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compact set &!2 ‚" &!, both À-1(&!2 ) )" K and À-1(&!2 ) )" K are compact.
1 2
80 4. Basic Theory of Pseudo Differential Operators
Here À1 and À2 are projections of &! × &! onto the first and second
factors.
For example the diagonal " = {(x, x) : x%EÅ‚&!} is proper. Most of the
proper subsets we will be considering are neighbourhoods of subsets of
the diagonal.
Figure 4.1: A proper set
"
Definition 4.23. An operator T : Co (&!) ’! C "(&!) is said to be prop-
erly supported, if its distribution kernel K has proper support.
Exercise. Let T = a±(x)D± be a differential operator on &!. Computer
the distribution kernel K of T and show that supp K is a subset of the
diagonal. Thus T is properly supported.
"
If T is properly supported then it maps Co into itself, since supp
Tu ‚" À1(À-1(supp u) )" supp K), as is easily seen from the formula
2
Tu(x) = K(x, y)u(y)dy. More generally, for any &!2 ‚"‚" &!, there exists
&!2 2 ‚"‚" &! such that the values of Tu on &!2 depend only on the values of
u on &!2 2 namely, &!2 2 = À2(À-1(&!2 ) )" supp K). From this it follows that
T can be extended to a map from C "(&!) to itself. In fact, if u%EÅ‚C "(&!)
and &!2 , &!2 2 are as above, we define Tu on &!2 by
2
Tu|&! = T(Æu)|&!,
95 where Æ%EÅ‚C "(&!) and Æ = 1 on &!2 2 . This definition is independent of the
choice of Æ and gives the same result on the intersection of two &!2 s; so
Tu is well defined on all of &!.
If T is a properly supported pseudo differential operator, so that T
extends to a map from E2 (&!) to D2 (&!), the same arguments show that T
maps E2 (&!) into itself and extends further to a map of D2 (&!) to itself.
Suppose T and S are properly supported operators on C "(&!) with
distribution kernels K and L which are C " off the diagonal. Then TS
5. Èdo2 s Defined by Multiple Symbols 81
is an operator on C "(&!) with distribution kernel M formally given by
M(x, y) = K(x, z)L(z, y)dz. In fact, if x y, since K(x, .) and L(., Y)
are smooth except at x and y, the product K(x, .)L(., Y) is a well defined
element of E2 , and the formula M(x, y) = displays M
as a C " function off the diagonal.
Proposition 4.24. Supp M is proper. Thus, TS is properly supported.
Proof. Clearly,
Supp M ‚" {(x, y) : À2(À-1(x) )" supp K) )" À1(À-1(y) )" supp L) Æ}
1 2
Suppose A ‚" &! is compact and set B = À2(À-1(A) )" supp K), then B is
1
compact and
À-1(A) )" supp M ‚" A × y : B )" À1(À-1(y) )" supp L) Æ
1 2
= A × y : À-1(B) )" À-1(y) )" supp L Æ
1 2
= A × À2(À-1(B) )" supp L)
1
which is compact. Likewise À-1(A) )" supp M is also compact.
2
Exercise . Suppose A ‚" (&! × &!) is proper. Show that there exists a 96
properly supported Æ%EÅ‚C "(&!×&!) such that Æ = 1 on A. (Hint: Let {Æj} ‚"
"
Co (&! × &!) be a partition of unity on &! × &! and let Æ = = ÆÆj).
A)"supp Æj
5 Èdo2 s Defined by Multiple Symbols
We have arranged our definition of pseudo differential operators to agree
with the usual convention for differential operators, according to which
differentiations are performed first, followed by multiplication by the
coefficients. However, in some situations (for example, computing ad-
joints), it is convenient to have a more flexible setup which allows mul-
tiplication operators both before and after differentiations. We there-
fore, introduce the following apparently more general class of operators
(which, however, turns out to coincide with the class of ÈDO, modulo
smoothing operators).
82 4. Basic Theory of Pseudo Differential Operators
Definition 4.25. For &! open in Rn and m real, we define the class of
multiple symbols of order m on &!,
m
2
S (&! × &!) = {a%EÅ‚C "(&! × Rn × &!) : "±, ², "&!2 ‚"‚" &!, "c = c±²&!
such that sup |D² D±a(x, ¾, y)| d" c(1 + |¾|)m-|±|}
x,y
¾
x,y%EÅ‚&!2
m
When a%EÅ‚S (&! × &!) we define the operator A = a(x, D, y) by
Au(x) = e2Ài(x-y).¾a(x, ¾, y)u(y)dyd¾
Here the integral must be interpreted as an iterated integral with in-
tegration performed first in y then in ¾ as it is not absolutely convergent
as a double integral.
97 We observe that if a(x, ¾, y) = a(x, ¾) is independent of y, then
A = a(x, D); thus this class of operators include the ÈDO2 s. We also
observe that different a2 s may give rise to same operator. For example,
"
if a(x, ¾, y) = Æ(x)È(y) with È, Æ%EÅ‚Co (&!) and supp È )" supp Æ = Æ, then
a%EÅ‚so(&! × &!) and a(x, D, y) = 0.
m
Definition 4.26. Given a%EÅ‚S (&! × &!) we define
= {(x, y)%EÅ‚&! × &! : (x, ¾, y)%EÅ‚ supp a for some ¾%EÅ‚Rn}.
a
m
Proposition 4.27. Let a%EÅ‚S (&! × &!) and let K be the distribution kernel
of A = a(x, D, y). Then
i) Supp K ‚" , and
a
m
ii) if the support of K is proper, then there exists a2 %EÅ‚S (&!×&!) such that
a2 (x, ¾, y) = a(x, ¾, y) when (x, y) is near the diagonal in &! × &!,
a,
is proper and a(x, D, y) = a2 (x, D, y).
Proof. The kernel K is given by
= e2Ài(x-y).¾a(x, ¾, y)w(x, y)dyd¾dx.
From this, we see . = 0 whenever supp w )" = Æ.
a
that
Therefore, supp K ‚" . This proves (i).
a
5. Èdo2 s Defined by Multiple Symbols 83
To prove (ii), choose a properly supported Æ%EÅ‚C "(&! × &!) with Æ = 1
on " *" K, " being the diagonal. Set a(x, ¾, y) = Æ(x, y)a(x, ¾, y).
supp
Then , ‚" supp Æ and hence , is proper. Also a2 (x, ¾, y) = a(x, ¾, y)
a a
when (x, y) is near the diagonal. Now,
=
=
=
= e2Ài(x-y)·¾a(x, ¾, y)Æ(x, y) × v(x)u(y)dyd¾dx
= e2Ài(x-y).¾a2 (x, ¾, y)u(y)dyd¾dx
= .
98
Since v is arbitrary, we have a(x, D, y) = a2 (x, D, y).
m
Theorem 4.28. Suppose a%EÅ‚S (&! × &!) and A = a(x, D, y) is prop-
m
erly supported. Let p(x, ¾) = e-2Àix.¾A(e2Ài(.).¾)(x). Then p%EÅ‚S (&!) and
p(x, D) = A. Further,
1
p(x, ¾)
¾ y
±!
a
"
Proof. For u in Co (&!), we have
u(x) = e2Àix.¾û(¾)d¾.
Therefore, by linearity and continuity of A,
Au(x) = A(e2Ài(.).¾)(x)û(¾)d¾
= e2Àix.¾ p(x, ¾)û(¾)d¾ = p(x, D)u(x).
84 4. Basic Theory of Pseudo Differential Operators
By the proposition above, we can modify a so that is proper with-
a
out affecting A or the behaviour of a along the diagonal x = y (which
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