MATHEMATICS

B. R. VAINBERG

ASYMPTOTIC REPRESENTATION OF FUNDAMENTAL SOLUTIONS OF HYPOELLIPTIC EQUATIONS AND A PROBLEM IN THE WHOLE SPACE WITH CONDITIONS AT INFINITY

(Presented by Academician I. G. Petrovskii on February 15, 1962)

§ 1. Introduction. In this paper we consider equations

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

where \(x=(x_1,x_2,\ldots,x_n)\), \(P\left(i\frac{\partial}{\partial x}\right)=P\left(i\frac{\partial}{\partial x_1}, i\frac{\partial}{\partial x_2},\ldots,i\frac{\partial}{\partial x_n}\right)\) is a hypoelliptic operator \((^1)\) in \(n\) variables with constant real coefficients, acting on functions defined in the whole \(n\)-dimensional real space \(R_x^n\).

By \(W\) we shall denote a class of functions, defined in the whole space \(R_x^n\), in which there exists a unique solution of equation (1). Usually the classes \(W\) are readily found after the asymptotics at infinity of the fundamental solutions for equation (1) have been obtained.

For \(n=2\), the asymptotics at infinity of the fundamental solutions and the classes \(W\) were obtained by the author for a fairly broad class of hypoelliptic operators \((^2)\). Sommerfeld \((^3)\), I. N. Vekua \((^4)\), and B. P. Paneiakh \((^5)\) found the asymptotics of fundamental solutions and the classes \(W\) for certain particular types of equations (1) with \(n>2\). In the work of V. P. Palamodov \((^6)\) the class \(W\) is found for any equation (1), not necessarily hypoelliptic, but whose characteristic polynomial \(P(s)\), \(s=(s_1,s_2,\ldots,s_n)\), has no real zeros.

In the present paper we shall find the asymptotics at infinity of certain fundamental solutions and the classes \(W\) for any hypoelliptic operator satisfying the following four conditions:

1. \(P(s)\) has only real coefficients.
2. \(P(s)\) has real zeros.
3. \(\operatorname{grad} P(s)\ne 0\) at the real zeros of \(P(s)\).

From the listed conditions it follows that, in the \(n\)-dimensional real space \(R_s^n\), the real zeros of the polynomial form several smooth closed surfaces. We denote them by \(K_j\), \(j=1,2,\ldots,m\).

1. The total curvature \(K(s)\) of the surfaces \(K_j\), \(j=1,2,\ldots,m\), is everywhere nonzero.*

The method of the present paper is a generalization of the method used by the author in \((^2)\).

§ 2. Construction of a fundamental solution. A fundamental solution for an operator satisfying the listed conditions will be obtained in the form of the integral

\[ E(x)=\frac{1}{(2\pi)^n}\int_H \frac{\exp[-i(x,s)]}{P(s)}\,dH, \tag{2} \]

where \(H\) is a set analogous to Hörmander’s ladder \((^7)\). We shall now construct the set \(H\).

* By the total curvature \(K\) of a surface we mean the ratio of the determinant of the second quadratic form of the surface to the determinant of the first quadratic form.

Let \(\alpha\) be some orthogonal matrix with real coefficients, let
\(s(\alpha)=(s_1(\alpha),s_2(\alpha),\ldots,s_n(\alpha))\) be the coordinate system obtained from
\(s=(s_1,s_2,\ldots,s_n)\) after the transformation with matrix \(\alpha\), and let \(R(\alpha)\) be the subspace of the subspace \(R_s^n\) determined by the conditions
\(\operatorname{Im}s_j(\alpha)=0,\ j=2,3,\ldots,n\).

Lemma 1. If \(P(s)\) satisfies conditions 1–3 and has no multiple factors, then for every transformation \(\alpha\), with the exception of a finite number of them,
\[ \operatorname{grad} P(s)\ne 0 \]
at the zeros of \(P(s)\) situated in the space \(R(\alpha)\).

This lemma makes it possible, without loss of generality, to assume that
\[ \operatorname{grad}P(s)\ne 0 \]
at the zeros of \(P(s)\) with real \(s_2,s_3,\ldots,s_n\).

Let \(s_j=\sigma_j+i t_j,\ j=1,2,\ldots,n\). Condition 4 ensures the convexity of the surfaces \(K_j,\ j=1,2,\ldots,m\). Therefore each \(K_j,\ j=1,2,\ldots,m\), can be divided into two parts: the “upper” part relative to the \(\sigma_1\)-axis and the “lower” part, i.e., into two such parts that, for fixed \(\sigma_2,\sigma_3,\ldots,\sigma_n\), at the points of one of them \(\sigma_1\) is not less than at the points of the other. We denote one of these parts by \(l_j^1\), the other by \(l_j^2,\ j=1,2,\ldots,m\). Let \(\delta_j=\pm 1,\ j=1,2,\ldots,m\), where \(\delta_j=1\) if, for fixed \(\sigma_2,\sigma_3,\ldots,\sigma_n\), \(\sigma_1\) on \(l_j^1\) is not greater than on \(l_j^2\). If, on the contrary, at the points of \(l_j^2\) \(\sigma_1\) is not greater than at the corresponding points of \(l_j^1\), then \(\delta_j=-1\). Denote
\[ \bigcup_j l_j^1=\Lambda_1 \]
and
\[ \bigcup_j l_j^2=\Lambda_2. \]
Depending on how we have divided the set of real zeros of \(P(s)\) into the sets \(\Lambda_1\) and \(\Lambda_2\), we shall construct different fundamental solutions of equation (1). We now fix one such division.

The set \(H\) is constructed in the space \((\sigma_1,\sigma_2,\ldots,\sigma_n,\tau_1)\) and consists of the lines \(C_{\bar\sigma}\). For each \(\bar\sigma=(\sigma_2,\sigma_3,\ldots,\sigma_n)\), \(-\infty<\sigma_j<\infty,\ j=2,3,\ldots,n\), the line \(C_{\bar\sigma}\) coincides with the straight line parallel to the \(\sigma_1\)-axis and passing through the point \((0,\sigma_2,\sigma_3,\ldots,\sigma_n,0)\), if on this straight line there are no zeros of \(P(s)\). If, however, on such a straight line there are zeros of \(P(s)\), then we go around them, remaining in the plane \(\sigma_j=\mathrm{const},\ j=2,3,\ldots,n\), and in such a way that the points of the set \(\Lambda_1\) remain above (in the direction \(\tau_1>0\)) our line \(C_{\bar\sigma}\), and the points of the set \(\Lambda_2\) below our line.

Theorem 1. If the polynomial \(P(s)\) is hypoelliptic, satisfies conditions 1–4, and has no multiple factors, then the integral (2) exists1 and gives a fundamental solution for equation (1).

If \(P(s)\) has factors \(Q_j^{\lambda_j}(s)\), \(\lambda_j>1\), then, by condition 3, the polynomials \(Q_j(s)\) have no real zeros, and then it is known that the operator
\[ \prod_j Q_j^{\lambda_j}\!\left(i\frac{\partial}{\partial x}\right) \]
has an exponentially decreasing fundamental solution. In this case the fundamental solution for equation (1) is obtained as the convolution of the exponentially decreasing fundamental solution for the operator
\[ \prod_j Q_j^{\lambda_j}\!\left(i\frac{\partial}{\partial x}\right) \]
and the fundamental solution constructed by the method described above for the operator
\[ \frac{P}{\prod_j Q_j^{\lambda_j}}\!\left(i\frac{\partial}{\partial x}\right). \]

§ 3. Asymptotics of fundamental solutions.

Just as in paper (2), the integral (2) with respect to the variable \(s_1\) is evaluated by means of the theory of residues, and the resulting integrals over the space \((\sigma_2,\sigma_3,\ldots,\sigma_n)\) are investigated by means of the saddle-point method (8).

Let
\[ r=\sqrt{x_1^2+x_2^2+\cdots+x_n^2},\qquad w_i=x_i/r,\qquad i=1,2,\ldots,n, \]

\(w=(w_1,w_2,\ldots,w_n)\). For all \(w\) (\(w_1\ne 0\)) and any \(j,\ j=1,2,\ldots,m\), the system

\[ w_1 P'_{s_i}(\sigma)-w_i P'_{s_1}(\sigma)=0,\qquad i=2,3,\ldots,n, \]

\[ P(\sigma)=0 \]

has exactly two real solutions belonging to \(K_j\), one of which belongs to \(\Lambda_1\), and the other to \(\Lambda_2\). We shall denote them respectively by
\[ \sigma_j^1=(\sigma_{1j}^1(w),\sigma_{2j}^1(w),\ldots,\sigma_{nj}^1(w)) \]
and
\[ \sigma_j^2=(\sigma_{1j}^2(w),\sigma_{2j}^2(w),\ldots,\sigma_{nj}^2(w)). \]
By \(\nu\) we shall denote the unit vector of the exterior normal to the surface \(K_j\).

Let \(D_p(x)=x_1^{p_1}x_2^{p_2}\cdots x_n^{p_n}\) (\(p_i\) are nonnegative integers), and let the symbol \(|p|\) denote the quantity \(p_1+p_2+\cdots+p_n\).

Theorem 2. If \(P\!\left(i\dfrac{\partial}{\partial x}\right)\) is a hypoelliptic operator satisfying conditions 1–4, then its fundamental solution constructed above has, as \(r\to\infty\), the following asymptotic form:

\[ D_p\!\left(\frac{\partial}{\partial x}\right)E(x)= \]

\[ =\sum_{j=1}^{m} \left\{ (2\pi)^{-\frac{n-1}{2}} \frac{\exp\left[-\delta_j\frac{\pi}{4}(n-1)\right]} {\sqrt{K(\sigma_j^\mu)}\,\dfrac{\partial P}{\partial \nu}(\sigma_j^\mu)} \,D(-i\sigma_j^\mu)\, \frac{\exp[-i(\sigma_j^\mu,w)r]}{r^{\frac{n-1}{2}}} \right\} +E_p(x), \]

where \(p\) is arbitrary; \(\mu=1\) if \(x_1<0\); \(\mu=2\) if \(x_1>0\); for \(E_p(x)\) in a neighborhood of infinity the estimate

\[ |E_p(x)|<\frac{C}{r^{n/2}}. \]

§ 4. Classes \(W\)

Theorem 2 enables us to obtain the classes \(W\) for the equations under consideration. Let \(l\) be the order of equation (1).

Theorem 3. If \(P\!\left(i\dfrac{\partial}{\partial x}\right)\) is a hypoelliptic operator satisfying conditions 1–4, and \(f(x)\) is any function having, in a neighborhood of infinity, \([n/2]+l\) derivatives that decrease as \(r\to\infty\) faster than \(C/r^{n+\varepsilon}\), then there exists a unique solution of equation (1) in the following class of functions \(W\): \(u\subset W\), if it is representable as a sum of functions \(u(x)=\)

\[ =\sum_{j=1}^{m}u_j(x), \]

for which, in a neighborhood of infinity, the inequalities hold:

\[ |u_j(x)|<\frac{C}{r^{(n-1)/2}}, \qquad \left| D_p\!\left(\frac{\partial}{\partial x}\right)u_j(x) - D_p(-i\sigma_j^\mu)u_j(x) \right| <\frac{C}{r^{n/2}} \]

with any \(p\) for which \(|p|<l\), where \(\mu=1,2\) respectively for \(x_1<0\) and \(x_1>0\).

Remark. Results analogous to those formulated in the present paper have simultaneously been obtained by V. V. Grushin by another method.

In conclusion, the author expresses deep gratitude to S. A. Galpern for his constant attention to the present work and to P. P. Mosolov for a number of useful suggestions.

Received
14 II 1962

References

1. G. E. Shilov, Uspekhi Mat. Nauk, 14, 5 (1959).
2. B. R. Vainberg, Dokl. Akad. Nauk SSSR, 144, No. 5 (1962).
3. A. N. Tikhonov, A. A. Samarskii, Equations of Mathematical Physics, Moscow, 1953.
4. I. N. Vekua, Trudy Tbilissk. Inst. Mat., 12 (1943).
5. B. P. Paneiakh, Vestnik Moskov. Univ., Ser. Mat., No. 5 (1959).
6. V. P. Palamodov, Dokl. Akad. Nauk SSSR, 132, No. 3 (1960).
7. I. M. Gel'fand, G. E. Shilov, Spaces of Basic and Generalized Functions, Moscow, 1958.
8. N. G. de Bruijn, Asymptotic Methods in Analysis, Moscow, 1961.

1. The existence of integral (2) may, for example, be understood as follows: for each \(\bar\sigma\) there exists an integral over \(C_{\bar\sigma}\) which can be integrated with respect to \(\sigma_2,\sigma_3,\ldots,\sigma_n\). ↩