UDC 517.43

MATHEMATICS

A. S. DYNIN

ON THE THEORY OF PSEUDODIFFERENTIAL OPERATORS ON A MANIFOLD WITH BOUNDARY

(Presented by Academician I. G. Petrovskii on 12 XI 1968)

A theorem is proved on the equivalence of the ellipticity and Fredholm properties for a certain class of pseudodifferential problems in spaces (\mathcal H^{s,p}) of Bessel potentials, which are sections of Hermitian vector bundles over a smooth compact Riemannian manifold with boundary. The problems considered are analogous to the pseudodifferential problems of M. I. Vishik and G. I. Eskin ((^{1})) in spaces of vector functions on a smooth compact Euclidean domain. In ((^{1})) a broad spectrum of such spaces is indicated in which the ellipticity of a problem implies its Fredholm property. However, Bessel potentials are contained in this spectrum only for (p=2).

1. Let (M) be a compact smooth Riemannian manifold with boundary. (By smoothness we everywhere mean belonging to the class (\mathcal C^\infty).) Let (M') be its boundary, (M_I=M\setminus M') its interior, (\pi:T\to M) the tangent bundle of the manifold (M), and (\pi_0:T_0\to M) the subbundle of nonzero vectors in (T).

By (E) we denote a Hermitian smooth vector bundle over (M). Let (\mathcal C^\infty(M_I;E)) be the space of smooth sections of this bundle over (M_I), and (\mathcal C_0^\infty(M_I;E)) the subspace of sections with compact supports in (M_I). For (1<p<\infty) there is defined the Banach space (\mathcal L^p(M;E)) of distributions—sections of the bundle—with norm
[
|u|_p=\left(\int |u|^p dM\right)^{1/p},
]
where (dM) is the Riemannian volume element on (M). With the aid of the Laplace operator on (M) with coefficients in (E), the space (\mathcal L^p(M;E)) generates an interpolation scale of Banach spaces (\mathcal H^{s,p}(M;E)), (-\infty<s<\infty), of distributions—sections of the bundle (E) (see the survey ((^{3}))). We note that (\mathcal H^{0,p}(M;E)=\mathcal L^p(M;E)), and for natural (m), (\mathcal H^{m,p}(M;E)) is the Sobolev space (\mathcal W^{m,p}(M;E)) of S. L. Sobolev. The closure of (\mathcal C_0^\infty(M_I;E)) in (\mathcal H^{s,p}(M;E)) is denoted by (\mathcal H_0^{s,p}(M;E)).

Let (E_1,E_2) be Hermitian smooth bundles over (M). The manifold (M_I) is open. Therefore, for (-\infty<r<\infty), there is defined a class (\mathfrak P^r(M_I;E_1,E_2)) of pseudodifferential operators (P) of order (r) ((^{4})). By definition, these operators act from (\mathcal C_0^\infty(M_I;E_1)) to (\mathcal C^\infty(M_I;E_2)). Their symbols are smooth homomorphisms (\sigma(P):\pi_0^(E_1|M_I)\to\pi_0^(E_2|M_I)). Here and below the vertical bar is the sign of restriction. We single out in (\mathfrak P^r(M_I;E_1,E_2)) the subclass (\mathfrak P^r(M;E_1,E_2)) of operators whose symbols (\sigma(P)) are restrictions of smooth mappings (\bar\sigma(P):\pi_0^(E_1)\to\pi_0^(E_2)).

Fix an arbitrary number (p) strictly between 1 and (\infty).

Lemma 1. If (P\in\mathfrak P^r(M;E_1,E_2)), then the operator (P:\mathcal C_0^\infty(M_I;E_1,E_2)\to\mathcal C^\infty(M_I;E_1,E_2)) (necessarily uniquely) extends to a continuous linear mapping
[
P_{(0)}^s:\mathcal H_0^{s,p}(M;E_1)\to\mathcal H^{s,p}(M;E_2).
]

Let (N) be the normal bundle over the boundary (M'), oriented into (M). Denote by (N^+) the subbundle of inward normals in (N).

Let (T') be the tangent bundle of the manifold (M'), (T'_0) the subbundle of nonzero elements in (T'), and (\rho:T'_0\to M') the corresponding projection. Setting (E'=E|M'), define over (T'_0) the direct product of bundles
[
E^+=\rho^N^+\times \rho^E',
]
whose space is naturally fibered over (\rho^N^+). With this Hermitian bundle (E^+\to \rho^N^+) are associated the Banach bundles of sections of the classes (\mathcal C_0^\infty,\mathcal C^\infty,\mathcal H_0^{s,p},\mathcal H^{s,p}).

Write the restriction (\bar\sigma(P)) to (T_0|M') in the form
[
\bar\sigma(P)(\xi)=\bar\sigma(P)(\nu,\tau),
]
where ((\nu,\tau)) are the normal and tangent components of the vector (\xi\in T_0|M). This restriction can be regarded as a family of symbols
[
\bar\sigma(P)(\cdot,\tau):(\rho^N^+)\tau\to \operatorname{Hom}(E^+)}^+,E_{2\tau
]
of Wiener–Hopf operators
[
\iota(P)_\tau:\mathcal C_0^\infty(\rho^N_\tau^+;E_{1\tau})\to \mathcal C^\infty(\rho^N_\tau^+;E_{2\tau}^+),\qquad \tau\in T'_0.
]
The totality of the latter defines a homomorphism of Banach bundles
[
\iota(P):\mathcal C_0^\infty(\rho^N^+;E_1^+)\to \mathcal C^\infty(\rho^N^+;E_2^+),
]
which we shall call the indicator* of the operator (P).

Lemma 2. For every real (s), the indicator of the operator
[
P\in \mathfrak P^r(M;E_1,E_2)
]
extends to a homomorphism of Banach bundles
[
\iota_0^s(P):\mathcal H_0^{s,p}(\rho^N^+;E_1^+)\to \mathcal H^{s,p}(\rho^N^+;E_2^+).
]

1. The restriction operator to the boundary
   [
   \delta:\mathcal C^\infty(M;E)\to \mathcal C^\infty(M';E')
   ]
   for (ps>1) extends to a continuous epimorphism of restriction
   [
   \delta:\mathcal H^{s,p}(M;E)\to \mathcal B^{s-1/p,p}(M';E'),
   ]
   where (\mathcal B^{s,p}(M';E')), (-\infty<s<\infty), is the scale of O. V. Besov spaces of sections—distributions of the bundle (E'=E|M') over (M') (see, for example, the survey ((^3))).

The adjoint monomorphism of corestriction (if one replaces (-s) by (s), (1-1/p) by (1/p))
[
\delta^*:\mathcal B^{s+1-1/p,p}(M';E')\to \mathcal H_0^{s,p}(M;E)
]
is defined for (ps<1-p).

Let
[
P\in \mathfrak P^r(M;E_1,E_2).
]
Introduce also an operator
[
P_2\in \mathfrak P^{r_2}(M;E_1|U,F_2),
]
where (U) is a tubular neighborhood of the boundary (M'), (F_2) is a Hermitian bundle over (U), and an operator
[
P_1\in \mathfrak P^{r_1}(M;F_1,E_2|U),
]
where (F_1) is a Hermitian bundle over (U). Let
[
\varphi\in \mathcal C_0^\infty(U),\qquad 0\le \varphi\le 1,\qquad \varphi|M'=1.
]

For every real (s) for which
[
s_1=s-r+r_1<-1+1/p,\qquad s_2=s-r_2>1/p,
]
define the fitting (\mathfrak P_{(0)}^s) of the operator (P) as the operator
[
\mathfrak P_0^{(s)}:\mathcal H_0^{s,p}(M;E_1)\times \mathcal B^{s_1+1-1/p,p}(M';F_1')
\to
\mathcal H^{s-r,p}(M;E_2)\times \mathcal B^{s_2-1/p,p}(M';F_2')
]
of the form
[
\mathfrak P_{(0)}^s(u,v)=\bigl(P_{(0)}^s u+\varphi P_{1(0)}^{s_1}\delta^*v,\ \delta P_{2(0)}^{s_2}(\varphi u)\bigr);
]
here
[
F_1'=F_1|M',\qquad F_2'=F_2|M'
]
(cf. (1), and also (2)).

Similarly one defines the fitting of the indicator (\iota(P))
[
\iota_{(0)}^s(P):\mathcal H_0^{s,p}(\rho^N^+;E_1^+)\times F_1'
\to
\mathcal H^{s-r,p}(\rho^N^+;E_2^+)\times F_2',
]
which is a homomorphism of the form
[
\iota_{(0)}^s(P)(u,v)=\bigl(\iota_{(0)}^s(P)u+\iota_{(0)}^{s_1}(P_1)\delta^v,\ \delta\,\iota_{(0)}^s(P)u\bigr);
]
here (\delta,\delta^) are the corresponding homomorphisms of restriction and corestriction on the bundle (N^+).

The fitting (\mathfrak P_{(0)}^s) is called elliptic if the homomorphisms (\sigma(P)) and (\iota_{(0)}^s(P)) are isomorphisms.

A linear operator is called Fredholm if its range is closed and the codimension of this range and the dimension of the null-space of the operator are finite.

Theorem 1. The fitting (\mathfrak P_{(0)}^s) is a Fredholm operator if and only if it is elliptic.

Corollary 1. The operator (P_{(0)}^s) is Fredholm if and only if the homomorphisms (\sigma(P)) and (\iota_{(0)}^s(P)) are isomorphisms.

1. If (P\in \mathcal P^r(M;E_1,E_2)), then its adjoint (P^*) belongs to (\mathcal P^r(M;E_2,E_1)). In this item we consider operators (P) satisfying the following condition:

((\mathcal D)) The operator (P^*) maps (\mathcal C_0^\infty(M;E_2)) into (\mathcal C_0^\infty(M;E_1)).

This condition is satisfied, for example, by differential operators.

Under this assumption the operator ((P^*)_{(0)}^{-s+r}) acts from
[
\mathcal H_0^{-s+r,p}(M;E_2)
]
to (\mathcal H_0^{-s,p}(M;E_1)). Therefore its adjoint is defined,
[
P^s:\mathcal H^{s,p}(M;E_1)\to \mathcal H^{s-r,p}(M;E_2).
]

Lemma 3. The operator (P) satisfies condition ((\mathcal D)) if and only if the indicator (\iota(P^)) maps (\mathcal C_0^\infty(\rho^N^+;E_2^+)) into (\mathcal C_0^\infty(\rho^*N^+;E_1^+)).

As above, the homomorphism dual to such an indicator defines, for every real (s), a mapping
[
\iota^s(P):\mathcal H^{s,p}(\rho^N^+;E_1^+)\to
\mathcal H^{s-r,p}(\rho^N^+;E_2^+).
]

Let now (P_1,P_2) be the same as in the preceding item, but let (P) satisfy condition ((\mathcal D)). Then, analogously to item 2, with the aid of (P^s), (P_{1(0)}^{s_1}), (P_2^{s_2}) one can define the ((\mathcal D))-equipment of the operator (P),
[
\mathfrak P^s:\mathcal H^{s,p}(M;E_1)\times
\mathcal B^{s_1+1-1/p}(M';F_1')
\to
\mathcal H^{s-r,p}(M;E_2)\times
\mathcal B^{s_2-r-1/p,p}(M';F_2'),
]
and, with the aid of (\iota^s(P)), (\iota_{(0)}^{s_1}(P_1)), (\iota^{s_2}(P_2)), the ((\mathcal D))-equipment of the indicator (\iota(P)),
[
\iota^s(P):\mathcal H^{s,p}(\rho^N^+;E_1^+)\times F_1'
\to
\mathcal H^{s-r,p}(\rho^N^+;E_2^+)\times F_2'.
]

We shall call the equipment (P^s) elliptic if the homomorphisms (\sigma(P)) and (\iota^s(P)) are isomorphisms.

Theorem 2. The equipment (\mathfrak P^s) is a Fredholm operator if and only if it is elliptic.

Corollary 2. The operator (P^s) is Fredholm if and only if the homomorphisms (\sigma(P)) and (\iota^s(P)) are isomorphisms.

1. M. I. Vishik and G. I. Eskin ((^1)) established under what conditions an operator (P\in Pr(M;E_1,E_2)) admits an elliptic equipment (P_{(0)}^s), when (M) is a Euclidean domain. A direct generalization of their result is the following

Proposition 1. In order that an operator (P\in \mathcal P^r(M;E_1,E_2)) admit a Fredholm equipment (P_{(0)}^s), it is necessary and sufficient that its symbol (\sigma(P)) be an isomorphism, that the indicator operators (\iota^s(P)_\tau), (\tau\in T_0'), be Fredholm, and that the index of the homomorphism (\iota^s(P)) in the sense of ((^5)) belong to the inverse image (\rho^*K(M')) in (K(T_0')) with respect to the canonical projection (\rho:T_0'\to M).

One can compute the index of the Wiener–Hopf family of operators. As a consequence one obtains the following simple sufficient conditions:

Proposition 2. An operator (P\in Pr(M;E_1,E_2)) admits a Fredholm equipment if the following set of conditions is satisfied: (a) (M) is a domain in (R^n); (b) (\dim E_1=\dim E_2<(n-1)/2); (c) the symbol (\sigma(P)) is an isomorphism; (d) for every (\nu\in N) the spectrum of the endomorphism
[
[\bar\sigma(P)(\nu,0)]^{-1}[\bar\sigma(P)(-\nu,0)]\in \operatorname{End}E_{1\pi(\nu)}
]
does not contain numbers with argument (2\pi/p).

Remark added in proof. If (n) is odd, condition (b) is superfluous.

Central Economics and Mathematics Institute
Academy of Sciences of the USSR

Received
12 XI 1968

References

1. M. I. Vishik, G. I. Eskin, Matem. sborn., 74, No. 3, 327 (1967).
2. B. Yu. Sternin, DAN, 172, No. 1, 44 (1967).
3. E. Madzhenes, UMN, 21, No. 2, 169 (1966).
4. M. F. Atiyah, I. M. Singer, UMN, 23, No. 5, 170 (1968).
5. L. Illusie, C. R. 260, 6499 (1965).