Humboldt-Universität zu Berlin Institut für Mathematik Wintersemester 2011/2012

Peetre's Theorem

Bodo Graumann

April 3, 2014

Contents

1 Basics 2

2 Peetre's Theorem 5

References 7

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1 Peetre’s Theorem 1 Basics Page 2

In 1959 Jaak Peetre first published his astonishing result, that any local (i.e. support-non- increasing) operator is a differential operator, [Pee59]. Unfortunately his proof contained an error and so in 1960 he had to publish a correction, [Pee60]. There, instead of just patching the error of the previous paper, he proposed a more general version of his theorem using distribution theory. We will take this second approach as it seems more natural. So before we can proof Peetre’s Theorem we need to introduce the necessary results of distribution theory. Most of those are taken from [Hö90]. For a more abstract context [RR73] is a good source.

1 Basics

Choose any open set 훺 ⊂ ℝ푛 and a field 핂 ∈ {ℝ, ℂ}.

1 Definition: “Test functions” Let 푈 ⊂ 훺, then we define (푈): = { 푓 ∈ 퐶∞(훺, 핂) | 푈 ⊃ supp 푓 compact } and call these functions test functions with support in 푈. For compact 푈 = 퐾, we take on (퐾) the topology defined by the semi-normes:

훼 ‖푓‖퐾,푚: = sup |퐷 푓(푥)| |훼|≤푚 푥∈퐾

for 푚 ∈ ℕ and 훼 denoting a multi-index. Then ‖푓‖퐾,푚 ≤ ‖푓‖퐾,푚+1.

Now we choose the topology on (훺) as the inductive limit under the inclusions (퐾) → (훺), 퐾 ⊂ 훺 compact. It is the coarsest topology which induces a finer topology on (퐾) than the given one or in other words the topology under which an operator is continuous iff its restriction to every (퐾) is continuous.

2 Definition: “Distribution” The continuous dual space of (훺) under this topology is called the space of distributions on 훺 and denoted by (훺). On that space in turn, we put the weak*-topology, i.e. the topology induced by the semi-norms

‖퐹 ‖푓 : = |퐹(푓)| 푓 ∈ (훺)

On every (푈) we can also consider the (not necessarily finite) operator norms

|퐹(푓)| ‖퐹 ‖푈,푚: = sup 푓∈(푈) ‖푓‖푚 Then we have ‖퐹 ‖푈,푚 ≥ ‖퐹 ‖푈,푚+1 and ∀compact 퐾, 퐹 ∈ (퐾)∃푚 ∈ ℕ: ‖퐹 ‖ < ∞ (1-1) 퐾 퐾,푚퐾 This (smallest possible) 푚 is called the order of the distribution 퐹 on 퐾.

Bodo Graumann Peetre’s Theorem 1 Basics Page 3

3 Definition: “Tensor Product” 푛푖 Let 푈푖 ⊂ ℝ , 푖 = 1, 2 open, then we define: For any 푓푖 ∈ 퐶(푈푖) their tensor product 푓1 ⊗ 푓2: 퐶(푈1 × 푈2) is

(푓1 ⊗ 푓2)(푥1, 푥2): = 푓1(푥1)푓2(푥2)

4 Theorem: Schwartz Kernel Theorem The equation

퐾(푔)(푓) = 푘(푓 ⊗ 푔), 푓 ∈ (푈1), 푔 ∈ (푈2) (4.1) defines a bijective relation between distributions 푘 ∈ (푈1 × 푈2) and continuous linear maps 퐾: (푈2) → (푈1). Compare [Hö90, 5.2.1, 128ff]

5 Remark: What we are actually interested in, are the distributions with compact support. They can in a natural way be seen as the continuous functionals on the space (훺). This space is 퐶∞(훺) with the topology given by the semi-norms ‖⋅‖퐾,푚, 퐾 ⊂ 훺 compact and 푚 ∈ ℕ. Notice that the induced topology on (훺) as a topological subspace of (훺) is coarser than the one above. Thus (훺) ⊃ 퐹 ∈ (훺) supp 퐹 compact ≃ (훺). The extension of a distribution 퐹 ∈ (훺) with compact support to functions without compact support can be done by using appropriate cut-off functions.

6 Theorem: Let 퐹 ∈ (훺) with order at most 푚 and 휙 ∈ 퐶푚+1(훺) such that ∀푥 ∈ supp 퐹, |훼| ≤ 푚: 퐷훼휙(푥) = 0 then already 퐹(휙) = 0, i.e. a distribution with compact support and order 푚 only depends on the derivatives up to order 푚. Compare [Hö90, 2.3.3, 46]

Proof (6) We accept at this point, that there are cut-off functions 휒휀 ∈ (훺) with 휒휀 ≡ 1 on a open neighbourhood of supp 퐹 and 휒휀 ≡ 0 on the complement of

푀휀 = supp 푢 + 퐵휀

denoting the standard 휀-Ball by 퐵휀 and + for element-wise addition. These 휒휀 can also be chosen in such a way, that

훼 −|훼| ∀훼, |훼| ≤ 푚: |퐷 휒휀| ≤ 퐶휀 holds. (See [Hö90, 1.4.1, 25]) Now on supp 퐹 we have 휙 = 휙휒휀 and so 푢(휙) = 푢(휙휒휀). Thus the following estimate holds: |퐹(휙)| ≤ 퐶 sup |퐷훼(휙휒 )| ≤ 퐶 sup |퐷훼휙||퐷훽휒 | ≤ 퐶 휀|훼|−푚 sup |퐷훼휙| 1 휀 2 휀 3 |훼|≤푚 |훼|+|훽|≤푚 |훼|≤푚 푀휀

Bodo Graumann Peetre’s Theorem 1 Basics Page 4

To complete the proof we will show that under the given preconditions

∀훼, |훼| ≤ 푚: lim 휀|훼|−푚 sup |퐷훼휙| = 0 휀→0 푀휀

훼 For |훼| = 푚 this follows because ∀푥 ∈ 푀휀: dist(푥, supp 퐹) ≤ 휀, 퐷 휙 is uniformly continuous and vanishes on supp 퐹 . For |훼| < 푚, 푦 ∈ 푀휀 and 푥 ∈ supp 퐹 s.t. |푥 − 푦| ≤ 휀 Taylor’s formula gives us: 1 푑 |퐷훼휙(푦)| ≤ sup |( )푚−|훼|(퐷훼휙)(푥 + 푡(푦 − 푥))| (푚 − |훼|)! 0<푡<1 푑푡

as all derivatives of order lower than 푚 of 휙 vanish at 푦 = 푥. According to the chain rule the differentiation wrt 푡 yields a sum of products of (푦 − 푥)푚−|훼| and some 푚-th order derivative of 휙. So because |푥 − 푦| ≤ 휀 and the derivatives of order 푚 of 휙 are continuous and vanish on supp 퐹 , so the claim is proven for any 훼.

7 Theorem: Splitting variables 푛 푛1 푛2 Regard ℝ as ℝ × ℝ and thus 푥 = (푥1, 푥2). If now the support of a distribution 퐹 ∈ (훺) of order 푚 is contained in the subspace {0} × ℝ푛2 , 퐹 has the form

퐹(휙) = 퐹 (휙 ) 훼 훼 |훼|≤푚

with certain distributions 퐹훼 in 푥2 having compact support and

훼 휙훼(푥2) = 퐷푥 휙(푥1, 푥2) 1 푥1=0 Notice that this can be done using any linear subspace and some complement. Compare [Hö90, 2.3.5, 47f]

Proof (7) Let 휙 ∈ (훺), then we can compute its Taylor expansion in 푥1 as 푥훼 휙(푥) = 퐷훼휙(0, 푥 ) 1 + 푟(푥) 2 |훼|≤푚 훼! 훼2=0

where the multi-index is split as 훼 = (훼1, 훼2) just like the space itself. 훼 Here 퐷 푟(푥) = 0 when 푥1 = 0 as long as |훼| ≤ 푘. So the conditions of Theorem 6 are fulfilled and so 퐹(푟) = 0. Also we have

푥훼 퐹 (휙) = 퐹(휙(푥 ) 1 ) 훼 2 훼!

Bodo Graumann Peetre’s Theorem 2 Peetre’s Theorem Page 5

8 Theorem: Diagonal support The kernel 푘 of a continuous map 퐾: (푈) → (푈) is supported in the diagonal iff it has the form 퐾(푓) = 푎 퐷훼푓 훼 훼 for 푎훼 ∈ (푈) and this sum is locally finite. Compare [Hö90, 5.2.3, 131] Proof (8) If 퐾 has the given form, the kernel is 푘(휙) = 푎 퐷훼(휙(푥, 푦)) 훼 푦 푥=푦 훼 because it meets Equation 4.1. This kernel is certainly supported in the diagonal. If on the other hand 퐾 is given with a kernel 푘 which is supported in the diagonal, we can apply Theorem 7 and get the above form for the kernel with a locally finite sum.

2 Peetre's Theorem

Let 푃 ∈ 퐿((훺), (훺)) not necessarily continuous, with the locality property: supp 푃푓 ⊂ supp 푓 ∀푓 ∈ (훺) (2-1) Define also: ‖푃푓‖푈,푚 푗(푚, 푥): = inf 푗 ∈ ℕ ∃푈 ∋ 푥: sup < ∞ 푓∈(푈) ‖푓‖푗

9 Definition: “Point of Continuity” We say that 푥 ∈ 훺 is a point of continuity of 푃 iff there is a neighbourhood 푈 ∋ 푥 such that 푃 |(푈) is continuous. Otherwise it is a point of discontinuity. The set of points of discontinuity let us denote by 훬. Then for 푥 ∈ 훬 and all 푚 we have 푗 = ∞ and for 푥 ∉ 훬 there is a 푚 such that 푗 < ∞.

10 Lemma: 훬 is a discrete set and the function 푗(푚, ⋅) is locally bounded on 훺 ∖ 훬 for 푚 sufficiently big.

Proof (10) It is enough to show that there is no sequence of pairwise different points (푥푖)푖∈ℕ, 푥푖 ∈ 퐾 compact and sequences (푗푖), (푚푖) diverging to +∞ with 푗푖 < 푗(푚푖, 푥푖). Assume such sequences are given, then we can can find disjoint 퐾푖 ∋ 푥푖 and functions 푓푖 ∈ (퐾푖) with ‖푓 ‖ = 2−푖 ∧ ‖푃푓 ‖ ≥ 2푖 푖 푗푖 푖 퐾,푚푖

Now define 푓(푥) = 푓푖(푥) for 푥 ∈ 퐾푖 and 푓(푥) = 0 otherwise. Then 푃푓 = 푃푓푖 in 퐾푖 and so we get ‖푃푓‖ ≥ ‖푃푓 ‖ ≥ 2푖 퐾,푚푖 푖 퐾푖,푚푖 for every 푖 which is contradiction to Equation 1-1

Bodo Graumann Peetre’s Theorem 2 Peetre’s Theorem Page 6

11 Theorem: (Peetre) For any such local 푃 as defined above there is a locally finitely supported family of distributions {푎훼}, unique on 훺 ∖ 훬, such that

∀푓 ∈ (훺): supp(푃푓 − 푎훼퐷훼푓) ⊂ 훬 훼

Proof (11) According to Lemma 10 there are 푗, 푚 ∈ ℕ such that for any 푥0 ∈ 훺 ∖ 훬 there is a neighbourhood 푈 of 푥0 with

|푃푓(푔)| ≤ 퐶‖푓‖푗‖푔‖푚

For all 푓, 푔 ∈ (푈) and some fixed constant 퐶. Thus Schwartz Kernel Theorem says, that there is a distribution 푝 ∈ (푈 × 푈) (called the kernel of 푃 ) with

푃푓(푔) = 푝(푓 ⊗ 푔)

Because of Equation 2-1 푝 is supported in the diagonal 퐷 of 푈×푈. In consequence we can apply Theorem 8: For all ℎ ∈ (푈 × 푈) we have

푃푓 = 푎 퐷훼푓 훼 훼

with 푎훼 distributions in (푈) and 훼 multi-indices up to a certain order. By considering a slightly smaller set 푈̃ ⊂ 푈 and choosing 푓 constant to 1 on 푈̃ , we get 훼 푎0 = 푃푓. Similar formulas hold for all 푎훼 when choosing the monomials 푥 . Thus all 푎훼 are uniquely determined only by 푃 . Finally the formula can be extended from the 푈̃ to all of 훺 ∖ 훬 by choosing a locally finite refinement and taking a subordinate partition of unity.

12 Corollary: If 퐻 ⊂ (훺) is a subspace which is closed under multiplication with (훺)-functions and im 푃 ⊂ 퐻, than the 푎훼 can be chosen to lie in 퐻. In particular 퐻 can be chosen to be the embedding of (훺) in (훺) and the theorem then says, that every local morphism of the sheaf (훺) is a differential operator.

13 Remark: The theorem can be applied to local linear operators between sections of vector bundles by choosing local trivializations. Further generalisations of the case 퐻 = (훺) for certain Banach spaces and non-linear oper- ators can be found in [WD73] and [KMS93, V.19, 176] respectively.

Bodo Graumann Peetre’s Theorem References Page 7

References

[Hö90] Hörmander, Lars: The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis. Second Edition. Springer, 1990 [KMS93] Kolář, Ivan ; Michor, Peter W. ; Slovák, Jan: Natural Operations in Differential Geometry. Springer, 1993. – 176ff S. [Pee59] Peetre, Jaak: Une caractérisation abstraite des opérateurs différentels. In: Mathematica Scandinavica 7 (1959), 211-218. http://www.mscand.dk/issue.php?year=1959&volume=7 [Pee60] Peetre, Jaak: Réctification a l’article »Une caractérisation abstraite des opérateurs différentels«. In: Mathematica Scandinavica 8 (1960), 116-120. http://www.mscand.dk/issue.php?year=1960&volume=8 [RR73] Robertson, A.P. ; Robertson, W.J.: Topological Vector Spaces. Second Edition. Cambridge University Press, 1973 [WD73] Wells, John C. ; DePrima, Charles R.: Local Automorphisms are Differential Operators on Some Banach Spaces. In: Proceedings of the AMS 40 (1973), October, Nr. 2, S. 453–457

Bodo Graumann