Pseudo-differential operators on a manifold, scalar case

distribution kernel K_p supported inside Ω_U := (ϕ×ϕ)⁻¹(Ω). Since ϕ∗(S) is a smoothing operator by the first part of the proof, we may as well assume that S= 0 so that P = Ψ_p. Under this assumption, we proceed as follows.

Let {χ_j} be a partition of unity on U such that suppχ_j ×suppχ_j ⊂ Ω_U for every j. For each j we select an open neighborhood U_j of suppχ_j with U_j ×U_j ⊂ Ω_U and a function χ⁰_j ∈C_c^∞(U_j) which is identically 1 on a neighborhood of suppχ_j. We consider the operator

P_j :=M_χj^◦P and observe thatPj =Pj◦M_χ⁰

j+Sj, withSj a smoothing operator supported in suppχj× U_j. We note that P_j◦M_χ⁰

j = Ψ_rj, wherer_j(x, ξ, y) = χ_j(x)p(x, ξ)χ⁰_j(y).

We now observe thatK_j = suppχ⁰_j is compact and thatϕ(K_j)×ϕ(K_j)⊂Ω.Moreover, r_j ∈ Σ^d_K

j. By application of Proposition 7.2.5 it follows that ϕ_∗(Ψ_rj) = Ψ_ϕ∗(rj). The supports of the kernels of the operators S_j form a locally finite set, so that S :=P

jS_j is a smoothing operator. Hence, so is ϕ∗(S).The sum P

jΨ_rj is locally finite, and therefore so is Q=P

jϕ∗(Ψrj). Hence Q∈Ψ^d(V). We now note that P =X

j

P_j =X

j

Ψ_rj +S

so that ϕ∗(P) =Q+ϕ∗(S) is a pseudo-differential operator in Ψ^d(V).

Its principal symbol equals the principal symbol of Q, which is represented by the symbol q ∈S^d(V) given by

q(x, ξ) = X

j

ϕ∗(rj)(x, ξ, x)

= X

j

r_j(ψ(x), dϕ(ψ(x))^tξ, ψ(x))

= X

j

p_j(ψ(x), dϕ(ψ(x))^tξ)

= X

j

χj(ψ(x))p(ψ(x), dϕ(ψ(x))^tξ)χ⁰_j(ψ(x))

= X

j

χ_j(ψ(x))p(ψ(x), dϕ(ψ(x))^tξ) = ϕ∗(p)(x, ξ).

7.3 Pseudo-differential operators on a manifold, scalar

Definition 7.3.1 Let d∈ R∪ {−∞}.A pseudo-differential operator P of order d on M is a continuous linear operator C_c^∞(M) → C^∞(M) given by a distribution kernel K_P ∈ Γ^−∞(M ×M,C^M D_M) such that the following conditions are fulfilled.

(a) The kernel K_P is smooth outside the diagonal of M ×M.

(b) For each a ∈ M there exists a chart (U_κ, κ) containing a such that the operator Pκ :C_c^∞(κ(U))→C^∞(κ(U)) given by

P_κ(f)(κ(x)) =P(f^◦κ)(x), (x∈U) belongs to Ψ^d(κ(U)).

Remark 7.3.2 Of course, by the Schwartz kernel theorem, each continuous linear oper-ator P :C_c^∞(M)→C^∞(M) is in particular continuous linear C_c^∞(M)→C^−∞(M), hence given by a distribution kernel KP ∈ Γ^−∞(M ×M, DM ⊗C^M). In the above formulation, the existence of the kernel is demanded in order not to rely on the kernel theorem.

Condition (a) asserts that K_P has singular support contained in the diagonal of M, whereas condition (b) stipulates that the singularity along the diagonal is of the same type as that of the kernel of a pseudo-differential operator on Rⁿ.

If ϕ : M → N is a diffeomorphism then by the nature of the definition the map ϕ∗ : Ψ^d(M) → Ψ^d(N) is readily seen to be a linear isomorphism. Before proceeding we will show that the definition of pseudo-differential operator coincides with the old one in case M is an open subset of Rⁿ.

Lemma 7.3.3 Let M be an open subset of Rⁿ and let P : C_c^∞(M) → C^∞(M) be a continuous linear operator with distribution kernel K_P. Then the following statements are equivalent.

(1) Conditions (a) and (b) of the above definition are fulfilled.

(2) P is a pseudo-differential operator in the sense of Definition 5.2.4.

Proof Clearly (2) implies (1), sinceM can be taken as the coordinate patch. We assume (1) and will prove (2).

We first assume that d = −∞. Then requirements (a) and (b) of Definition 7.3.1 guarantee that K_P is smooth on all of M, hence that P is an integral operator with smooth kernel KP ∈Γ^∞(M ×M,C^M DM). Thus,P is an operator in Ψ^−∞ in the sense of Definition 5.2.4.

We now assume that d ∈ R. By Lemma 6.1.6 each a ∈ M has an open neighborhood Ua in M such that the operator Pa : C_c^∞(Ua) → C^∞(Ua) given by f 7→ (P f)|Ua may be written asP_a = Ψ_pa+T_a,withp_a a symbol in Ψ^d(U_a) and with T_a∈Ψ^−∞(U_a) a smoothing operator.

By paracompactness of M there exists a partition of unity {χj} on M such that for eachj the support of χ_j is contained in some set U_aj as above. We put P_j =P_aj, p_j =p_aj

and T_j =T_aj.Then P_j = Ψ_pj +T_j.For each j we choose aχ⁰_j ∈C_c^∞(U_j) such thatχ⁰_j = 1 on an open neighborhood of suppχ_j. Then χ_j(1−χ⁰_j) = 0 so T_j⁰ := M_χjP M(1−χ⁰_j) has kernel [χ_j ⊗(1−χ⁰_j)]K_P which is smooth. The supports of these kernels form a locally finite collection, so thatT⁰ =P

jT_j⁰ is a smoothing operator.

Moreover, we may write

M_χj^◦P ^◦M_χ⁰

j = Ψ_qj+M_χj^◦T_j^◦M_χ⁰

j, withq_j ∈S^d(M) supported by suppψ_j.It follows thatq =P

jq_j is a locally finite sum and defines an element ofS^d(M).Moreover, the smooth kernels of the operatorsM_χj^◦T_j^◦M_χ⁰

j

are locally finitely supported in M ×M so that the operators sum up to a smoothing operator T.For f ∈C_c^∞(M) we now have that

P f =X

j

χ_jP(χ⁰_jf) +T⁰(f) = Ψ_q(f) +T(f) +T⁰(f)

and (2) follows.

If M is a smooth manifold, P : C_c^∞(M) → C^∞(M) a continuous linear operator, and U ⊂M an open subset, we agree to write P_U for the operator C_c^∞(U)→C^∞(U) given by

P_Uf = (P f)|_U, (f ∈C_c^∞(U)).

The following results are now easy consequences of the definitions.

Exercise 7.3.4 If K ∈ Γ^−∞(M ×M,C^M D_M) is the distribution kernel of P, then the restriction K|U×U is the distribution kernel of P_U.

Exercise 7.3.5 LetM be a smooth manifold, andU ⊂M an open subset. Then for each P ∈Ψ^d(M), the operator P_U belongs to Ψ^d(U).

In the sequel we shall make frequent use of the following results.

Lemma 7.3.6 Let P ∈Ψ^d(M) and let χ∈C^∞(M).

(a) Let ψ ∈C^∞(M) be such that suppψ∩suppχ=∅. Then M_χ◦P◦M_ψ ∈Ψ^−∞(M).

(b) Let χ⁰ ∈C^∞(M) be such that χ⁰ = 1 on an open neighborhood of supp χ. Then M_χ◦P −M_χ◦P ◦M_χ⁰ ∈Ψ^−∞(M).

Likewise, P ◦M_χ−M_χ⁰◦P ◦M_χ ∈Ψ^−∞(M).

(c) Let {P_j} is a collection of operators from Ψ^d(M) such that the supports suppK_Pj of the distribution kernels form a locally finite collection of subsets of M×M. Then

X

j

P_j ∈Ψ^d(M).

Proof Let K_P denote the distribution kernel of P. The distribution kernel K⁰ of the operator M_χ^◦P ^◦M_ψ equals K⁰ = (χ⊗ψ)K_P. Since χ⊗ψ = 0 on an open neighborhood of the diagonal in M ×M,the kernel K⁰ is smooth. Hence (a).

We turn to (b). It follows from the hypothesis that (1−χ⁰) and χ have disjoint supports. Hence

Mχ◦P −Mχ◦P ^◦Mχ⁰ =Mχ◦P^◦(I −Mχ⁰) = Mχ◦P^◦M1−χ⁰ ∈Ψ^−∞(M).

The second statement of (b) is proved in a similar way.

It remains to prove (c). Let Q = P

jP_j. Then Q is a well defined continuous linear operator C_c^∞(M) → C^∞(M) with distribution kernel K_Q = P

K_Pj. On the complement of the diagonal in M×M the kernel K_Q is a locally finite sum of smooth functions, hence a smooth function. Let a∈M.There exists a coordinate patch U 3a whose closure in M is compact. The collection J of indices j for which suppK_Pj ∩(U ×U) 6= ∅ is finite. It follows that the kernel of Q_U equals

K_Q|U×U =X

j∈J

K_Pj|U×U =X

j∈J

K_PjU. Hence, Q_U equals the finite sum P

j∈JP_jU and belongs to Ψ^d(U). It follows that Q ∈

Ψ^d(M).

Exercise 7.3.7 Let M be a smooth manifold, and P : C_c^∞(M)→ C^∞(M) a continuous linear operator with a distribution kernel that it smooth outside the diagonal in M ×M.

Let{U_j} be an open covering ofM. IfP_Uj ∈Ψ^d(U_j) for each j, then P ∈Ψ^d(M).

The following result indicates that pseudo-differential operators modulo smoothing op-erators behave like sections of a sheaf.

Lemma 7.3.8 Let {U_j} be an open covering of the manifold M.

(a) Let P, Q∈Ψ^d(M) be such that PUj =QUj for all j. Then P −Q∈Ψ^−∞(M).

(b) Assume that for each j a pseudo-differential operator P_j ∈Ψ^d(U_j) is given. Assume furthermore that P_i =P_j on C_c^∞(U_i∩U_j) for all indices i, j with U_i∩U_j 6=∅. Then there exist a P ∈Ψ^d(M) such that P_Uj −P_j ∈Ψ^−∞(U_j) for all j. The operator P is uniquely determined modulo Ψ^−∞(M).

Proof Let K_P and K_Q denote the distribution kernels of P and Q, respectively. Then K_P|_Uj×U_j andK_Q|_Uj×U_j are the distribution kernels ofP_Uj and Q_Uj,respectively. It follows that K_P−Q = K_P −K_Q is zero hence smooth on each of the sets U_j ×U_j, hence on the diagonal ofM×M.AsK_P−Qis already smooth outside the diagonal, it follows thatK_P−Q

is smooth on M ×M.Hence, P −Q∈Ψ^−∞(M).

We turn to (b). Let Ω =∪jUj×Uj.Then Ω is an open neighborhood of the diagonal in M×M.LetK_j ∈C^−∞(U_j×U_j) denote the distribution kernel ofP_j.LetU_ij :=U_i∩U_j 6=∅.

Then from the assumption it follows that K_i =K_j on (U_i ×U_i) ∩ (U_j ×U_j) =U_ij ×U_ij. From the gluing property of the sheafC^−∞ on Ω it follows that there exists aK ∈C^−∞(Ω) such that K =K_j onU_j ×U_j for all j.

We will now use a cut off function to extendK to all ofM×M,leaving it unchanged on an open neighborhood of the diagonal. Let{χ_ν}be a partition of unity of M,subordinate to the covering {U_j}. For each ν we select j(ν) such that suppχ_ν ⊂ U_j(ν) and we fix a function χ⁰_ν ∈ C_c^∞(U_j(ν)) which is identically 1 on an open neighborhood of suppχ_ν. The functions χ_ν⊗χ⁰_ν form a locally finitely supported family of functions in C_c^∞(Ω).Put

ψ :=X

ν

χ_ν ⊗χ⁰_ν.

Let x ∈ M and let N_x be the finite collection of indices ν with x ∈ suppψ_ν. Then the functionsχ⁰_ν,forν ∈N_xare all 1 on a common open neighborhoodV_x ofxinM.Moreover, P

ν∈Nxχ_ν equals 1 on an open neighborhood U_x of x in M. It follows that ψ = 1 on U_x ×V_x. Hence, ψ = 1 on an open neighborhood of the diagonal in M ×M. Put P = P

νM_χν◦P_j(ν)◦M_χ⁰_ν. Then P is a pseudo-differential operator on M with kernel equal to KP =X

ν

(χν ⊗χ⁰_ν)KPj =ψK.

For each j ∈U we have thatP_Uj has kernel

K_P|_Uj×U_j =ψK|_Uj×U_j =ψK_j.

It follows thatK_P−K_Pj is smooth onU_j×U_j,henceP_Uj−P_j ∈Ψ^−∞(U_j).The uniqueness

statement follows from (a).

In fact, with a bit more effort it can be shown that Ψ^d/Ψ^−∞ defines a sheaf of vector spaces on M. More precisely, for two open subsets U ⊂ V of M the map P 7→ P_U, Ψ^d(V) → Ψ^d(U) induces a restriction map Ψ^d(V)/Ψ^−∞(V) → Ψ^d(U)/Ψ^−∞(U) which we claim to define a sheaf. The following exercise prepares for the proof of this fact.

Exercise 7.3.9 Let Ω be smooth manifold, and let{Ω_j}j∈J be an open cover of Ω.Assume that for each pair of indices (i, j) with Ωij := Ωi∩Ωj 6=∅a smooth function gij ∈C^∞(Ωij) is given such that

g_ij+g_jk +g_kj = 0 on Ω_ijk := Ω_i∩Ω_j∩Ω_k

for alli, j, k with Ω_ijk6=∅.Show that there exist functionsg_j ∈C^∞(U_j) such thatg_i−g_j = g_ij for all i, j. Hint: select a partition of unity {ψ_α}α∈A on Ω which is subordinate to the covering {Ω_j}. Thus, a map j :A → J is given such that suppψ_α ⊂ U_j(α). Now consider g_j :=P

αψ_αg_jj(α). Exercise 7.3.10

(a) Show that with the restriction maps defined above the assignmentU 7→Ψ^d(U)/Ψ^−∞(U) defines a presheaf onM.

(b) Show thatU 7→Ψ^d(U)/Ψ^−∞(U) satisfies the restriction property of a sheaf.

(c) Use the previous exercise combined with the arguments of the proof of Lemma 7.3.8 to show that U 7→Ψ^d(U)/Ψ^−∞(U) has the gluing property.

No documento 1.3 Ellipticity; preliminary version of the Atiyah-Singer index theorem (páginas 139-144)