V. V. Grushin

ON ONE PROPERTY OF SOLUTIONS OF A HYPOELLIPTIC EQUATION

(Presented by Academician P. S. Aleksandrov, 25 X 1960)

Let \(u(x)\) be a solution of the partial differential equation with constant coefficients

\[ P\left(i\frac{\partial}{\partial x}\right)u(x)=0. \tag{1} \]

In the present note we consider the question of the relation between the growth of the function \(u(x)\) at infinity and its smoothness. In § 1 two theorems are proved which show what conditions must be imposed on the polynomial \(P(s)\) in order that every solution \(u(x)\) of equation (1), defined for all \(x\), under one or another restriction on growth at infinity, be infinitely differentiable. In § 2 the smoothness of solutions of a hypoelliptic equation is discussed. Hypoelliptic equations were introduced by L. Hörmander in \((^1)\), where it was proved that, if \(u(x)\) is a solution of such an equation, then \(u(x)\) is infinitely differentiable, and if \(W\) is a bounded domain, then

\[ \max_{x\in W}\left|D^b u(x)\right|\le c^b\Gamma\left(\frac{k}{\gamma}\right), \]

where \(\Gamma(z)\) is Euler’s function, and \(\gamma\) is the genus of the hypoelliptic equation. In § 2 it is shown that, under certain restrictions on the growth of \(u(x)\) at infinity, this estimate of the smoothness of the function \(u(x)\) can be substantially improved.

1. We first impose on \(u(x)\) the strongest condition. Let \(u(x)\), as \(x\to\infty\), grow no faster than a polynomial. Under these assumptions it is proved in \((^2)\), p. 165, that if all real solutions of the algebraic equation

\[ P(\sigma)=0 \tag{2} \]

lie in a bounded part of the plane, then \(u(x)\) is an entire analytic function of order of growth not higher than one. We formulate an assertion which, in a certain sense, is the converse of this.

Theorem 1. If every continuous bounded solution of equation (1) has continuous first-order derivatives, then all real solutions of equation (2) lie in a bounded part of the plane.

Proof. Consider the normed space \(B\) of continuous bounded functions, defined for all \(x\), with norm

\[ \|\varphi(x)\|_B=\sup |\varphi(x)|, \]

and the normed space \(D\) of continuous functions, defined on the whole real plane and having continuous first derivatives in the circle \(|x|\le 1\). Define the norm in the space \(D\) by

\[ \|\varphi(x)\|_D=\sup |\varphi(x)|+\sum_{i=1}^{n}\max_{|x|\le 1}\left|\frac{\partial\varphi(x)}{\partial x_i}\right|. \]

All functions from \(B(D)\) that are solutions of equation (1) (in the generalized sense) form in the normed space \(B(D)\) a linear subspace,

which is, obviously, closed. Denote this subspace by \(\widetilde B(\widetilde D)\). It is clear that \(\widetilde B(\widetilde D)\) itself is a complete normed space. Moreover, from the conditions of the theorem it follows that each function in \(\widetilde B\) has continuous first derivatives and, consequently, belongs to \(\widetilde D\). Thus we have a natural continuous mapping of the space \(\widetilde D\) onto the whole space \(\widetilde B\). Applying now the well-known Banach theorem, we obtain that the inverse mapping is continuous, and, consequently, there exists a constant \(C>0\) such that
\[ \|u(x)\|_D \le C\|(ux)\|_B \]
for every \(u(x)\in \widetilde B\). Let \(\sigma\) be a real root of equation (2). It is easy to verify that \(u(x)=e^{-i\langle x\sigma\rangle}\) is, in this case, a solution of equation (1). Since
\[ \|e^{-i\langle x\sigma\rangle}\|_B=1 \quad\text{and}\quad \|e^{-i\langle x\sigma\rangle}\|_D=1+\sum_{i=1}^n |\sigma_i|, \]
we obtain
\[ 1+\sum_{i=1}^n |\sigma_i| \le C. \]
Thus the theorem is proved.

Lemma. If every continuous solution \(u(x)\) of equation (1) such that
\[ |u(x)|\le e^{a|x|}, \tag{3} \]
is a continuously differentiable function, then there exists a constant \(C>0\) such that from \(P(s)=0\), \(s=\sigma+i\tau\), \(|\tau|\le a\), it follows that \(|\sigma|\le C\).

The last condition means that the strip \(|\tau|\le a\) cuts out from the variety of all complex roots of the equation
\[ P(s)=0,\qquad s=\sigma+i\tau, \tag{4} \]
a bounded set. The proof of this lemma can be obtained by an almost verbatim repetition of the preceding one, if as the spaces \(B\) and \(D\) one takes the spaces of continuous functions with the norms
\[ \|\varphi\|_B=\sup |\varphi(x)|e^{-a|x|},\qquad \|\varphi(x)\|_D=\sup |\varphi(x)|e^{-a|x|} +\sum_{i=1}^n \max_{|x|\le 1}\left|\frac{\partial\varphi(x)}{\partial x_i}\right|. \]
A direct consequence of the lemma is the following theorem:

Theorem 2. If every continuous solution of equation (1) which, for some \(a>0\), satisfies inequality (3) is a continuously differentiable function, then equation (1) is hypoelliptic.

Recall that equation (1) is called hypoelliptic if, for the roots of equation (4), from \(|\sigma|\to\infty\) it follows that also \(|\tau|\to\infty\). Thus, for nonhypoelliptic equations no information about the smoothness of the solution \(u(x)\) can be obtained from estimate (3).

1. Suppose now that (1) is a hypoelliptic equation of type \(\gamma\). This means that the variety of roots of equation (4) lies in the region
   \[ |\tau|\ge c|\sigma|^\gamma-c_1. \]

Theorem 3. If \(u(x)\) is a solution of the hypoelliptic equation (1) and
\[ |u(x)|\le Ce^{a|x|^{1/\beta}},\qquad 0<\beta\le 1, \tag{5} \]
then in any bounded domain
\[ |D^k u(x)|\le C^k\Gamma\left[\frac{(1-\beta)}{\gamma}\,k\right] \]
with some constant \(C>0\).

Proof. As shown in (3), there exists a constant \(M>0\) such that for \(l\le p\), where \(p\) is the order of equation (1),
\[ |D^l u(x)|\le M\max_{|\xi|\le 1}|u(x+\xi)|. \]
From this inequality and estimate (5) we obtain that
\[ |D^l u(x)|\le Ce^{a|x|^{1/\beta}},\qquad 0\le l\le p, \tag{6} \]
with other constants \(C\) and \(a\).

Let \(\alpha(x)\) be an infinitely differentiable function such that \(\alpha(x)=1\) for \(|x|\leqslant 1\) and \(\alpha(x)=0\) for \(|x|\geqslant 2\). Consider the function
\[ f_k(x)=P(D)u(x)\alpha(k^{-\beta}x). \]
It is clear that \(f_k(x)\) is different from zero only when
\[ k^\beta<|x|<2k^\beta. \]
Moreover, from (6) it follows that \(|f_k(x)|\leqslant A^k\). Let us now consider a fundamental solution \(\mathcal E(x)\) of equation (1). Since \(P(D)\mathcal E(x)=\delta(x)\), for \(|x|<k^\beta\)
\[ D^k u(x)=P(D)u(x)\alpha(k^{-\beta}x)*D^k\mathcal E(x)=f_k(x)*D^k\mathcal E(x), \]
\[ |D^k u(x)|=\left|\int_{k^\beta<|\xi|<2k^\beta} f_k(\xi)D^k\mathcal E(x-\xi)\,d\xi\right| \leqslant Ck^{n\beta}A^k\max_{k^\beta\leqslant|\xi|\leqslant 2k^\beta}|D^k\mathcal E(x-\xi)|. \tag{7} \]

If \(x\in W\) lies in a bounded domain, then for sufficiently large \(k\)
\[ \max_{k^\beta\leqslant|\xi|\leqslant 2k^\beta}|D^k\mathcal E(x-\xi)| \leqslant \max_{0.5k^\beta\leqslant|y|\leqslant 3k^\beta}|D^k\mathcal E(y)|. \tag{8} \]

Thus, in order to obtain an estimate for the growth of the derivatives \(D^k u(x)\), \(x\in W\), it is necessary to estimate \(D^k\mathcal E(x)\). As was shown in (4), \(\mathcal E(x)\) can be represented in the form
\[ \mathcal E(x)=G(x)+F(x), \tag{9} \]
where \(G(x)\) is an entire analytic function of order of growth not exceeding one, and
\[ |D^k F(x)|\leqslant C^k e^{l|x|}\int_1^\infty e^{-L|x|r^\gamma}r^{k+n-1}\,dr \leqslant C^k e^{b|x|}\int_0^\infty e^{-L|x|r^\gamma}r^{k+n-1}\,dr. \tag{10} \]

Making the change of variable \(L|x|r^\gamma=y\), \(dr=\dfrac{1}{\gamma}(L|x|)^{-1/\gamma}y^{1/\gamma-1}\,dy\), we obtain
\[ |D^k F(x)|\leqslant C_1^k e^{b_1|x|}|x|^{-k/\gamma} \int_0^\infty e^{-y}y^{\frac{k+n}{\gamma}-1}\,dy = C_1^k e^{b_1|x|}|x|^{-k/\gamma}\Gamma\left(\frac{k+n}{\gamma}\right). \]

Applying Stirling’s formula, we shall have, for \(|x|>1\),
\[ |D^k F(x)|\leqslant C_2^{k+|x|}\left(\frac{k}{|x|}\right)^{k/\gamma}. \tag{11} \]

From (11) we obtain that
\[ \max_{0.5k^\beta\leqslant|x|\leqslant 3k^\beta}|D^k F(x)| \leqslant C_2^{4k}(2k)^{\frac{1-\beta}{\gamma}k} \leqslant C_3^k\Gamma\left[\frac{(1-\beta)k}{\gamma}\right]. \tag{12} \]

Let us now estimate \(D^kG(x)\) for \(|x|<3k\). Since \(G(x)\) is an entire function of order of growth not exceeding one, \(|G(z)|\leqslant Ce^{a|z|}\). Applying Cauchy’s formula with respect to one of the variables,
\[ D^kG(x)=\frac{k!}{2\pi i}\int_L \frac{G(z)}{(z_1-x_1)^{k+1}}\,dz_1, \]
where as the contour \(L\) we take the circle \(|z_1|=4k\). In this case we obtain that for \(|x|<3k\)
\[ |D^kG(x)|\leqslant \frac{4Ck!}{k^k}e^{b_2k}\leqslant B^k. \tag{13} \]

Relations (7), (8), (9), (12), and (13) prove the theorem.

Let us derive several consequences from this theorem.

Corollary 1. If \(u(x)\) satisfies the conditions of the preceding theorem and \(\beta\geqslant 1-\gamma\), then \(u(x)\) is an analytic function of order of growth not exceeding
\[ \frac{\gamma}{\gamma+\beta-1}. \]

Corollary 2. If \(u(x)\) is a solution of a hypoelliptic equation and satisfies inequality (3), then \(u(x)\) is an entire analytic function of order of growth not exceeding one.

Corollary 3. If \(u(x)\) is a solution of an elliptic equation that satisfies inequality (5), then \(u(x)\) is an entire function whose order of growth does not exceed \(1/\beta\).

This assertion follows from Corollary 1 if one notes that for elliptic equations \(\gamma=1\). In particular, for the Laplace equation we obtain that any harmonic function \(u(x)\) which at infinity grows no faster than \(C e^{a|x|^\alpha}\) is an entire analytic function of several complex variables which in the complex plane satisfies the inequality \(|u(z)|\leq C_1 e^{a_1|z|^{\alpha_1}}\), if \(\alpha\geq 1\).

Corollary 4. If the hypoelliptic equation (1) has genus \(\gamma\), then there exists no solution \(u(x)\) of equation (1) which vanishes in some domain and satisfies the inequality
\[ |u(x)|\leq C e^{a|x|^{\frac{1}{1-\gamma}}}. \]

As an example let us consider the heat equation
\[ \frac{\partial u(x_1,x_2)}{\partial x_1} = \frac{\partial^2 u(x_1,x_2)}{\partial x_2^2}. \]
It is not hard to compute that the genus of this equation is equal to \(1/2\). Thus, every solution \(u(x)\) of the heat equation which grows at infinity no faster than
\[ C e^{a(x_1^2+x_2^2)} \]
is an analytic function in both variables.

1. An infinitely differentiable function \(u(x)\), defined in a domain \(W\), is called a function of class \(\rho\) in the direction \(y\), if for every compact set \(K\subset W\) there exists a constant \(C\) such that
   \[ \max_{x\in K}\left|\langle yD\rangle^k u(x)\right|\leq C^k \Gamma(\rho k) \]
   (see (\(^{1}\), p. 106). It is well known that all solutions of elliptic equations are analytic and, consequently, are functions of class 1 in all directions. We shall formulate an assertion which in a certain sense is converse to this.

Theorem 4. If every infinitely differentiable solution of equation (1) is a function of class 1 in the direction \(y\), then the vector \(y\) is orthogonal to all real solutions of the equation \(P_0(\xi)=0\), where \(P_0(s)\) is the principal part of the polynomial \(P(s)\).

The proof of this theorem follows from the fact that for any characteristic \(\xi\), \(P_0(\xi)=0\), there exists an infinitely differentiable solution \(u(x)\) which is identically zero in the half-space \(\langle x,\xi\rangle\geq 0\) ((\(^{1}\), p. 86). Such a solution \(u(x)\) can be a function of class 1 only for those \(y\) for which \(\langle y,\xi\rangle=0\).

Theorem 5. Suppose that in some domain \(W\) every infinitely differentiable solution of equation (1) is a function of class \(\rho\) in the direction \(y\). If \(y\) is not orthogonal to any real solution of the equation \(P_0(\xi)=0\), where \(P_0(s)\) is the principal part of the polynomial \(P(s)\), then \(P(s)\) is a hypoelliptic polynomial and every solution of equation (1) is a function of class \(\rho\) in any other direction.

In conclusion I express my gratitude to Prof. G. E. Shilov for his great attention to this work and for a number of valuable comments.

Moscow State University
named after M. V. Lomonosov

Received
11 X 1960

References

1. L. Hörmander, On the theory of general differential operators, IIL, 1959.
2. I. M. Gelfand, G. E. Shilov, Spaces of fundamental and generalized functions, Moscow, 1958.
3. G. E. Shilov, UMN, 14, no. 5 (89) (1959).
4. V. V. Grushin, UMN, 16, no. 3 (1961).