UDC 517.944

MATHEMATICS

V. S. FEDII

ESTIMATES IN THE NORMS (H_{(s)}) AND HYPOELLIPTICITY

(Presented by Academician I. G. Petrovskii, 31 XII 1969)

Let (P) be a linear differential operator with infinitely differentiable coefficients, defined in a domain (\Omega \subseteq \mathbf{R}^n) (or on a manifold of class (C^\infty)). The operator (P) is called hypoelliptic if, for every distribution (u) in (\Omega), one has
[
\operatorname{sing\,supp} u=\operatorname{sing\,supp} Pu.
]

In this paper we give sufficient conditions for hypoellipticity that are more general than those in [1]. With their help it is possible to prove the hypoellipticity of certain second-order differential operators with real coefficients and a semidefinite principal part which do not satisfy the conditions obtained in [4, 6].

For convenience of formulation, let us introduce the conditions A, B, and C written below. As usual, ([P,l]) denotes the commutator of the operators (P) and (l).

A. For every point (x_0 \in \Omega) there exist a neighborhood (U(x_0)) and real numbers (s_0) and (a) such that, whatever the real number (N), with some constant (C_N) independent of (u), the inequality
[
|u|{(s_0+a)}^2 \leq C_N{|Pu|^2},\qquad}^2+|u|_{(-N)
u\in C_0^\infty(U(x_0)).
\tag{1}
]
holds.

Since in inequality (1), for all (x) belonging to some compact set (K), one may take one and the same (a), we shall assume that (a) does not depend on (x).

B. For every point (x_0 \in \Omega) there exist a neighborhood (U(x_0)) and a real number (s_1) such that, whatever the linear differential operator (l) of order zero or one with infinitely differentiable coefficients in (\Omega), and whatever the real (s>s_1), (\varepsilon>0), and (N), with some constant (C_{\varepsilon,l,N,s}) independent of (u), the inequality
[
|[P,l]u|{(s)}^2
\leq
\varepsilon |Pu|^2}l)
+
C_{\varepsilon,l,N,s}|u|_{(-N)}^2,
\qquad
u\in C_0^\infty(U(x_0)).
\tag{2}
]
holds.

C. For every point (x_0 \in \Omega) and every neighborhood (U(x_0)) there exist a real number (s_2) and a sequence of functions ({\psi_j(x)}) ((\psi_j(x)\in C_0^\infty(U(x_0)),\ \psi_1(x_0)\ne 0,\ \psi_{j+1}) is equal to one in a neighborhood of the support of (\psi_j)) such that, for arbitrary real (s>s_2) and (N), with some (\varkappa>0) and constants (C_{j,N,s}) independent of (u), the inequalities
[
|[P,\psi_j]u|{(s)}^2
\leq
C{|Pu|{(s-\varkappa)}^2+|u|^2},
\qquad
u\in C_0^\infty(U(x_0)).
\tag{3}
]
hold.

Theorem 1. If the operator (P) satisfies conditions A and B in (\Omega), then from (u\in \mathcal{E}'(\Omega)), (Pu\in H_{(s)}(\Omega)), it follows that (u\in H_{(s+a)}(\Omega)).

Consider the operator (P), equal to
[
\sum_{j=1}^{r} X_j^2+X_0+c,
]
where all (X_j) are homogeneous first-order differential operators with real infinitely differentiable coefficients in (\Omega).

Definition. We shall say that the operator (P) satisfies Hörmander’s condition at the point (x\in\Omega) if at this point the Lie algebra generated by the operators (X_0,\ldots,X_r) has rank (n). In [4] it is proved that,

that if (P) satisfies this condition at every point of (\Omega), then it is hypoelliptic.

The following lemma gives an example of operators satisfying the conditions of Theorem 1.

Lemma. Let (M) be a smooth submanifold in (\Omega) of dimension (\leq n-1). Suppose that at all points (x \in \Omega \setminus M) the operator

[
P=\sum_{j=1}^{r} X_j^2+X_0+c
]

satisfies the Hör condition, while at those points (x \in M) where this condition is violated, at least one of the operators (X_j) ((j=1,\ldots,r)) is transversal to (M). Then the operator (P) satisfies condition A with (a=0) and condition B.

If (\Omega) is a domain in (\mathbb{R}^n), then condition B is equivalent to condition B′, which is more convenient for use.

B′. For any point (x_0 \in \Omega) there exists a neighborhood (U(x_0)) such that for any multi-indices (\alpha) and (\beta), (|\alpha|+|\beta|\ne 0), and certain real (s_\beta^\alpha), whatever (\varepsilon>0) and (N) may be, the inequalities

[
\left|P_{(\beta)}^{(\alpha)}u\right|{(s\beta^\alpha)}^2
\leq
\varepsilon |Pu|{(s\beta^\alpha+|\beta|)}^2
+
C|u|_{(-N)}^2,
\qquad
u\in C_0^\infty(U(x_0)).
\tag{4}
]

hold with certain constants independent of (u), but possibly depending on (x_0,\alpha,\beta,\varepsilon), and (N).

Here (P_{(\beta)}^{(\alpha)}) is the differential operator corresponding to the symbol (D_\xi^\alpha D_x^\beta p(x,\xi)), where (p(x,\xi)) is the symbol of the operator (P).

Remark. It can be shown that from the validity of B′ it follows that the inequalities (4) hold with arbitrary (s) in place of (s_\beta^\alpha) in the exponents of the norms. Conversely, if the inequalities (4) hold only for (|\alpha|+|\beta|=1), but for all (s), then condition B′ is fulfilled. From the validity of condition B′ there follows the validity of condition B″.

B″. For any compact (K\subset \Omega), any multi-indices (\beta), (|\beta|\ne 0), and certain real (s_\beta), whatever (\varepsilon>0) and (N) may be, the inequalities

[
\left|P_{(\beta)}u\right|{(s\beta)}^2
\leq
\varepsilon |Pu|{(s\beta+|\beta|)}^2
+
C|u|_{(-N)}^2,
\qquad
u\in C_0^\infty(K).
\tag{5}
]

hold with certain constants independent of (u).

Theorem 1′. Let the operator (P) satisfy conditions A and B″ in some domain (\Omega\subseteq\mathbb{R}^n). Then from (u\in D'(\Omega)), (Pu\in H_{(s)}^{\mathrm{loc}}(\Omega)), and (\operatorname{sing\,supp} u) being compact in (\Omega), it follows that (u\in H_{(s+a)}^{\mathrm{loc}}(\Omega)).

Let us note that conditions A and B″ are satisfied for all differential operators with constant coefficients. Therefore Theorem 1′ may be regarded as a certain generalization of a theorem of M. S. Agranovich ((^7)).

The following theorem is the main result of the paper.

Theorem 2. If the operator (P) satisfies conditions A, B, and C in (\Omega), then from (u\in D'(\Omega)), (Pu\in H_{(s)}^{\mathrm{loc}}(\Omega')), where (\Omega') is an arbitrary open subset in (\Omega), it follows that (u\in H_{(s+a)}^{\mathrm{loc}}(\Omega')). Consequently, the operator (P) is hypoelliptic.

In the proof of Theorems 1 and 2, mollifiers studied in ((^2,^3)) are used.

Theorem 3. If the operator

[
P=\sum_{j=1}^{r} X_j^2+X_0+c
]

at every point of (\Omega), except for a certain set (Q) of isolated points, satisfies the Hör condition, and at every point (x\in Q) at least one of the operators (X_j) ((j=0,1,\ldots,r)) is nonzero, then from (u\in D'(\Omega)), (Pu\in H_{(s)}^{\mathrm{loc}}(\Omega')), it follows that (u\in H_{(s)}^{\mathrm{loc}}(\Omega')). Consequently, the operator (P) is hypoelliptic.

Theorem 4. Consider the differential operator (P=-\partial^2/\partial x^2-\varphi^2(x)\partial^2/\partial y^2). Suppose that (\varphi(x)) belongs to (C^\infty(\mathbf R^1)), is equal to zero at (x=0) together with all its derivatives, and is different from zero for (x\ne 0). Then from (u\in D'(\Omega)), (Pu\in H^{\mathrm{loc}}{(s)}(\Omega')), it follows that (u\in H^{\mathrm{loc}}(\Omega')). Consequently, the operator (P) is hypoelliptic.

Theorems 3 and 4 are proved by using Theorem 2. The greatest difficulty is the verification that the operator (-\partial^2/\partial x^2-\varphi^2(x)\partial^2/\partial y^2) satisfies condition C. By Fourier transformation with respect to (y), this problem is reduced to proving the estimate

[
\left||\eta|^{\varkappa}\frac{d}{dx}\varphi^2 u\right|{(0)}^2+
\left||\eta|^{1+\varkappa}\varphi^2 u\right|^2
\le
C\left{\left|Q_\eta u\right|{(0)}^2+|u|^2\right},
\qquad
u\in C_0^\infty(K).
\tag{6}
]

Here (K) is some compact set in (\mathbf R^1), (Q_\eta=-d^2/dx^2+\varphi^2(x)\eta^2), the constant (C) does not depend on (\eta), and (\varkappa) is some real number greater than zero. First, estimate (6) is proved when the support of (u) belongs to the set ({x:\ |\varphi(x)|>b/\sqrt{|\eta|}}), where (b) is a certain constant depending on the function (\varphi(x)). In doing so, the method of paper ((^5)) is used. Then estimate (6) is proved by integration by parts when the support of (u) belongs to the set ({x:\ |\varphi(x)|<2b/\sqrt{|\eta|}}). The proof is completed by using a certain special partition of unity.

Remark. Theorems 1 and 2 are also valid for pseudodifferential operators with compact supports.

The author expresses his deep gratitude to V. V. Grushin for posing the problem and for his constant attention to the work.

Moscow State University
named after M. V. Lomonosov

Received
25 XII 1969

REFERENCES

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