UDC 517.43

MATHEMATICS

Yu. V. EGOROV

ON A CLASS OF PSEUDODIFFERENTIAL OPERATORS

(Presented by Academician I. G. Petrovskii, February 28, 1968)

1. In this paper we continue the study of pseudodifferential operators of principal type, begun in \((^{1-3})\). As L. Hörmander showed (see \((^1)\)), the estimate

\[ |u|_s \leq C(K)\bigl(|Pu|_{s-m+\delta}+|u|_{s-1}\bigr),\quad u\in C_0^\infty(K) \tag{1} \]

(\(m\) is the order of the operator \(P\), \(K\) is a compact set in \(\Omega\subset \mathbf{R}^n\)) holds for \(0\leq \delta<1/2\) only for elliptic operators, but for \(\delta=1/2\) this estimate defines a new class of “subelliptic” operators, broader than the class of elliptic operators. In our paper \((^3)\) it was shown that for \(1/2\leq \delta<2/3\) estimate (1) holds only for subelliptic operators. In the same paper examples were given of operators for which estimate (1) holds for \(\delta=k/(k+1)\) (and does not hold for smaller values of \(\delta\)), and a hypothesis was stated concerning the absence of operators corresponding to intermediate values of \(\delta\).

In the present paper this hypothesis is proved for the interval \(2/3<\delta<3/4\). Here we make essential use of the exact conditions of an algebraic nature obtained by us, which determine the class of operators for which estimate (1) holds with \(\delta=2/3\). These conditions are also of independent interest.

Unlike subelliptic operators, whose characteristic points always form a manifold of dimension \(2n-2\) in the space of variables \((x_1,\ldots,x_n,\xi_1,\ldots,\xi_n)\), operators of the indicated class may have a characteristic manifold of any dimension from 1 to \(2n-2\). For example, this class includes operators with symbol

\[ p(x,\xi)=i\xi_1+(x_1^2+\cdots+x_k^2)|\xi|+(\xi_1^2+\cdots+\xi_l^2)|\xi|^{-1} \]

\[ (1\leq k\leq n,\ 1\leq l\leq n-1), \]

for which the dimension of the characteristic manifold is equal to \(2n-k-l\). In addition, the class of operators under consideration contains some scalar differential operators, for example the operator

\[ P=x_1\Delta+ia\,\partial^2/\partial x_1^2\quad (a\neq 0). \]

2. Theorem 1 below gives exact conditions characterizing the operators for which

\[ |u|_s\leq C(K)\bigl(|Pu|_{s-m+2/3}+|u|_{s-1}\bigr),\quad u\in C_0^\infty(K). \tag{2} \]

Let us note the following simple assertion:

Lemma 1. If for an operator \(P\) estimate (1) holds with some \(\delta<1\), then at a characteristic point \((x_0,\xi_0)\) all first-order partial derivatives \(\partial p^0(x_0,\xi_0)/\partial x_k\), \(\partial p^0(x_0,\xi_0)/\partial \xi_j\) cannot vanish simultaneously.

We now give the main result of the present paper.

Theorem 1. In order that the estimate (2) hold for the operator \(P\), it is necessary and sufficient that:

1) at the characteristic points
\[ \{(x_0,\xi_0),\ p^0(x_0,\xi_0)=0,\ |\xi_0|=1\} \]
the condition
\[ \operatorname{Im} p_x^0(x_0,\xi_0)p_\xi^0(x_0,\xi_0) = \operatorname{Im}\sum_1^n \frac{\partial p^0(x_0,\xi_0)}{\partial x_k} \frac{\overline{\partial p^0(x_0,\xi_0)}}{\partial \xi_k} \geq 0 \]
be satisfied;

2) if at a characteristic point \((x_0,\xi_0)\)
\[ \operatorname{Im} p_x^0(x_0,\xi_0)p_\xi^0(x_0,\xi_0)=0, \]
then
\[ \partial p^0(x_0,\xi_0)/\partial x_j=\kappa\alpha_j,\qquad \partial p^0(x_0,\xi_0)/\partial \xi_j=\kappa\beta_j, \]
where \(\alpha_j,\beta_j\) are real numbers, \(j=1,\ldots,n\);

3) if at a characteristic point \((x_0,\xi_0)\)
\[ \operatorname{Im} p_x^0(x_0,\xi_0)p_\xi^0(x_0,\xi_0)=0, \]
then
\[ I_1=\operatorname{Im}\,\bar\kappa\big[(Ap_\xi'(x_0,\xi_0),\overline{p_\xi'(x_0,\xi_0)}) -2(Bp_x^0(x_0,\xi_0),p_\xi^0(x_0,\xi_0))+ \]
\[ +(Cp_x^0(x_0,\xi_0),\overline{p_x^0(x_0,\xi_0)})\big]\ne 0, \]
where
\[ A=(\partial^2 p^0(x_0,\xi_0)/(\partial x_i\partial x_j)),\qquad B=(\partial^2 p^0(x_0,\xi_0)/\partial x_i\partial \xi_j), \]
\[ C=(\partial^2 p^0(x_0,\xi_0)/\partial \xi_i\partial \xi_j). \]

Moreover,
\[ I_1\operatorname{Im}\kappa\big[(Ax,x)+2(B\xi,x)+(C\xi,\xi)\big]\geq 0 \]
for all real \(x\) and \(\xi\) such that
\[ (p_x^0(x_0,\xi_0),x)+(p_\xi^0(x_0,\xi_0),\xi)=0. \]

For the proof of Theorem 1 we use Theorem 1 from our paper (3) and the following lemmas.

Lemma 2. For no complex \(a_{ij}\) can one specify such a constant \(C\) that for every \(\psi\in C_0^\infty(\mathbf R^n)\) the inequality
\[ \int |\psi(y)|^2\,dy \leq C\int\left| \frac{\partial\psi}{\partial y_1} +i\frac{\partial\psi}{\partial y_2} +\sum_1^n a_{kj}y_ky_j\psi \right|^2dy \]
holds.

Lemma 3. There does not exist a constant \(C\), independent of the parameter \(\lambda\), for which the inequality
\[ \int_{-\infty}^{\infty}|f(t)|^2\,dt \leq C\int_{-\infty}^{\infty}|f'(t)+(t^2-\lambda^2)f(t)|^2\,dt, \qquad f\in C_0^\infty(\mathbf R^1), \]
would hold.

Lemma 4. In order that the estimate
\[ \int |\psi(y)|^2\,dy \leq C_1\int\left| \frac{\partial\psi}{\partial y_1} +\sum a_{jk}y_jy_k\psi +2\lambda^{-1/2}\sum b_{jk}y_jD_k\psi(y) +\right. \]
\[ \left. +\lambda^{-1/3}\sum c_{jk}D_jD_k\psi(y) \right|^2dy + C_2\lambda^{-\varepsilon} \int_{|\alpha+\beta|\leq N} \sum |y^\beta D^\alpha\psi(y)|^2\lambda^{-2|\alpha|/3}\,dy, \]
\[ \varepsilon>0,\qquad \psi\in C_0^\infty(\mathbf R^n), \]
where \(C_1\) and \(C_2\) do not depend on \(\lambda\), and \(N\) is any number \(\geq 3\), it is necessary and sufficient that
\[ \operatorname{Re}a_{11}\ne 0;\qquad \operatorname{Re}a_{11}\cdot \operatorname{Re}\left( \sum_1^n a_{jk}\alpha_j\alpha_k +2\sum_{\substack{j=1\\ k=2}}^n b_{jk}\alpha_j\beta_k +\sum_2^n c_{jk}\beta_j\beta_k \right)\geq 0 \]
for all real \(\alpha_1,\ldots,\alpha_n,\beta_2,\ldots,\beta_n\).

1. Theorem 2. If for the operator \(P\) the estimate

\[ |u|_s \leq C(K)\bigl(|Pu|_{s-m+\delta}+|u|_{s-1}\bigr), \qquad u\in C_0^\infty(K), \]

holds, and \(2/3\leq \delta < 3/4\), then there exists a constant \(C_1(K)\), depending on \(K\) but not depending on the function \(u\), such that

\[ |u|_s \leq C_1(K)\bigl(|Pu|_{s-m+2/3}+|u|_{s-1}\bigr), \qquad u\in C_0^\infty(K). \]

For the proof of Theorem 2 one uses not only the result formulated in Theorem 1, but also its proof. In particular, Lemmas 2–4 are used essentially.

1. Analogous theorems are valid for broader classes of operators \(L_{\rho,\delta}^{0}\) with \(\rho>\delta\), introduced by L. Hörmander in [4].

As examples of pseudodifferential operators satisfying the conditions of Theorem 1, we mention, in addition to those indicated in item 1, the following:

1) the operator with symbol

\[ p^0(x,\xi)=(ax_1^2+bx_2^2)|\xi|-ic\xi_1,\qquad ac\ne 0,\qquad ab\geq 0, \]

arising in the solution of the problem with oblique derivative

\[ \partial^2 u/\partial x_1^2+\partial^2 u/\partial x_2^2+\partial^2 u/\partial x_3^2=0,\qquad x_3>0, \]

\[ (ax_1^2+bx_2^2)\partial u/\partial x_3+c\,\partial u/\partial x_1=f(x_1,x_2),\qquad x_3=0; \]

2) the operator with symbol

\[ p^0(x,\xi)=ix_1|\xi|^2+a\xi_1^2+b\xi_2^2,\qquad a>0,\qquad b\geq 0, \]

arising in the solution of the problem:

\[ \partial^2 u/\partial x_1^2+\partial^2 u/\partial x_2^2+\partial^2 u/\partial x_3^2+\partial^2 u/\partial x_4^2=0,\qquad x_4>0, \]

\[ ix_1\partial^2 u/\partial x_4^2-a\,\partial^2 u/\partial x_1^2-b\,\partial^2 u/\partial x_2^2=f(x_1,x_2,x_3),\qquad x_4=0. \]

Moscow State University
named after M. V. Lomonosov

Received
23 II 1698

CITED LITERATURE

1. L. Hörmander, Ann. Math., 83, No. 1, 129 (1966).
2. Yu. V. Egorov, DAN, 171, No. 4, 778 (1966).
3. Yu. V. Egorov, Mat. sborn., 73 (115), No. 3, 75 (1967).
4. L. Hörmander, Collection: Pseudodifferential Operators, 1968, p. 297.