Introduction to Microlocal Analysis First Lecture: Basics

Introduction to Microlocal Analysis First lecture: Basics

Dorothea Bahns (Gottingen)¨

Third Summer School on “Dynamical Approaches in Spectral Geometry” “Microlocal Methods in Global Analysis”

University of Gottingen¨ August 27-30, 2018

D. Bahns (Gottingen)¨ Introduction to Microlocal Analysis, I August 2018 1 / 20 1 (Tempered) distributions – basic properties

2 Tensor product, convolution, Schwartz kernel theorem

3 Analysis on phase space

4 Differential operators with constant coefﬁcients

5 Structure theorems

D. Bahns (Gottingen)¨ Introduction to Microlocal Analysis, I August 2018 2 / 20 ∞ Topological dual = (sequ.) continuous linear functionals u : Cc (X) → C ∞ Cc (X) 3 φ 7→ u(φ) =: hu, φi ∞ n 1 n Examples: the bump function φ is in Cc (R ). Any f ∈ Lloc (R ) deﬁnes a distribution, Z hf , φi = f (x)φ(x)dx ,

φ 7→ δx0 (φ) = φ(x0) is a distribution. Hence the name “generalized functions”.

(Tempered) distributions – basic properties Distributions

∞ Deﬁnition (Topology on Cc (X), the space of test functions) n ∞ Let X ⊆ R be open. A sequence (φj )j ⊂ Cc (X) is said to converge to ∞ 0 in Cc (X) iff there is a ﬁxed compact K ⊂ X s.t. supp φj ⊆ K for all j α and all derivatives ∂ φj converge uniformly to 0 as j tends to ∞.

Deﬁnition 0 ∞ The space of distributions D (X) is the topological dual of Cc (X).

D. Bahns (Gottingen)¨ Introduction to Microlocal Analysis, I August 2018 3 / 20 (Tempered) distributions – basic properties Distributions

Deﬁnition 0 ∞ The space of distributions D (X) is the topological dual of Cc (X).

∞ Topological dual = (sequ.) continuous linear functionals u : Cc (X) → C ∞ Cc (X) 3 φ 7→ u(φ) =: hu, φi ∞ n 1 n Examples: the bump function φ is in Cc (R ). Any f ∈ Lloc (R ) deﬁnes a distribution, Z hf , φi = f (x)φ(x)dx ,

φ 7→ δx0 (φ) = φ(x0) is a distribution. Hence the name “generalized functions”.

D. Bahns (Gottingen)¨ Introduction to Microlocal Analysis, I August 2018 3 / 20 (Tempered) distributions – basic properties Distributions II

Observe (not obvious!): ∞ n 0 n A linear functional on Cc (R ) is in D (R ) iff for every compact K ⊂ Rn, there exist C and k such that

X α ∞ |hu, φi| ≤ C sup |∂ φ(x)| , for all φ ∈ Cc (K ) |α|≤k

Deﬁnition (order of a distribution) If the same k can be chosen for all K , u is said to be of order ≤ k.

Example α n φ 7→ ∂ φ(x0) for ﬁxed x0 ∈ R is of order |α|.

D. Bahns (Gottingen)¨ Introduction to Microlocal Analysis, I August 2018 4 / 20 (Tempered) distributions – basic properties Localization and (singular) support

Remark (localization) Let Y ⊆ X ⊆ Rn be open. ∞ ∞ ∞ n Then Cc (Y ) ⊆ Cc (X) ⊆ Cc (R ) Let u ∈ D0(X), then ∞ uY : Cc (Y ) 3 φ 7→ hu, φi

is a distribution on Y , called the restriction (localization) of u to Y .

Deﬁnition Let u ∈ D0(X), then the support of u is the set of all x such that

there is no open nbhd Y of x s.t. uY = 0 Its singular support singsupp u is the set of all x such that ∞ there is no open nbhd Y of x s.t. uY ∈ C

D. Bahns (Gottingen)¨ Introduction to Microlocal Analysis, I August 2018 5 / 20 (Tempered) distributions – basic properties Compactly supported distributions

Deﬁnition (Topology on C ∞(X)) ∞ ∞ A sequence (φj )j ⊂ C (X) is said to converge to 0 in C (X) iff all α derivatives ∂ φj converge to 0 uniformly on any compact set K ⊂ X as j tends to ∞.

Deﬁnition The topological dual of C ∞(X) (with this topology) is denoted E 0(X).

Proposition E 0(X) is identical with the set of distributions in X with compact support.

D. Bahns (Gottingen)¨ Introduction to Microlocal Analysis, I August 2018 6 / 20 (Tempered) distributions – basic properties Tempered distributions

Schwartz space = the space of rapidly decreasing smooth functions, n ∞ n α β S (R ) = {φ ∈ C (R ) | kφkα,β := sup |x ∂ φ(x)| < ∞ for all α, β} Deﬁnition (Topology on S (Rn)) n A sequence of Schwartz functions (φj )j ⊂ S (R ) is said to converge to n 0 in S (R ) iff kφj kα,β → 0 as j tends to ∞ for all α, β.

Deﬁnition (Tempered distributions) 0 The space of tempered distributions S (Rn) is the topological dual of S (Rn).

∞ n n −x2 n Examples: Cc (R ) is dense in S (R ). e ∈ S (R ). 0 n ∞ n Restricting u ∈ S (R ) to Cc (R ) gives a distributions. By the denseness above, 0 n 0 n n 0 n S (R ) can be identiﬁed with a subspace of D (R ). Moreover, E (R ) ⊂ S (R ). Any polynomially bounded continuous function f is a tempered distribution with Z φ 7→ f (x)φ(x)dx =: hf , φi

D. Bahns (Gottingen)¨ Introduction to Microlocal Analysis, I August 2018 7 / 20 (Tempered) distributions – basic properties Operations on distributions

0 0 Deﬁnition (Topology on D (X) and S (Rn): pointwise convergence)

A sequence of (tempered) distributions (uj ) converges to 0 as j tends ∞ n to ∞ iff huj , φi → 0 for all φ ∈ Cc (X) (or S (R ), resp.)

Corollary (Extending operations by duality) ∞ ∞ Let g be a linear continuous map Cc (Y ) → Cc (X) (or S (Rn) → S (Rn)), then the transpose gt deﬁned by

hgt (u), φi := hu, g(φ)i

0 0 0 0 is a (seq.) continuous map D (Y ) → D (X) (or S (Rn) → S (Rn)).

∞ 0 n t To extend a contin. operation f on Cc (X) to one in D (R ), calculate hfu, φi = hu, f φi ∞ t for u ∈ Cc (X) and set g = f .

D. Bahns (Gottingen)¨ Introduction to Microlocal Analysis, I August 2018 8 / 20 Similarly, one deﬁnes: Given a diffeomorphism f : X → Y , the pullback of u ∈ D0(Y ) along f is hf ∗u, φi = hu, g∗φ |detg0|i with g = f −1 , g∗φ = φ ◦ g ∗ ∞ This way, f u = u ◦ f for u ∈ Cc (Y ) (by the transf. formula). Multiplication with ψ ∈ C ∞(X): hψu, φi = hu, ψφi Observe: supp ψu ⊂ supp ψ. 0 Analogously for S (Rn) (with X = Y = Rn and ψ polyn. bdd)

(Tempered) distributions – basic properties Operations on distributions – Examples

D. Bahns (Gottingen)¨ Introduction to Microlocal Analysis, I August 2018 9 / 20 (Tempered) distributions – basic properties Operations on distributions – Examples

The derivative ∂α : D0(X) → D0(X) is deﬁned as the transpose of |α| α ∞ ∞ (−1) ∂ : Cc (X) → Cc (X), h∂αu, φi = hu, (−1)|α|∂αφi ∞ For u ∈ Cc (X), this is the usual derivative (by partial integration). Similarly, one deﬁnes: Given a diffeomorphism f : X → Y , the pullback of u ∈ D0(Y ) along f is hf ∗u, φi = hu, g∗φ |detg0|i with g = f −1 , g∗φ = φ ◦ g ∗ ∞ This way, f u = u ◦ f for u ∈ Cc (Y ) (by the transf. formula). Multiplication with ψ ∈ C ∞(X): hψu, φi = hu, ψφi Observe: supp ψu ⊂ supp ψ. 0 Analogously for S (Rn) (with X = Y = Rn and ψ polyn. bdd)

D. Bahns (Gottingen)¨ Introduction to Microlocal Analysis, I August 2018 9 / 20 (Tempered) distributions – basic properties Operations on distributions – Fourier transform

The Fourier transform Z n n −ixξ F : S (R ) → S (R ) , F φ(ξ) = e φ(x)dx =: φˆ(ξ) is a continuous isomorphism, with inverse Z F −1φ(x) = φ(ξ)eixξd¯ξ , where d¯ξ = (2π)−ndξ

0 0 Hence, F : S (Rn) → S (Rn), u 7→ uˆ,

huˆ, φi := hu, φˆi

is continuous.

n Again, if u ∈ S (R ), we recover the ordinary Fourier transform.

D. Bahns (Gottingen)¨ Introduction to Microlocal Analysis, I August 2018 10 / 20 (Tempered) distributions – basic properties Properties of the Fourier transform

Proposition 0 0 F : S (Rn) → S (Rn) is an isomorphism. It holds (D = −i∂): α |α| α Ddαu = ξ uˆ and xdαu = (−1) D uˆ If u has compact support, its Fourier transform is a smooth function, and Z uˆ(ξ) = hu, e−iξ·i = hu(x), e−iξx i notation= u(x)e−iξx dx

Example: For the δ distribution, δx0 (φ) = φ(x0), we have Z ˆ ˆ −ix0ξ −ix0· hδcx0 , φi = hδx0 , φi = φ(x0) = e φ(ξ)dξ = he , φi , so Z −ix0ξ −iξ· notation −iξx δcx0 (ξ) = e = hδx0 , e i = δx0 (x)e dx

D. Bahns (Gottingen)¨ Introduction to Microlocal Analysis, I August 2018 11 / 20 Tensor product, convolution, Schwartz kernel theorem Tensorproduct

Let X ⊆ Rn and Y ⊆ Rm be open. Theorem (tensor product) Let u ∈ D0(X), v ∈ D0(Y ). Then there is a unique element of D0(X × Y ), called the tensor product u ⊗ v, such that ∞ ∞ hu ⊗ v, φ ⊗ ψi = hu, φihv, ψi for φ ∈ Cc (X) , ψ ∈ Cc (Y ).

∞ ∞ The proof relies on the fact that {φ ⊗ ψ|φ ∈ Cc (X), ψ ∈ Cc (Y )} is dense in ∞ Cc (X × Y ).

∞ ∞ For u ∈ Cc (X), v ∈ Cc (Y ), u ⊗ v as above is the tensor product of functions, u ⊗ v(x, y) = u(x)v(y).

D. Bahns (Gottingen)¨ Introduction to Microlocal Analysis, I August 2018 12 / 20 Tensor product, convolution, Schwartz kernel theorem Convolution

Proposition 0 Let u, v ∈ D (Rn), v compactly supported. Then hu ∗ v, φi := hu ⊗ v(x, y), φ(x + y)i deﬁnes a distribution. It holds ∂α(u ∗ v) = u ∗ ∂αv = ∂αu ∗ v and supp u ∗ v ⊂ supp u + supp v. ∞ n ∞ n If v = ρ ∈ Cc (R ), then u ∗ ρ ∈ C (R ) (regularization). ∞ If u ∈ C (Rn), then u ∗ ρ is the convolution.

Notation: hu ∗ v, φi = R u(x)v(y)φ(x + y)dxdy.

0 n u ∗ v is still deﬁned by the above for general u, v ∈ D (R ) if (x, y) 7→ x + y is proper on supp u × supp v.

0 n δ ∗ u = u for all u ∈ D (R ).

D. Bahns (Gottingen)¨ Introduction to Microlocal Analysis, I August 2018 13 / 20 Tensor product, convolution, Schwartz kernel theorem Schwarz kernel theorem

R An integral transform f 7→ Y K (x, y)f (y)dy with kernel K (function on X × Y ) maps a suitable class of functions on Y to functions on X. Generalization to distributions: Theorem There is an one-to-one correspondence between linear continuous ∞ 0 0 operators A: Cc (Y ) → D (X) and kernels K ∈ D (X × Y ). For ∞ ∞ φ ∈ Cc (X) and ψ ∈ Cc (Y ),

hAψ, φi = hK , φ ⊗ ψi.

Proof idea “⊇” For K ⊂ X, K 0 ⊂ Y compact, there are C and N s.t.

X α X β ∞ ∞ 0 |hK , φ ⊗ ψi| ≤ C k∂ φk∞ k∂ ψk∞ for all φ ∈ Cc (K ), ψ ∈ Cc (K ) |α|≤N |β|≤N

∞ “⊆” uniqueness: from denseness of {φ ⊗ ψ} in Cc (X × Y ). Existence: show that estimate above holds for B(ψ, φ) := hAψ, φi and that this sufﬁces for B to extend to a distribution on X × Y .

D. Bahns (Gottingen)¨ Introduction to Microlocal Analysis, I August 2018 14 / 20 Analysis on phase space High frequency cone

Analysis on T ∗X (phase space): in addition to local information (x-space) analyse behaviour w.r.t. covariables (ξ-space). Main idea: Fourier transform of a compactly supported smooth function is a Schwartz function; Fourier transform of a compactly supported distribution v is smooth. Hence: if there are directions where F (v) does not decay quickly, there are singularities.

Deﬁnition Let v be cptly supported distribution. Its high frequency cone Σ(v) is the set of all η 6= 0 that have no conic nbhd Vη such that

−N for all N = 1, 2,... ∃CN s.t. |vˆ(ξ)| ≤ CN hξi for all ξ ∈ Vη

2 1 where hξi = (1 + |ξ| ) 2 is the Japanese bracket.

n Σ(v) is a closed cone. Example: Σ(δ) = R \ 0

D. Bahns (Gottingen)¨ Introduction to Microlocal Analysis, I August 2018 15 / 20 Analysis on phase space Wavefront set

For u ∈ D 0(X) localize:

T ∞ For x ∈ X, set Σx = φ Σ(φu) where φ ∈ Cc (X) with φ(x) 6= 0.

Deﬁnition The wavefront set of u ∈ D0(X) is

˙ ∗ n WF(u) = {(x, ξ) ∈ T X = X × (R \ 0) | ξ ∈ Σx (u)}

The projection of WF(u) onto the ﬁrst variables is the singular support of u.

D. Bahns (Gottingen)¨ Introduction to Microlocal Analysis, I August 2018 16 / 20 Analysis on phase space Previous constructions revisited

P α For a linear differential operator P = |α|≤m aαD be with smooth coefﬁcients aα on X, we have

WF(Pu) ⊆ WF(u)

If the set of normals of a smooth map f : Y → X has empty intersection with WF(u), the pullback of u along f can be uniquely deﬁned and WF(f ∗u) ⊆ f ∗WF(u). ∞ 0 0 For contin. lin. A: Cc (Y ) → D (X) and its kernel K ∈ D (X × Y ) from Schwartz’ kernel theorem, we have

WF(Aψ) ⊂ {(x, ξ) | (x, y, ξ, 0) ∈ WF(K ) for some y ∈ supp ψ}

D. Bahns (Gottingen)¨ Introduction to Microlocal Analysis, I August 2018 17 / 20 Differential operators with constant coefﬁcients Fundamental solutions

Deﬁnition P α A fundamental solution for a differential operator P(D) = |α|≤m aαD 0 on Rn is a distribution E ∈ D (Rn), s.t. P(D)E = δ.

Why is this interesting? If E exists, the existence of smooth solutions ∞ n to P(D)u = f with f ∈ Cc (R ) is guaranteed (and can be calculated explicitly if E is known explicitly): u := E ∗ f is a smooth function with P(D)u = (P(D)E) ∗ f = δ ∗ f = f

Example 3 1 On R , E(x) = − 4π|x| is a fundamental sol’n to Poisson’s equation ∆u = f .

D. Bahns (Gottingen)¨ Introduction to Microlocal Analysis, I August 2018 18 / 20 Differential operators with constant coefﬁcients Malgrange-Ehrenpreis theorem

Theorem Every constant coefﬁcient differential operator on Rn possesses a fundamental solution.

Remarks: Classic proof idea relies on Hahn-Banach argument, were not constructive.

The theorem fails in general for nonconstant smooth coefﬁcients (Lewy’s example: there is an equation P(D)u = f (everything smooth) that does not have a smooth solution).

D. Bahns (Gottingen)¨ Introduction to Microlocal Analysis, I August 2018 19 / 20 Structure theorems Structure theorems

Theorem Every tempered distribution is a derivative of ﬁnite order of some continuous function of polynomial growth.

Theorem 0 If u ∈ D (X) then there are continuous functions fα, such that X α u = ∂ fα α≥0 where the sum is locally ﬁnite, i.e. on any compact set K ⊂ X, only a ﬁnite number of the fα do not vanish identically.

Theorem

If supp u = {x0}, then there are N ∈ N0 and cα ∈ C, s.t. X α u = cα ∂ δx0 |α|≤N D. Bahns (Gottingen)¨ Introduction to Microlocal Analysis, I August 2018 20 / 20