UDC 517.946

MATHEMATICS

E. V. RADKEVICH

A PRIORI ESTIMATES AND HYPOELLIPTIC OPERATORS WITH MULTIPLE CHARACTERISTICS

(Presented by Academician I. G. Petrovskii on February 7, 1969)

In the paper ((^{1})) L. Hörmander, for a second-order differential operator of the form

[
Pu=\sum_{j=1}^{N}X_j^2u+iX_0u+\gamma(x)u,
\tag{1}
]

where (X_j(x,D)) ((j=0,1,\ldots,N)) are first-order differential operators with infinitely differentiable coefficients, obtained sufficient conditions for the existence of estimates

[
\sum_{j=1}^{N}|X_j u|{(\varepsilon)}^2+|u|^2
\leq C(K){|Pu|{(0)}^2+|u|^2},\qquad
u\in C_0^\infty(K),
\tag{2}
]

where (K) is any compact set in the domain (\Omega\subset R^n) and the constant (\varepsilon=\varepsilon(K)>0) (see ((^{2}))); it can be shown that Hörmander’s conditions are also necessary in the class of operators of the form (1) for the existence of the estimate (2) with some (\varepsilon(K)>0) ((D_j=-i\partial/\partial x_j,\ j=1,\ldots,n;\ i^2=-1)). In the paper ((^{2})) a simple proof of Hörmander’s theorem was obtained. Using the same method, we shall consider the general case of second-order differential operators with nonnegative characteristic form, namely:

[
Lu\equiv \sum_{k,j=1}^{n}a^{kj}(x)D_kD_j u+iX_0(x,D)u+\gamma(x)u,
\tag{3}
]

where (a^{kj}(x)\xi_k\xi_j\geq 0) for any point ((x,\xi)\in \Omega\times S^{n-1}).

The question of global smoothness of generalized solutions of boundary-value problems for equation (3) was considered in the papers ((^{3-6})). We shall give sufficient conditions for local smoothness of generalized solutions of equation (3).

In the domain (\Omega\subset R^n) consider the system of differential operators

[
{L^{0(j)},\ j=1,\ldots,n;\quad L^0_{(j)},\ j=1,\ldots,n;\quad X_0+L^{0(j)}_{(j)}},
]

where

[
L^{0(j)}(x,D)\equiv \sum_k a^{kj}(x)D_k\qquad (j=1,\ldots,n);
]

[
L^0_{(s)}(x,D)\equiv \sum_{k,j}\frac{\partial a^{kj}(x)}{\partial x_s}D_kD_j
\quad (s=1,\ldots,n);\qquad
L^{0(j)}{(j)}\equiv \sumD_k.}\frac{\partial a^{kj}}{\partial x_j
]

For any multi-index (I=(\alpha_1,\ldots,\alpha_k)) ((k\geq 1)), where (\alpha_s=0,1,\ldots,2n), for any (s=1,\ldots,k) put: (|I|=\sum_{s=1}^{k}\lambda_s), where (\lambda_s=2) if (\alpha_s=0), and (\lambda_s=1) if (\alpha_s=1,\ldots,2n), and let

[
A_I(x,D)\equiv \operatorname{ad} A_{\alpha_1}\cdots \operatorname{ad} A_{\alpha_{k-1}}A_{\alpha_k},
]

where (A_s(x,D)\equiv L^{0(s)}(x,D)) ((s=1,\ldots,n)); (A_{s+n}(x,D)\equiv L^0_{(s)}(x,D)) ((s=1,\ldots,n)); (A_0(x,D)\equiv X_0+L^{0(j)}_{(j)}) and (\operatorname{ad}AB=AB-BA) for any pseudodifferential operators (A,B) (see, for example, ((^{7}))).

Definition. The system of operators ((A_0,\ldots,A_{2n})) has at the point (x_0\in\Omega) rank (R_{(A_0,\ldots,A_{2n})}(x_0)=k), if

[
\sum_{|I|\leq k-1}\left|A_I^0(x_0,\xi)\right|=0
\quad \text{for some } \xi\in S^{n-1};
\tag{4}
]

[
\sum_{|I|\leq k}\left|A_I^0(x_0,\xi)\right|\ne 0
\quad \text{for every } \xi\in S^{n-1},
\tag{5}
]

where (A_I^0) is the principal part of the symbol (\sigma(A_I)(x,\xi)) of the operator (A_I) (see, for example, (7)), (S^{n-1}={\xi\in R^n:\sum \xi_j^2=1}).

Remark. It is easy to show that the definition of (R_{(A_0,\ldots,A_{2n})}(x)) is invariant under infinitely differentiable changes of independent variables; moreover, if (R_{(A_0,\ldots,A_{2n})}(x)<\infty) for every point (x\in\Omega), then

[
R_L(K)=\sup_K R_{(A_0,\ldots,A_{2n})}(x)=k(K)<\infty
]

for every compact set (K\subset\Omega).

Theorem 1. If (R_L(x)<\infty) for every point (x\in\Omega), then for every compact (K\subset\Omega) there exists a constant (C(K)) such that the inequality

[
\sum_{j=1}^{n}\left(\left|L^{0(j)}u\right|{(\varepsilon(K))}^{2}
+\left|Lu\right|}^{0{(\varepsilon(K)-1)}^{2}\right)
+|u|})}^{2
\leq
]

[
\leq C(K)\left(|Pu|{(0)}^{2}+|u|\right),}^{2
\quad
u\in C_0^\infty(K),
\tag{6}
]

holds, where

[
\varepsilon(K)=\min\left(1,\;2^{1-R(K)}\left(2^{R(K)-1}-1\right)^{-1}\right).
]

Theorem 2. If (R_L(x)<\infty) for every point (x\in\Omega), then for every compact (K\subset\Omega), every (s\in R^1), there exists a constant (C(K,s)) such that for every function (u\in D'(\Omega)) such that (Lu\in H_{(s)}^{\mathrm{loc}}), an estimate of the form

[
|\varphi u|{(s+\varepsilon(K)+2^{1-R_L(K)})}^{2}
\leq
C(K,s)\left(|\varphi_1Pu|\right),}^{2}+|\varphi_1u|_{(\gamma)}^{2
\tag{7}
]

holds, where the functions (\varphi,\varphi_1\in C_0^\infty(K)) and (\varphi_1\equiv 1) in a neighborhood of (\operatorname{supp}\varphi), (\gamma=\mathrm{const}<s+\varepsilon(K)+2^{1-R_L(K)}).

Theorem 3. If (R_L(x)<\infty) for every point (x) of the manifold (M), where (L) is a second-order differential operator with nonnegative characteristic form, defined on (M), then (L) is a hypoelliptic operator.

The proof of Theorem 1, from which Theorems 2 and 3 can be obtained by known methods (see, for example, (10)), is based on the following auxiliary assertions.

Lemma 1 (energy estimate). For every compact (K\subset\Omega) and every (s\geq 0) there exists a constant (C(K,s)) such that for every (\mu) ((0<\mu<1)) the inequality

[
\sum_{j=1}^{n}\left(\left|L^{0(j)}u\right|{(s)}^{2}
+\left|Lu\right|}^{0{(s-1)}^{2}\right)
+\left|\left(X_0+L}^{0(j)}\right)u\right|_{(s-1/2)}^{2
\leq
]

[
\leq
C(K,s)\left{
\frac{1}{\mu}|Lu|{(0)}^{2}
+\mu|u|}^{2
+C_\mu|u|_{(0)}^{2}
\right},
\quad
u\in C_0^\infty(K).
\tag{8}
]

For every (s\in R^1) in the domain (\Omega) introduce the pseudodifferential operator (\mathcal E_{(s)}) with symbol

[
\sigma(\mathcal E_{(s)})(x,\xi)=\varphi(x)(1+|\xi|^2)^{s/2},
]

where the function (\varphi(x)\in C_0^\infty(\Omega)) and (\varphi\equiv 1) in a neighborhood of the compact set (K).

Consider the system ((Q_0,\ldots,Q_{2n})) of first-order pseudodifferential operators, where (Q_j=L^{0(j)}) ((j=1,\ldots,n)); (Q_0=X_0+L_{(j)}^{0(j)}), (Q_{n+j}=L_{(j)}^{0}\mathcal E_{(-1)}) ((j=1,\ldots,n)).

Lemma 2. For every compact (K\subset\Omega), every integer (k\geq 1), and every (s) ((0\leq s<1/2^{k-1})) there exists a constant (C(K,k,s)) such that for

for any (\mu) ((0<\mu<1)) the inequality
[
\sum_{|I|=k}|Q_Iu|{(s-1+2^{1-k})}^{2}\leq C(K,k,s)\times
]
[
\times\left{\frac1\mu|Lu|+\mu|u|}^{2{(2s)}^{2}+\mu|u|\right},\quad}s)}^{2}+C_\mu|u|_{(0)}^{2
u\in C_0^\infty(K).
\tag{9}
]

From the condition (R_L(K)<\infty) it follows that there exists a neighborhood (O_x\subset\Omega) of the point (x) and a finite system of operators ((Q_{I_1},\ldots,Q_{I_l})), ((|I_j|\leq R) for any (j\leq l)), which is elliptic in the domain (U_x). Therefore the following is valid.

Lemma 3. If (R_L(x)<\infty), then there exists a neighborhood (U_x\subset\Omega) of the point (x) such that an estimate of the form
[
|u|{(s+2^{1-R_L(x)})}^{2}\leq C_1\left{
\sum|Q_Iu|{(s-1+2^{1-|I|})}^{2}+|u|\right},}^{2
\quad u\in C_0^\infty(O_x).
\tag{10}
]

From Lemmas 1–3 follows the proof of Theorem 1.

In papers ((^8,^9)), for a scalar pseudodifferential operator (P) and any compact set (K\subset\Omega), algebraic necessary and sufficient conditions were obtained for the existence of estimates of the form:
[
|u|{(m-\delta)}^{2}\leq C(K){|Pu|},\quad}^{2}+|u|_{(0)}^{2
u\in C_0^\infty(K),
\tag{11}
]
where (m) is the order of the operator (P) and (0\leq\delta<3/4).

We shall consider the class of pseudodifferential operators (P) of order (m), satisfying the following conditions:

1. (p^0(x,\xi)\geq0) for any point ((x,\xi)\in\Omega\times S^{n-1}).

2. (p^1(x,\xi)=-\overline{p^1(x,\xi)}) for any point ((x,\xi)\in N), where
   [
   N={(x,\xi)\in\Omega\times S^{n-1};\ p^0(x,\xi)=0},
   ]
   and (p^\nu(x,\xi)) are the terms homogeneous of order (m-\nu) in the asymptotic expansion in (\xi) of the symbol (\sigma(P)(x,\xi)) of the operator (P) as (\xi\to\infty) (see, for example, (7)).

Applying the method of localization of pseudodifferential operators proposed in paper ((^8)), we obtain algebraic necessary and sufficient conditions for the existence, for any compact set (K\subset\Omega), in the class of operators satisfying conditions 1, 2, of estimates of the form
[
|\varphi u|{(m+s-1)}^{2}\leq C(K,s){|\varphi_1Pu|+|\varphi_1u|}^{2{(\gamma)}^{2}}
\tag{12}
]
for any function (u\in D'(\Omega)) such that (Pu\in H^{\mathrm{loc}}<m+s-1).}), where the functions (\varphi,\varphi_1\in C_0^\infty(K)); (\varphi_1\equiv1) in a neighborhood of (\operatorname{supp}\varphi); (\gamma=\mathrm{const

For any point ((x,\eta)) of the characteristic manifold (N), consider the symmetric matrix
[
\mathfrak A(x,\eta)=
\begin{pmatrix}
p^{0(kj)}(x,\eta) & p_{(j)}^{0(k)}(x,\eta)\
p_{(k)}^{0(j)}(x,\eta) & p_{(kj)}^{0}(x,\eta)
\end{pmatrix}
\quad (k,j=1,\ldots,n),
]
where for any multiindices (\alpha,\beta) ((|\alpha|+|\beta|=2))
[
p_{(\beta)}^{\nu(\alpha)}(x,\xi)=\partial_x^\beta\partial_\xi^\alpha p^\nu(x,\xi).
]

By virtue of condition 1 the matrix (\mathfrak A(x,\eta)) is positive semidefinite at any point ((x,\eta)\in N). Let
[
{Y_j=(a_1^j,\ldots,a_n^j,b_1^j,\ldots,b_n^j),\ j=1,\ldots,2n}
]
be an orthonormal system of eigenvectors of the matrix (\mathfrak A(x,\eta)), and let
[
{Y_j^+,\ j\leq s(x,\eta)}
]
be the set of eigenvectors corresponding to the positive eigenvalues of this matrix;
[
I(x,\eta)\equiv
\left|p_{(j)}^{0(j)}(x,\eta)-ip^1(x,\eta)\right|
+
\sum_{j,k}^{S(x,\eta)}
\left|
\sum_{l=1}^{n}\left(a_l^j b_l^k-a_l^k b_l^j\right)
\right|.
]

Theorem 4. A necessary and sufficient condition for the existence, for the operator (P) with conditions 1, 2, for every compact (K \subset \Omega), of the estimate

[
|u|{(m-1)}^{2} \leq C(K){|Pu|+|u|}^{2{(0)}^{2}}, \qquad u \in C(K),}^{\infty
\tag{13}
]

is the fulfillment of the inequality

[
I(x,\eta)>0 \quad \text{for every point } (x,\eta)\in N.
\tag{14}
]

Theorem 5. If the pseudodifferential operator (P) satisfies conditions 1, 2 and (14), then for every compact (K \subset \Omega) there exists a constant (C(K)) such that the inequality

[
|u|{(m-1)}^{2}+\sumu|}^{n}|P^{(s){(1/2)}^{2}+|P \leq}u|_{(-1/2)}^{2
]

[
\leq C(K){|Pu|{(0)}^{2}+|u|(K).}^{2}}, \qquad u \in C_{0}^{\infty
\tag{15}
]

From estimate (15) (see, for example, ({}^{10})) it follows:

Theorem 6. If the differential operator (P) with infinitely differentiable coefficients satisfies conditions 1, 2 and (14), then for every compact (K \subset \Omega) and every (s \in R^{1}) there exists a constant (C(K,s)) such that for any function (u \in D'(\Omega)) such that (Pu \in H_{(s)}^{\mathrm{loc}}), the estimate

[
|\varphi u|{(s+m-1)}^{2} \leq C(K,s){|\varphiPu|{(s)}^{2}+|\varphi},}u|_{(\gamma)}^{2
]

holds, where the functions (\varphi,\varphi_{1}\in C_{0}^{\infty}(K)) and (\varphi_{1}\equiv 1) on (\operatorname{supp}\varphi); (\gamma=\mathrm{const}<s+m-1).

Theorem 7. A differential operator (P) satisfying conditions 1, 2 and (14) is hypoelliptic in (\Omega).

Remark. For a differential operator (P) of second order of the form (1), condition (14) is equivalent to the following inequality:

[
|X_{0}(x,\xi)|+\sum_{j,k=1}^{N}|X_{j},X_{k}|>0
\quad \text{for every point } (x,\xi)\in N.
]

In conclusion I take this opportunity to express my gratitude to Prof. O. A. Oleinik for her constant attention to my work.

Moscow State University
named after M. V. Lomonosov

Received
28 I 1969

CITED LITERATURE

({}^{1}) L. Hörmander, Sbornik: Mathematics, 12, no. 2, 88 (1968).
({}^{2}) E. V. Radkevich, Uspekhi Mat. Nauk, 24, no. 2 (1969).
({}^{3}) O. A. Oleinik, DAN, 163, no. 3, 577 (1965).
({}^{4}) O. A. Oleinik, Mat. Sb., 69, no. 1, 111 (1966).
({}^{5}) L. Kohn, L. Nirenberg, Sbornik: Pseudodifferential Operators, 1967, p. 88.
({}^{6}) J. J. Kohn, L. N. Nirenberg, Comm. Pure and Appl. Math., 20, 797 (1967).
({}^{7}) J. Kohn, L. Nirenberg, Sbornik: Pseudodifferential Operators, Moscow, 1967, p. 10.
({}^{8}) L. Hörmander, ibid., p. 166.
({}^{9}) Yu. V. Egorov, Mat. Sb., 73, no. 3, 356 (1967).
({}^{10}) E. V. Radkevich, Uspekhi Mat. Nauk, 24, no. 1 (1969).