology. Let \(\Omega\subset R^n\) be some open set and let \(K_1\subset K_2\subset\cdots\) be an increasing sequence of compact subsets of \(\Omega\) tending to \(\Omega\). Put \(S_0^\beta(\Omega)=\lim\operatorname{ind} S_0^\beta(K_i)\). By \(\mathscr E_0^\beta(\Omega)\) we denote the space of all functions in the domain \(\Omega\) that are infinitely differentiable for \(\beta=\infty\) and belong to the Gevrey class of order \(\beta\) for \(1<\beta<\infty\).

Let \(p(s)\) be the polynomial corresponding to the operator in (1). Construct a factorization \(p(s)=p_1(s)\cdots p_r(s)\) of this polynomial such that each polynomial \(p_i(s)\) has no multiple factors and is divisible by \(p_{i+1}(s)\). Such a factorization is unique. Let \(N_i\) be the complex variety of roots of the polynomial \(p_i(s)\), and let \(N_i^0\) be the difference between the set \(N_i\) and a 1-neighborhood of the set of its singular points; \(n_i(s)\) is the normal vector to \(N_i^0\) at the point \(s\).

By \(s_a\) we denote the open ball of radius \(a\) with center at the origin. We formulate the main theorem.

Theorem 1. Each generalized function \(u\in S_0^{\beta'}(s_{na})\) (infinitely differentiable function \(u\in\mathscr E_0^\beta(s_{na})\)) that is a solution of equation (1) admits the representation

\[ u(x)=\sum_{i\leq r}\int_{N_i^0} e^{ixs}\frac{\partial^i}{\partial n_i^i(s)}\,d\mu_i(s), \tag{2} \]

* The theorem announced by L. Ehrenpreis in \((^1)\) also covers systems of equations of the form (1); however, in the general case this theorem is false.

in which the integral converges and coincides with \(u\) as a functional belonging to \(S_0^{\beta'}(s_a)\) \((\mathscr E_0^\beta(s_a))\). In addition, the measure

\[ \exp\bigl(a|\tau|_{(+)} b|\sigma|^{1/\beta}\bigr)\,d\mu_i(s), \]

where for \(\beta=\infty\) we put \(|\sigma|^{1/\beta}=\log(|\sigma|+1)\), has bounded variation for every \(i\) and some \(b\) (any \(b\)).

Thus, if \(u\) is a solution of (1) in some ball, then in the concentric ball of radius \(n\) times smaller the functional \(u\) admits the representation (2). Hence it follows:

Theorem 1a. Let \(\Omega\subset R^n\) be an open set. In order that a function \(u\in S_0^{\beta'}(\Omega)\) \((u\in\mathscr E_0^\beta(\Omega))\) satisfy equation (1), it is necessary and sufficient that for every point \(x_0\in\Omega\) there be a number \(a>0\) such that the functional \(u(x-x_0)\) admits the representation (2).

In the case \(\Omega=R^n\) this result can be formulated more simply. We note that \(S_0^\beta(R^n)\) and \(\mathscr E_0^\beta(R^n)\) coincide with the spaces \(S_0^\beta\) and \(\mathscr E_0^\beta\), defined in \((^{2,4})\).

Theorem 1b. The general form of the solutions of equation (1) in the space \(S_0^{\beta'}\) \((\mathscr E_0^\beta)\) is given by equality (2), where the integral converges and coincides with \(u\) as a functional belonging to \(S_0^{\beta'}\) \((\mathscr E_0^\beta)\), and the measure
\[ \exp\bigl(a|\tau|_{(+)} b|\sigma|^{1/\beta}\bigr)\,d\mu_i(s) \]
has bounded variation for all \(i\), \(a\), and some \(b\) (any \(b\)).

The representation (2) turns out to be convenient for solving a number of problems; in particular, from Theorems 1, 1a, 1b, the results \((^4)\), and the duality formulas \((^{2,3})\), all the results of the theory of hypoelliptic operators can be obtained. Here we note only one consequence of the formulated theorems and the results \((^4)\), which gives the solution of a known problem posed by Hadamard.

Corollary. Let \(\Phi\) be one of the spaces \(S_\alpha^{\beta'}\), \(\mathscr E_\alpha^\beta\), \(\alpha<1\) (see \((^{2,4})\)).

1) Suppose that \(p(s)=q(s)r(s)\), where the polynomials \(q\) and \(r\) are relatively prime. Then every solution of equation (1) belonging to \(\Phi\) is the sum of solutions of the equations

\[ q(D)w=0,\qquad r(D)v=0, \]

belonging to \(\Phi\).

2) Suppose that \(p(s)=q^k(s)\), \(k>0\), and that the polynomial \(q\) has no multiple factors. Then every solution of equation (1) belonging to \(\Phi\) is represented in the form

\[ u=\sum_{|\nu|\le k-1} x^\nu u_i, \]

where all \(u_i\in\Phi\) satisfy the equation \(q(D)u_i=0\).

We shall outline the proof of Theorem 1 for the case \(u\in S_0^\infty(s_{na})\). For simplicity suppose that the polynomial \(p(s)\) has no multiple factors. Let \(N_\nu\) denote the complex algebraic variety of roots of the polynomial \(p(s)\) of multiplicity not less than \(\nu\), i.e. the set of solutions of the system \(D^i p(s)=0\), \(|i|\le \nu-1\). These varieties form a decreasing sequence and
\[ N_1=\overline{N}_1,\quad C^n=N_0\supset N_1\supset\cdots\supset N_m\supset N_{m+1}=\Lambda, \]
where \(m\) is the degree of the polynomial \(p(s)\). Obviously
\[ C^n=\bigcup N_\nu\setminus N_{\nu+1}. \]
Let \(\theta(s)\) denote the function defined in \(C^n\) which on the set \(N_\nu\setminus N_{\nu+1}\) is equal to
\[ \rho(s,N_{\nu+1}/(|s|^2+1)). \]

Lemma 1. There exists a countable set of differential operators \(\mathcal D_\alpha(s,D)\) with polynomial coefficients on each set \(N_\nu\setminus N_{\nu+1}\) and of functions \(\chi_{i\alpha}(s)\), satisfying the inequalities \(|\chi_{i\alpha}(s)|\le\)

\(\leq \theta^{-q_i,\alpha}(s)\), such that for any entire function \(\varphi(s)\) and any point \(s \in N_1\), the series

\[ \varphi_s(s+\xi)=\sum \varkappa_{i\alpha}(s)\,\xi^i \mathscr{D}_{\alpha}(s,D)\varphi(s) \]

converges absolutely in some neighborhood \(U_s\) of the form \(U_s=\{s': |s'-s|\leq c\theta^q(s)\}\). Moreover, \(\varphi-\varphi_s=p\psi\), where \(\psi\) is a function analytic in \(U_s\),

\[ \max_{|s'-s|\leq \frac{c}{2}\theta^q(s)} |\varphi_s(s')|\leq C \max_{s'\in U_s\cap N_1}|\varphi(s')| \]

and, with certain measures \(\mu_s^\alpha(s')\),

\[ \mathscr{D}_{\alpha}(s,D)\varphi(s)= \int_{U_s\cap N_1}\varphi(s')\,d\mu_s^\alpha(s'). \]

The idea of the proof of Lemma 1 is as follows. Let \(s_0\) be a point belonging to \(N_1\); let \((\xi,\eta)\) be a splitting of the variables \(s\) into two groups, with \(\xi\) a single variable and \(s_0=(\xi_0,\eta_0)\). Further, let \(\xi_i(\eta)\), \(i=1,\ldots,k\), be the roots of the polynomial \(p(s)\) close to \(\xi_0\) when \(\eta\) is close to \(\eta_0\). Then the function \(\varphi_{s_0}(s)\), for each fixed \(\eta\), is a polynomial in \(\xi\) of degree \(k-1\), coinciding with \(\varphi(s)\) at the points \(\xi_i(\eta)\), \(i=1,\ldots,k\).

Remark. The system of operators \(\mathscr{D}_{\alpha}(s,D)\) constructed in this lemma has the following property at each point \(s\): if \(\psi[\xi]\) is a formal power series, then, in order that \(\psi[\xi]\) belong to the ideal generated by the series

\[ \sum \xi^i \frac{D^i}{i!}p(s) \]

in the space of all power series, it is necessary and sufficient that \(\mathscr{D}_{\alpha}(s,\delta)\psi[\xi]=0\) for every \(\alpha\). This system of operators coincides with the system of operators, bearing the same notation, constructed in (5).

The space dual to \(S_0^\infty(s_a)\) will be denoted by \(Z_a\). By the Paley—Wiener theorem, \(Z_a\) is the space of entire functions of first order, rapidly decreasing along the real subspace, with topology determined by the countable set of norms

\[ \|\varphi\|^{a,j}=\sup_{s\in C^n} e^{-a|\tau|}\,|s^j\varphi(s)|. \tag{3} \]

By \(C_a\) we denote the space of all continuous functions in \(C^n\) for which all the norms (3) are defined.

Definition. The completion of the space \(Z_a\) in the topology determined by the countable set of norms

\[ \|\varphi\|_{N_1}^{a,j}=\sup_{s\in N_1} e^{-a|\tau|}\,|s^j\varphi(s)|, \tag{4} \]

will be denoted by \(Z_a(N_1)\). In view of Lemma 1, every element \(f\in Z_a(N_1)\) is a continuous function given on the set \(N_1\), having the property that, for each point \(s\in N_1\), the series

\[ f_s(s+\xi)=\sum \varkappa_{i\alpha}(s)\,\xi^i \int_{U_s\cap N_1} f(s')\,d\mu_s^\alpha(s') \]

converges absolutely in \(U_s\) and coincides with \(f\) in \(U_s\cap N_1\).

In the following lemma we shall give a description of the complex measure that determines the zero functional on the space \(Z_a\).

Lemma 2. Every measure \(\mu\), given in \(C^n\), determining a continuous functional on \(C_{pa}\) and equal to zero on the subspace \(Z_{a+\varepsilon}\), for any \(\varepsilon>0\), admits the representation

\[ (\mu,\varphi)= \sum_{|j|\leq k}\int_{|x|\leq a} e^{-2a|\tau|} \int_{\Gamma_\tau} e^{itxs}s^j\varphi(s)\,ds\,d\gamma(x,\tau)+ \]

\[ +\sum_{|j|\leq k}\int_{|x|>a} e^{-a|\tau|} \int_{\gamma_\tau} e^{ixs}s^j\varphi(s)\,ds\,d\lambda(x,\tau), \tag{5} \]

where the right-hand side converges and coincides with \(\mu\) as a functional belonging to \(C'_a\); \(d\gamma\) and \(d\lambda\) are complex measures of bounded variation; \(T_\tau\) is a contour consisting of the manifold \(R^n_+ + i\tau\); \(\Gamma_\tau\) is a contour consisting of the manifold \(R^n_- + i\tau\) with positive orientation and the manifold \(R^n\) with negative orientation.

Lemma 2 follows from the easily verified inequality

\[ \|\varphi-\varphi * \chi_a\|^{na,j} \leq C_j \|\varphi\|^{a,j}_0, \]

where \(\chi_a(x)\) is a certain infinitely differentiable function equal to zero for \(|x|\geq a\) and to one for \(|x|\leq a-\varepsilon\), and

\[ \|\varphi\|^{a,j}_0 = \]

\[ =\max\left\{ \sup_{\tau,\,|x|\leq a} e^{-2a|\tau|} \left|\int_{\Gamma_\tau} e^{ixs}s^j\varphi(s)\,ds\right|, \sup_{\tau,\,|x|>a} e^{-a|\tau|} \left|\int_{\Gamma_\tau} e^{ixs}s^j\varphi(s)\,ds\right| \right\}. \]

The proof of Theorem 1 is completed as follows. Let the functional \(u\in S^{\infty}_a(s_{na})\) be a solution of equation (1). Then the functional \(\tilde u\) belongs to \(Z'_{na}\) and is a solution of the equation \(p(s)\tilde u=0\). Extend this functional to the space \(C_{na}\), represent it in the form of a measure by means of the Riesz theorem, and multiply this measure by \(p(s)\). The resulting measure \(\mu\) belongs to \(C'_{na}\), gives the zero functional on \(Z_{na}\), and, consequently, admits the representation (5). Substitute this representation into the scalar product \((\mu,\varphi/p)\), where \(\varphi\in Z_a\), and deform the contours \(\Gamma_\tau\) and \(T_\tau\) in (5) near \(N_1\) so that they do not intersect \(N_1\). Then contract these contours into a neighborhood of the set \(N_1\), which is the union of all the domains \(U_s\). In this way we have extended the functional \(\mu\) to the space \(Z_a(N_1)\). But, by construction, \((\tilde u,\varphi)=(\mu,\varphi/p)\), whence also \(\tilde u\in Z'_a(N_1)\). The space \(Z_a(N_1)\) is complete in the topology determined by the norms (4); therefore, applying the Riesz theorem once again, we obtain the representation (2).

Remark. Denote

\[ Z(Z(N_1))=\lim_{a\to\infty}\operatorname{ind} Z_a(Z_a(N_1)). \]

It follows from the arguments given above that the natural embedding \(Z\) into \(Z(N_1)\) determines a topological isomorphism

\[ Z/pZ \simeq Z(N_1), \]

where \(pZ\) is the ideal generated by the polynomial \(p(s)\) in \(Z\). An analogous assertion is valid for any basic space of entire functions.

Moscow State University
named after M. V. Lomonosov

Received
29 XI 1961

CITED LITERATURE

1. L. Ehrenpreis, The Fundamental Principle and some of its Applications, Collection of Varšava Congress on Functional Analysis, 1960.
2. I. M. Gelfand, G. E. Shilov, Spaces of Basic and Generalized Functions, Moscow, 1958.
3. V. P. Palamodov, UMN, 15, 4 (94), 208 (1960).
4. V. P. Palamodov, DAN, 137, No. 4, 774 (1961).
5. V. P. Palamodov, DAN, 141, No. 6, 1302 (1961).