MATHEMATICS

L. R. VOLEVICH

ON HYPOELLIPTIC SYSTEMS WITH VARIABLE COEFFICIENTS

(Presented by Academician I. G. Petrovskii, 24 I 1964)

According to L. Schwartz, a differential equation (system) is called hypoelliptic in a domain \(\Omega\) if every generalized solution is infinitely differentiable whenever the right-hand side of the equation has this property.

L. Hörmander \((^1)\) established a criterion for the hypoellipticity of equations (systems) with constant coefficients. Naturally, the problem arose of extending these results to equations with variable coefficients. For a single equation \((^{2,3})\) a naturally broad class was found (formally hypoelliptic operators)\(^*\). As for systems with variable coefficients, hypoellipticity had been proved only for certain classes (elliptic, \(p\)-parabolic, quasi-elliptic \((^7)\)), possessing a special definite principal part. Below we propose a new class of hypoelliptic systems which includes all the above-mentioned systems and equations with variable coefficients.

1. Notation. \(x=(x^1,\ldots,x^n)\) is a point in Euclidean space \(R^n\), \(\xi=(\xi_1,\ldots,\xi_n)\) are variables dual to \(x\) with respect to the scalar product \(x\cdot \xi=x^1\xi_1+\cdots+x^n\xi_n\), \(D=(D_1,\ldots,D_n)\), \(D_k=-i\,\partial/\partial x^k\). If \(\alpha=(\alpha_1,\ldots,\alpha_n)\) is an integer multi-index, then \(D^\alpha=D_1^{\alpha_1}\cdots D_n^{\alpha_n}\), \(|\alpha|=\alpha_1+\cdots+\alpha_n\). A polynomial \(Q(\xi)\) may be written in the form \(\sum a_\alpha \xi^\alpha\) (a differential operator is written similarly as \(Q(x;D)=\sum a_\alpha(x)D^\alpha\)). Put \(Q^{(\alpha)}(\xi)=\partial^{|\alpha|}Q/\partial \xi_1^{\alpha_1}\cdots \partial \xi_n^{\alpha_n}\); \(\widetilde Q(\xi)=\left[\sum_\alpha |Q^{(\alpha)}(\xi)|^2\right]^{1/2}\). By \(\mathscr D,\mathscr D(\Omega)\) we shall denote the space of infinitely differentiable functions in \(R^n\), \(\Omega\), with compact supports, endowed with the natural topology. By \(\mathscr D',\mathscr D'(\Omega)\) are denoted the spaces of generalized functions over \(\mathscr D,\mathscr D(\Omega)\); \(C^\infty(\Omega)\) is the space of infinitely differentiable functions in \(\Omega\).

2. Formulation of the main results. We shall consider the system

\[ \mathscr P(x;D)u(x)=f(x); \tag{1} \]

here \(u(x)=\{u_1(x),\ldots,u_m(x)\}\), \(f(x)=\{f_1(x),\ldots,f_m(x)\}\) are column vectors of height \(m\), and \(\mathscr P(x;D)=\|P_{ij}(x;D)\|_{i,j=1,\ldots,m}\) is a square matrix of linear differential operators with infinitely differentiable coefficients.

Suppose that the following conditions are satisfied:

A. At each fixed point \(x\in\Omega\) the polynomial \(Q(x;\xi)=\det \mathscr P(x;\xi)\) is hypoelliptic (in the sense of \((^1)\)).

B. There exists a constant \(C>0\) such that for all \(x',x''\in\Omega\)

\[ \widetilde Q(x';\xi)<C\widetilde Q(x'';\xi). \]

\(^*\) Somewhat more general hypoelliptic operators are contained in papers \((^{4-6})\).

C. There exist nonnegative numbers \(s_1,\ldots,s_m,\ t_1,\ldots,t_m\), and \(\sigma>0\) such that

\[ \left|P_{ij}^{(\alpha)}(x;\xi)\right|<C[\widetilde Q(x;\xi)]^{t_j-s_i}(1+|\xi|)^{-\sigma|\alpha|}, \]

where

\[ \sum_{i=1}^{m}(t_i-s_i)=1 \]

and \(P_{ij}\equiv 0\) if \(t_j-s_i<0\).

We note that A and B are conditions of formal hypoellipticity of \(\det \mathscr P\), while condition C is an analogue of the “nondegeneracy” condition \((^7)\). As the example in \((^7)\) (p. 4) shows, conditions A and B are not sufficient for hypoellipticity of the system (1).

Main theorem. Let the system (1) in the domain \(\Omega\) satisfy conditions A, B, C. Let \(u\in\mathscr D'(\Omega)\) and \(f\in C^\infty(\Omega)\). Then \(u\in C^\infty(\Omega)\); in other words, the system (1) is hypoelliptic.

We shall prove this theorem according to the scheme proposed in \((^8)\) (for the case of a single equation). As usual, the center of gravity will lie in the derivation of a priori estimates.

3. Some spaces of generalized functions. Denote by \(U^l\) the space of generalized vector-functions \(u=\{u_1,\ldots,u_m\}\), \(u_j\in\mathscr D'\), for which the Fourier transform \(\hat u(\xi)\) is an ordinary locally integrable function and

\[ \|u,U^l\|=\left[\sum_{j=1}^{m}|\hat u_j(\xi)|^2(1+|\xi|)^{2l}\widetilde Q(\xi)^{2t_j}\,d\xi\right]^{1/2}<\infty . \tag{2} \]

Here \(Q(\xi)=Q(0;\xi)=\det\mathscr P(0;\xi)\) \((0\in\Omega)\). Similarly we define the space \(\mathscr F^l\), consisting of vector-functions \(f=\{f_1,\ldots,f_m\}\), \(f_j\in\mathscr D'\), for which

\[ \|f,\mathscr F^l\|=\left[\sum_{i=1}^{m}|\hat f_i(\xi)|^2(1+|\xi|)^{2l}Q(\xi)^{2s_i}\,d\xi\right]^{1/2}<\infty . \tag{3} \]

Lemma 1. For any \(l\), \(u\in U^l\) \((f\in\mathscr F^l)\) if and only if \(u\in U^{l-1}\) \((f\in\mathscr F^{l-1})\), and the expressions

\[ |h|^{-1}\|u_h-u,U^l\|,\qquad |h|^{-1}\|f_h-f,\mathscr F^l\| \]

are uniformly bounded as \(|h|\to0\). Here \(h\) is a vector in \(R^n\); \(u_h,f_h\) are the translates of the generalized functions \(u,f\) by this vector.

Lemma 2. Let \(a\in\mathscr D,\ f\in\mathscr F^l\). Then \(af\in\mathscr F^l\) and the estimate

\[ \|au,\mathscr F^l\|\leq \sup |a(x)|\,\|f,\mathscr F^l\|+C\|f,\mathscr F^{l-\gamma}\|, \tag{4} \]

\(\gamma>0\), holds; the constant \(C>0\) depends on \(a(x)\), but does not depend on \(u\).

The proof of this lemma is based on Lemma 3 and is analogous to the proof of Lemma 2 in \((^8)\).

Lemma 3. Let \(Q(\xi)\) be a hypoelliptic polynomial of order \(\mu\) and let

\[ \mu(\xi)=(1+|\xi|)^l[\widetilde Q(\xi)]^a,\qquad a>0. \]

Then there exists \(\sigma>0\) such that

\[ |\mu(\xi+\eta)-\mu(\xi)|<C(1+|\eta|)^{a\gamma}(1+|\xi|)^{-a\sigma}\mu(\xi). \]

This lemma is based on the fact that for every hypoelliptic polynomial \(Q(\xi)\), for sufficiently large \(\xi\), the inequality \((^2)\)

\[ |Q^{(\alpha)}(\xi)|<C|\xi|^{-\sigma|\alpha|}|Q(\xi)| \tag{5} \]

holds.

1. The fundamental inequality. The properties B, C of the operator (1) make it possible to write it in the form (cf. (2, 3))

\[ \mathscr{P}(x;D)=\mathscr{P}(D)+\sum_{\omega} a_{\omega}(x)\mathscr{P}_{\omega}(D), \tag{6} \]

where \(a_{\omega}(x)\in C^\infty,\ a_{\omega}(0)=0,\ \mathscr{P}(D)=\mathscr{P}(0;D)\).

If \(\mathscr{P}_{\omega}(D)=\|P_{\omega ij}(D)\|\), then, by virtue of C:

\[ \left|P_{\omega ij}^{(\alpha)}(\xi)\right|\leq C(1+|\xi|)^{-\sigma|\alpha|}\widetilde Q(\xi). \tag{7} \]

Since our aim is to prove the local regularity of solutions of (1), we may assume that \(a_{\omega}(x)\in \mathscr{D}\) and that the quantity \(\delta\)

\[ \delta=\sum_{\omega}\sup |a_{\omega}(x)| \]

is sufficiently small. Under these assumptions we shall establish an estimate.

Theorem 1. Let \(\delta\) be sufficiently small, \(u\in U^l,\ l'<l\). Then there exists a constant \(C>0\) (independent of \(u\)) such that

\[ \|u,U^l\|\leq C\bigl(\|\mathscr{P}(x;D)u,\mathcal F^{\,l'}\|+\|u,U^{l'}\|\bigr). \tag{8} \]

Proof. This theorem must be established for constant coefficients \((\mathscr{P}(x;D)=\mathscr{P}(D))\). The standard passage to variable coefficients is based on representation (6) and Lemma 2. Since smooth finite functions are dense in \(U^l\), we may assume that \(u_j\in\mathscr D,\ j=1,\ldots,m\). Thus, let \(\mathscr{P}(D)u=f\). Passing to the Fourier transform, we obtain the algebraic system

\[ \sum P_{jk}\hat u_k(\xi)=\hat f_j(\xi),\qquad j=1,\ldots,m, \]

from which it follows that

\[ Q(\xi)u_i(\xi)=\sum_{j=1}^{m} P^{ji}(\xi)\hat f_j(\xi), \tag{9} \]

where \(\|P^{ij}(\xi)\|\) is the matrix of algebraic complements of the matrix \(\mathscr P=\|P_{ij}(\xi)\|\). By property C, \(|P^{ji}(\xi)|\leq C\widetilde Q(\xi)^{1-t_j+s_i}\). Since the polynomial \(Q(\xi)\) is hypoelliptic (i.e. (5) is satisfied), for \(l'<l\)

\[ (1+|\xi|)^l\widetilde Q(\xi)\leq C\bigl[(1+|\xi|)^{l'}|Q(\xi)|+(1+|\xi|)^{l'}\widetilde Q(\xi)\bigr]. \]

From these estimates and equality (9) we obtain inequality (8) for \(\mathscr{P}(x;D)=\mathscr{P}(D)\).

1. Theorem 2 (on the regularity of solutions in the whole space). Let \(u\in U^\lambda\) and \(\mathscr{P}(x;D)u=f\in\mathcal F^l\). Then \(u\in U^l\).

Proof. Let \(\lambda\leq l-1\). We shall show that \(u\in U^{\lambda+1}\). By Lemma 1 it suffices for us to establish the uniform boundedness (in \(h\)) of the norms \(\|\Delta_hu,U^\lambda\|\), where \(\Delta_hu=|h|^{-1}(u_h-u)\). The function \(\Delta_hu\) will satisfy the system

\[ \mathscr{P}(x;D)\Delta_hu=\Delta_hf+\sum \Delta_{-h}a_{\omega}\mathscr{P}_{\omega}(D)u_h. \tag{10} \]

Applying Theorem 1 with \(l=\lambda,\ l'=\lambda-1\), we establish the uniform boundedness of \(\|\Delta_hu,U^\lambda\|\), and together with it also that \(u\in U^{\lambda+1}\). If \(l-1<\lambda<l\), then, applying Theorem 1 with \(l=l-1,\ l'=\lambda-1\), by means of (10) we at once establish the uniform boundedness of \(\|\Delta_hu,U^l\|\), i.e. that \(u\in U^l\). The theorem is proved.

6. Local regularity of solutions.

Let \(u=\{u_1,\ldots,u_m\}\), \(u_j\in \mathcal D'(\Omega)\), where \(\Omega\) is a bounded domain. We shall say that \(u\in U^l_{\mathrm{loc}}(\Omega)\) if \(\varphi u\in U^l\) for every function \(\varphi(x)\in\mathcal D\). The space \(F^l_{\mathrm{loc}}(\Omega)\) is defined analogously.

Lemma 4. Let \(\alpha\) be a multi-index and \(|\alpha|>0\). If \(u\in U^l_{\mathrm{loc}}(\Omega)\), then \(\mathcal P^{(\alpha)}(x;D)u\in F^{l+\sigma}_{\mathrm{loc}}(\Omega)\).

Proof. If \(v\in U^l\), then, by virtue of (7), \(\mathcal P^{(\alpha)}_\omega(D)v\in U^{l+\sigma|\alpha|}\). Using (6) and Lemma 2, we find that \(\mathcal P^{(\alpha)}(x;D)v\in U^{l+\sigma}\). Now let \(u\in U^l_{\mathrm{loc}}(\Omega)\), \(\varphi\in\mathcal D(\Omega)\). Choose a function \(\psi\in\mathcal D(\Omega)\) so that \(\psi(x)=1\) for \(x\in \operatorname{sup} p\,\varphi\). Then
\[ \varphi \mathcal P^{(\alpha)}(x;D)u = \varphi \mathcal P^{(\alpha)}(x;D)\psi u \in U^{l+\sigma}. \]
The lemma is proved.

Theorem 3. Let \(u=\{u_1,\ldots,u_m\}\), \(u_j\in\mathcal D'(\Omega)\); let \(\mathcal P(x;D)u\in F^l_{\mathrm{loc}}(\Omega)\). Then \(u\in U^l_{\mathrm{loc}}(\Omega)\).

Proof. Let \(\Omega'\) be any compact subdomain of \(\Omega\). Then there is such a \(\lambda\) that \(u\in U^\lambda_{\mathrm{loc}}(\Omega)\). We shall show that if \(l\ge \lambda+\sigma\), then \(u\in U^{\lambda+\sigma}_{\mathrm{loc}}(\Omega)\). Indeed, by Leibniz’ formula, for \(\varphi\in\mathcal D(\Omega)\),
\[ \mathcal P(x;D)(\varphi u) = \varphi \mathcal P(x;D)u + \sum D^\alpha \varphi\, \mathcal P^{(\alpha)}(x;D)/|\alpha|! \tag{11} \]
From Lemma 4 it follows that the right-hand side of (11) belongs to \(U^{\lambda+\sigma}\). Then, according to Theorem 2, \(\varphi u\in U^{\lambda+\sigma}\), i.e. \(u\in U^{\lambda+\sigma}_{\mathrm{loc}}(\Omega')\). Repeating these arguments, we find that \(\varphi u\in U^l\), i.e. \(u\in U^l_{\mathrm{loc}}(\Omega')\), and in view of the arbitrariness of \(\Omega'\), \(u\in U^l_{\mathrm{loc}}(\Omega)\). The theorem is proved.

The main theorem formulated above follows immediately from Theorem 3.

Received
21 I 1964

References

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