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Reports of the Academy of Sciences of the USSR
- Volume 186, No. 5
MATHEMATICS
Yu. V. EGOROV
ON NONDEGENERATE SUBELLIPTIC PSEUDODIFFERENTIAL OPERATORS
(Presented by Academician I. G. Petrovsky on 25 XI 1968)
In this note conditions are formulated on the symbol \(p^0(x,\xi)\) of a pseudodifferential operator \(P\) of order \(m\), under which the estimate
\[ |u|_{s-(k-1)/k} \leq C(K)\bigl(|Pu|_{s-m}+|u|_{s-1}\bigr),\qquad u\in C_0^\infty(K), \tag{1} \]
holds, where \(|u|_s\) is the norm of the function \(u\) in the L. N. Slobodetskii space \(H^s\), \(k\) is a natural number, and \(K\) is a compact set in \(\Omega\subset \mathbb{R}^n\).
These conditions are exact, necessary and sufficient, if the operator is nondegenerate. Here we call an operator \(P\) nondegenerate if its vector
\[ \nabla_{x,\xi}p^0(x,\xi) \equiv (\partial p^0/\partial x_1,\ldots,\partial p^0/\partial x_n, \partial p^0/\partial \xi_1,\ldots,\partial p^0/\partial \xi_n) \]
is not proportional to a real vector at each characteristic point \((x,\xi)\in K\times S^{n-1}\), i.e. at a point where \(p^0(x,\xi)=0\).
From the conditions obtained it follows that if, for an operator \(P\), the estimate
\[ |u|_{s-1+\delta}\leq C_1(K)\bigl(|Pu|_{s-m}+|u|_{s-1}\bigr),\qquad u\in C_0^\infty(K), \tag{2} \]
is valid and \(1/(k+1)<\delta\leq 1/k\), then there exists a constant \(C=C(K)\) such that for all functions \(u\in C_0^\infty(K)\) inequality (1) is valid.
Such results were obtained earlier by L. Hörmander in \((^1)\) for the case \(k=1\) and by the author in \((^{4,5})\) for the cases \(k=2\) and \(k=3\).
Without loss of generality, in what follows one may assume that \(m=1\), and that \(P\) is an operator mapping functions from \(C_0^\infty(K)\) into \(C^\infty(K')\), where \(K'\) is a compact set containing \(K\) in its interior (see \((^1)\)).
Let \(P=P_1+iP_2+P_3\), where \(P_1\) and \(P_2\) are first-order operators with positively homogeneous in \(\xi\) real symbols \(p_1(x,\xi)\) and \(p_2(x,\xi)\), respectively, and \(P_3\) is an operator of order zero. The nondegeneracy condition means that the \(2n\)-dimensional vectors \(\nabla_{x,\xi}p_1(x,\xi)\) and \(\nabla_{x,\xi}p_2(x,\xi)\) are not collinear if \(p^0(x,\xi)=0\).
Denote by \(C\) the first-order pseudodifferential operator equal to \(i[P_2,P_1]=i(P_2P_1-P_1P_2)\), and by \(c(x,\xi)\) the principal part of the symbol of this operator. As L. Hörmander showed in \((^1)\), if \(c(x,\xi)<0\) at a characteristic point \((x,\xi)\), then estimate (2) is impossible for the operator \(P\) for any real \(\delta\). If, however, \(c(x,\xi)>0\) at all characteristic points \((x,\xi)\), then estimate (2) holds for \(\delta=1/2\). Suppose that at some point \((x,\xi)\in K\times S^{n-1}\) we have \(p^0(x,\xi)=c(x,\xi)=0\). Consider at this point the values of the principal parts of the symbols of the operators which are successive commutators of the operators \(P_1\) and \(P_2\). Let the first (with minimal \(s\)) nonzero value correspond to the commutator \([\ldots[[P_{j_1},P_{j_2}],P_{j_3}],\ldots,P_{j_s}]\). Then we set \(k(x,\xi)=s-1\). In this way we define the function \(k(x,\xi)\) at all characteristic points \((x,\xi)\in K\times S^{n-1}\) of the operator \(P\).
We shall now formulate the main results of this work.
Theorem 1. Estimate (1) is valid for a nondegenerate operator \(P\) if and only if:
1) \(c(x,\xi) \geqslant 0\) at those points \((x,\xi) \in K \times S^{n-1}\) where \(p^0(x,\xi)=0\);
2) \(k(x,\xi)\) assumes odd values not exceeding \(k+1\).
Theorem 2. If for a nondegenerate operator \(P\) estimate (2) holds and \(1/2(l+1)<\delta\leqslant 1/2l\), then there exists a constant \(C_2=C_2(K)\) such that
\[ |u|_{s-1+1/2l}\leqslant C_2(K)\left(|Pu|_{s-m}+|u|_{s-1}\right) \]
for all functions \(u\in C_0^\infty(K)\).
It is not difficult to see that Theorem 2 follows directly from Theorem 1.
Let us note some consequences of Theorem 1. As is easy to show (see \((^5)\)), the set of points \(x,\xi\) for which \(p^0(x,\xi)=0\) forms, in the case of a nondegenerate operator, a manifold of dimension \(2n-2\). If for the operator \(P\) estimate (2) holds with some \(\delta>0\), then \(c(x,\xi)\) cannot have a zero of infinite order at a characteristic point. Therefore the set of points \((x,\xi)\) for which \(p^0(x,\xi)=c(x,\xi)=0\) has dimension \(\leqslant 2n-3\). The example of an operator \(P\) with symbol
\[ p(x,\xi)=i\xi_1+\xi_2-x_1(\xi_1^2+\ldots+\xi_l^2)|\xi|^{-1} -x_1(x_1^2+\ldots+x_m^2)|\xi| \]
\[ (1\leqslant m\leqslant n,\qquad 2\leqslant l\leqslant n-1), \]
for which, by Theorem 1, estimate (1) holds with \(k=4\), shows that the dimension of the set \(p^0(x,\xi)=c(x,\xi)=0\) can assume any values between \(1\) and \(2n-3\).
Another example of an operator satisfying the conditions of Theorem 1 was noted in \((^6)\). This is the operator with symbol
\[ p(x,\xi)=i\xi_1+\xi_2+ax_1^l|\xi|, \]
where \(l\) is odd and \(a<0\). It is easy to verify that for this operator estimate (2) is fulfilled with \(\delta=1/(l+1)\).
The proof of Theorem 1 uses the conditions, obtained in \((^2,^4)\), of explicit form, necessary and sufficient for the validity of inequality (1). The necessity of condition 1) was proved by L. Hörmander in \((^1)\). The necessity of condition 2) can be proved with the aid of the results of our work \((^6)\), in which conditions were obtained that are necessary and sufficient for the validity of inequalities of the form
\[ \int |u|^2 e^{Q(x,\mu)}\,dx \leqslant C\int \left|\sum_{j=1}^{n} a_j\frac{\partial u}{\partial x_j}\right|^2 e^{Q(x,\mu)}\,dx, \qquad u\in C_0^\infty(\mathbb{R}^n), \]
where \(a_j\) are complex numbers; \(Q(x,\mu)\) is a homogeneous polynomial in \(x\in\mathbb{R}^n\), \(\mu\in\mathbb{R}^r\), with \(\mu_j\geqslant0,\ j=1,\ldots,r\). From this latter work, as well as from \((^5)\), it is clear why the conditions of Theorem 1 are not necessary in the general case.
For the proof of the sufficiency of the conditions of Theorem 1, the results obtained in \((^2,^4)\), as well as the methods developed by L. Hörmander in \((^3)\) in connection with the study of hypoelliptic equations of second order, are used. These methods are connected with the use of certain facts from the theory of noncommutative Lie algebras.
Moscow State University
named after M. V. Lomonosov
Received
20 XI 1968
CITED LITERATURE
\(^1\) L. Hörmander, Ann. Math., 83, No. 1, 129 (1966).
\(^2\) L. Hörmander, Pseudo-differential operators, Moscow, 1967, p. 297.
\(^3\) L. Hörmander, Acta Math., 119, 147 (1968).
\(^4\) Yu. V. Egorov, DAN, 171, No. 4, 778 (1966).
\(^5\) Yu. V. Egorov, DAN, 182, No. 6, 1261 (1968).
\(^6\) Yu. V. Egorov, DAN, 185, No. 3 (1969).