Peetre’s Theorem
On a C ∞ manifold, one can define the notion of a differential operator
on C ∞ -functions by appealing to the natural definition on open subsets of
Rn and using local coordinates to give this meaning on the manifold. While
such a definition is useful for calculations, it seems a bit clumsy and inelegant. The theorem by Peetre proven below gives a simple characterization
of differential operators on manifolds without appealing to local coordinates.
The proof below is adapted from the proof in [He84, pp. 233–239] which itself is adapted from the proof given in [Na68, pp. 171–176]. We begin with
some standard notation and propositions before coming to the main course
of Peetre’s Theorem.
Notation and Basic Propositions
Let V ⊂ Rn be open. Denote by Cc∞ (V ) the set of functions compactly
supported inside V . Let ∂i denote the partial derivative with respect to the
i-th coordinate in Rn , and for a multi-index α = (α1 , . . . , αn ) of non-negative
integers we put
Dα = ∂1α1 · · · ∂nαn , xα = xα1 1 · · · xαnn
|α| = α1 + · · · + αn , α! = α1 ! · · · αn !
In addition, if β ≤ α (i.e. βi ≤ αi for every i) then we can also define
α − β = (α1 − β1 , . . . , αn − βn )
µ
¶
α!
α
.
=
β
β!(α − β)!
With this very convenient notation, the generalized Leibniz rule for the differentiation of two (sufficiently differentiable) functions f and g can be written
as
X µ α ¶
α
D (f g) =
Dµ (f )Dν (g).
µ
µ+ν=α
For any subset S ⊂ V and m ∈ Z≥0 we define
X
kf kSm =
sup |(Dα f )(x)|.
|α|≤ m
x∈S
When S = V we will drop the superscript. Clearly, these form a countable
family of semi-norms on C ∞ (V ).
Proposition 1. Let C ⊂ U ⊂ V where U and V are open and C a closed
subset of Rn . Then there is a function φ ∈ C ∞ (V ) such that φ ≡ 1 on C,
φ ≡ 0 on V \ U , and 0 ≤ φ ≤ 1 everywhere on V .
Proposition 2. Let C ⊂ Rn be any closed subset. Then there exists a
φ ∈ C ∞ (Rn ) such that
C = {x ∈ Rn : φ(x) = 0}.
For the construction of such functions, see [Sp05, pp. 32–32] and [He84,
pp. 234-235].
Lemma 1. Let m > 0, and suppose f ∈ C ∞ (Rn ) has all derivatives of order
less than or equal to m vanish at the origin. Then given ² > 0 there exists a
g ∈ C ∞ (Rn ) vanishing in a neighborhood of the origin and satisfying
kg − f km < ².
Proof: Let φ ∈ C ∞ (Rn ) be such that φ(x) = 0 for |x| ≤ 21 , φ(x) = 1 for
|x| ≥ 1, and 0 ≤ φ ≤ 1 everywhere. For δ > 0 define
³x´
gδ (x) = φ
f (x).
δ
Note that gδ ≡ 0 on |x| ≤ 2δ and gδ ≡ f on |x| ≥ δ. Hence, the theorem will
follow if we can show that for each multi-index |α| ≤ m
sup |Dα gδ (x) − Dα f (x)| → 0 as δ → 0.
|x|≤ δ
By the assumption that f and its derivatives up to order m vanish at the
origin, we know that
sup |Dα f (x)| → 0 as δ → 0.
|x|≤ δ
Using the Leibniz rule and the chain rule shows that
³x´
X µ α ¶
α
(Dν f )(x).
D gδ (x) =
δ −|µ| (Dµ φ)
µ
δ
µ+ν=α
Since φ is a fixed function, bounded in absolute value by 1, and constant
outside a compact set (so that all of its derivatives have compact support),
we can get an upper bound on (Dµ φ)(x). Thus, we can conlcude that there
is a constant C1 so that
X
|Dα gδ (x)| ≤ C1
δ −|µ| |(Dν f )(x)|.
µ+ν=α
But Dν f is such that all of its derivatives up to order m − |ν| vanish at the
origin. So, expanding Dν f about the origin, we can show that
sup |Dν f (x)| ≤ C2 δ m+1−|ν| .
|x|≤ δ
Hence, we conclude that
sup |Dα gδ (x)| ≤ C1 C2
|x|≤ δ
X
δ m+1−|ν|−|µ| .
µ+ν=α
Since |ν| + |µ| = |α| ≤ m we see that
sup |Dα gδ (x)| → 0 as δ → 0. ¤
|x|≤ δ
Corollary. The lemma above holds if the origin is replaced by any point
y ∈ Rn .
Peetre’s Theorem on Rn
Defintion. Let V ⊂ Rn be an open set. A differential operator on V is a
linear mapping L : Cc∞ (V ) → Cc∞ (V ) such that for each open set U compactly
contained in V there exists a finite family of functions aα ∈ C ∞ (U ) so that
X
Lφ =
aα D α φ
α
for every φ ∈ Cc∞ (U ).
Proposition 3. Let L be a differential operator on V , then we have
supp(Lψ) ⊂ supp(ψ) for all ψ ∈ Cc∞ (V ).
(†)
Peetre’s Theorem proves the amazing result that (†) is also sufficient for
a linear operator on Cc∞ (V ) to be a differential operator! We first establish
this on Rn and then extend the result to smooth manifolds.
Lemma 2. Suppose ψ1 and ψ2 in Cc∞ (V ) agree on some neighborhood of
x ∈ V and the linear operator L satisfies (†), then Lψ1 (x) = Lψ2 (x).
Proof: Clearly x ∈
/ supp(ψ1 − ψ2 ). So, by assumption x ∈
/ supp(L(ψ1 − ψ2 ))
and so Lψ1 (x) = Lψ2 (x) by linearity of L. ¤
Corollary. If L is a linear operator defined on Cc∞ (V ) satisfying (†), then
L can be extended to all of C ∞ (V ).
Proof: Let φx be any element of Cc∞ (V ) that is identically 1 on a neighborhood of x. For any f ∈ C ∞ (V ) define
Lf (x) = L(φx (x) · f (x)).
By the lemma above, the particular choice of φx is irrelevant. ¤
The next two lemmas are the main technical results that facilitate the
proof of the theorem.
Lemma 3. Suppose L : Cc∞ (V ) → Cc∞ (V ) is linear and satisfies (†). Then
every a ∈ V has an open neighborhood U compactly contained in V together
with a non-negative integer m and a positive constant C so that
kLuk0 ≤ Ckukm for all u ∈ Cc∞ (U \ {a}).
Proof: Suppose this were false for some a ∈ V . Let U0 ⊂ V be an open
neighborhood of a that is compactly contained in V . Then by assumption,
there must be a function u1 ∈ Cc∞ (U0 \ {a}) such that kLu1 k0 > 22 ku1 k1 .
Let U1 = {x : u1 (x) 6= 0}. Then U0 \ Ū1 is an open neighborhood of a.
Again by our assumption, there must be a u2 ∈ Cc∞ (U0 \ (Ū1 ∪ {a})) such
that kLu2 k0 > 24 ku2 k2 . Continue this process by induction to get a sequence
of open sets U1 , U2 , . . . with
U¯k ⊂ U0 \ {a}
U¯k ∩ Ūl = ∅ for k 6= l
and a sequence of functions uk ∈ Cc∞ (U0 \ (Ū1 ∪ · · · ∪ Ūk−1 ∪ {a})) ⊂ Cc∞ (V )
satisfying
Uk = {x : uk (x) 6= 0}
kLuk k0 > 22k kuk kk .
Now define
∞
X
2−k
u(x) =
ui (x).
kui ki
i=1
It is easy to show that this is a well-defined element of Cc∞ . Moreover, by
construction
u|Uk = 2−k kuk k−1 uk
which by linearity and our support condition (†) implies that
Lu|Uk = 2−k kuk k−1 Luk .
Since kLuk k0 > 22k kuk kk , there must be xk ∈ Uk so that Lu(xk ) > 2k . Hence,
Lu is unbounded which is impossible since Lu ∈ Cc∞ (V ) by assumption. ¤
Lemma 4. Suppose L : Cc∞ (V ) → Cc∞ (V ) is linear and satisfies (†). Let
U ⊂ V be any open set, and assume there is a constant C > 0 and a nonnegative integer m so that
kLuk0 ≤ Ckukm for all u ∈ Cc∞ (U ).
Then there are functions aα ∈ C ∞ (V ) such that
X
aα (x)(Dα u)(x)
(Lu)(x) =
|α|≤m
for all x ∈ U and u ∈ Cc∞ (U ).
Proof: By the corollary to Lemma 2, we can extend L to all of C ∞ (V ). For
each a ∈ V and each multi-index α, define
Qα,a (x) = (x1 − a1 )α1 · · · (xn − an )αn
bα (a) = (LQα,a )(a).
Clearly, the functions bα belong to C ∞ (V ). For any u ∈ Cc∞ (U ) and
a ∈ U , consider the function fa,m defined by
fa,m = u −
X (Dα u)(a)
Qα,a
α!
|α|≤m
(1)
(note that fa,m is just the difference between u and its Taylor Polynomial
of order m centered at a). By construction, fa,m has all derivatives or order
|α| ≤ m vanishing at a. By the proof of Lemma 1, we can approximate fa,m
in the semi-norm k· km by functions gν which exactly equal fa,m outside some
neighborhood of a but vanish identically near a. By (†) and the corollary to
Lemma 2, we must have Lgν vanish identically near a for all ν. Thus, by our
assumption
kL(fa,m − gν )k0 ≤ kfa,m − gν km → 0 as ν → ∞
and so
(Lfa,m )(a) = lim (Lgν )(a) = 0.
ν→∞
Applying L to both sides of (1) and using (Lfa,m )(a) = 0 gives
(Lu)(a) =
X (LQα,a )(a)
(Dα u)(a).
α!
|α|≤m
Taking aα (x) = bαα!(x) we get the desired result. ¤
We finally have all the necessary technical elements to prove
Theorem 1 (Peetre’s Theorem). Suppose L : Cc∞ (V ) → Cc∞ (V ) is linear and
satisfies supp(Lψ) ⊂ supp(ψ) for all ψ ∈ Cc∞ (V ). Then L is a differential
operator.
Proof: Let U ⊂ V be open with Ū compact and Ū ⊂ V . Applying Lemma
3 to every point of Ū gives us an open cover. By the compactness of Ū we
get a finite cover of open sets U1 , . . . , Ur neighborhoods of points a1 , . . . , ar
in Ū , respectively; by the construction in the lemma, we can take each Ūi
to be compact. For each i, we know that there are constants Ci > 0 and
positive integers mi so that kLuk0 ≤ Ci kukmi for every u ∈ Cc∞ (Ui \ {ai })
Set C = max{Ci } and m = max{mi }. Then we have
kLuk0 ≤ Ckukm
(2)
P
∞
for each i and every u ∈ Cc∞ (Ui \ {ai }). Let 1 = r+1
i=1 φi be a C -partitionof-unity subordinate to the covering U1 , . . . , Ur , V \ Ū of V (c.f. [Sp05, pp.
51 – 52]).
Let u ∈ Cc∞ (U \ {a1 , . . . , ar }), then we can write u =
holds for each φi u. Hence, we have
kLuk0 = k
r
X
i=1
L(φi u)k0 ≤
r
X
kL(φi u)k0 ≤ C
i=1
r
X
Pr+1
i=1
φi u and (2)
kφi ukm .
i=1
Using the Leibniz rule to expand the derivatives of φi u and realizing that the
φi are fixed functions of compact support allows us to conclude that for any
u ∈ Cc∞ (U \ {a1 , . . . , ar })
kLuk0 ≤ C 0 kukm
for some (possibly different) fixed constant C 0 .
By Lemma 4, we can conclude that
X
aα (x)(Dα u)(x)
(Lu)(x) =
|α|≤m
for all x ∈ U \ {a1 , . . . , ar } and some collection of functions aα ∈ C ∞ (V ).
But, both sides are continuous! Hence they must be equal on all of U . ¤
Peetre’s Theorem and Differential Operators on Manifolds
As is typical, we can use the local result in Rn to prove a similar result on
smooth manifolds. To that end, we formulate the following (non-standard)
definition.
Defintion. Let M be a C ∞ -manifold. A differential operator on M is a linear
operator L : Cc∞ (M ) → Cc∞ (M ) satisfying supp(Lψ) ⊂ supp(ψ) for all ψ ∈
Cc∞ (M )
The great advantage of this formulation is that it makes no reference to
local coordinates. We now show that this definition is justified.
Theorem 2. The definition above is equivalent to the usual definition of
differential operators on manifolds.
Proof: Let (U, φ) be a chart on M . Consider the mapping Lφ : Cc∞ (φ(U )) →
Cc∞ (φ(U )) given by
Lφ F = (L(F ◦ φ)) ◦ φ−1 .
An easy check shows that this satisfies the hypotheses of Peetre’s Theorem.
Hence, for any open set compactly contained in φ(U ) we have
X
(Lφ F )(x) =
aα (x)(Dα F )(x).
α
Thus, setting F = f ◦ φ−1 where f ∈ Cc∞ (U ) and y = φ−1 (x) ∈ U gives
X
aα (φ(y))(Dα (f ◦ φ−1 ))(φ(y))
Lf (y) =
α
for every open W compactly contained in U . This is just the usual definition
of a differential operator on a smooth manifold. That the usual definition
implies our (non-standard) definition is trivial! ¤
References
[He84] Helgason, S. (1984). Groups and Geometric Analysis. Orlando, FL,
USA: Academic Press, Inc.
[Na68] Narasimhan, R. (1968). Analysis on Real and Complex Manifolds.
Amsterdam, The Netherlands: Elsevier Science Publishers B.V.
[Sp05] Spivak, M. (2005). A Comprehensive Introduction to Differential Geometry I. Houston, TX, USA: Publish or Perish, Inc.