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Introduction to Microlocal Analysis First Lecture: Basics

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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 1 Topological dual = (sequ.) continuous linear functionals u : Cc (X) ! C 1 Cc (X) 3 φ 7! u(φ) =: hu; φi 1 n 1 n Examples: the bump function φ is in Cc (R ). Any f 2 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 1 Definition (Topology on Cc (X), the space of test functions) n 1 Let X ⊆ R be open. A sequence (φj )j ⊂ Cc (X) is said to converge to 1 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 1. Definition 0 1 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 1 Definition (Topology on Cc (X), the space of test functions) n 1 Let X ⊆ R be open. A sequence (φj )j ⊂ Cc (X) is said to converge to 1 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 1. Definition 0 1 The space of distributions D (X) is the topological dual of Cc (X). 1 Topological dual = (sequ.) continuous linear functionals u : Cc (X) ! C 1 Cc (X) 3 φ 7! u(φ) =: hu; φi 1 n 1 n Examples: the bump function φ is in Cc (R ). Any f 2 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!): 1 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 1 jhu; φij ≤ C sup j@ φ(x)j ; for all φ 2 Cc (K ) jα|≤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 2 R is of order jαj. 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. 1 1 1 n Then Cc (Y ) ⊆ Cc (X) ⊆ Cc (R ) Let u 2 D0(X), then 1 uY : Cc (Y ) 3 φ 7! hu; φi is a distribution on Y , called the restriction (localization) of u to Y . Definition Let u 2 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 1 there is no open nbhd Y of x s.t. uY 2 C D. Bahns (Gottingen)¨ Introduction to Microlocal Analysis, I August 2018 5 / 20 (Tempered) distributions – basic properties Compactly supported distributions Definition (Topology on C 1(X)) 1 1 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 1. Definition The topological dual of C 1(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 1 n α β S (R ) = fφ 2 C (R ) j kφkα,β := sup jx @ φ(x)j < 1 for all α; βg 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 1 for all α, β. Definition (Tempered distributions) 0 The space of tempered distributions S (Rn) is the topological dual of S (Rn). 1 n n −x2 n Examples: Cc (R ) is dense in S (R ). e 2 S (R ). 0 n 1 n Restricting u 2 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 1 n to 1 iff huj ; φi ! 0 for all φ 2 Cc (X) (or S (R ), resp.) Corollary (Extending operations by duality) 1 1 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)). 1 0 n t To extend a contin. operation f on Cc (X) to one in D (R ), calculate hfu; φi = hu; f φi 1 t for u 2 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 2 D0(Y ) along f is hf ∗u; φi = hu; g∗φ jdetg0ji with g = f −1 ; g∗φ = φ ◦ g ∗ 1 This way, f u = u ◦ f for u 2 Cc (Y ) (by the transf. formula). Multiplication with 2 C 1(X): h u; φi = hu; φi Observe: supp u ⊂ supp . 0 Analogously for S (Rn) (with X = Y = Rn and polyn. bdd) (Tempered) distributions – basic properties Operations on distributions – Examples The derivative @α : D0(X) ! D0(X) is defined as the transpose of jαj α 1 1 (−1) @ : Cc (X) ! Cc (X), h@αu; φi = hu; (−1)jαj@αφi 1 For u 2 Cc (X), this is the usual derivative (by partial integration). 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 jαj α 1 1 (−1) @ : Cc (X) ! Cc (X), h@αu; φi = hu; (−1)jαj@αφi 1 For u 2 Cc (X), this is the usual derivative (by partial integration). Similarly, one defines: Given a diffeomorphism f : X ! Y , the pullback of u 2 D0(Y ) along f is hf ∗u; φi = hu; g∗φ jdetg0ji with g = f −1 ; g∗φ = φ ◦ g ∗ 1 This way, f u = u ◦ f for u 2 Cc (Y ) (by the transf. formula). Multiplication with 2 C 1(X): h u; φi = hu; φi Observe: supp u ⊂ supp . 0 Analogously for S (Rn) (with X = Y = Rn and 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 =: φ^(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! u^, hu^; φi := hu; φ^i is continuous. n Again, if u 2 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 jαj α Ddαu = ξ u^ and xdαu = (−1) D u^ If u has compact support, its Fourier transform is a smooth function, and Z u^(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 ^ ^ −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 2 D0(X), v 2 D0(Y ). Then there is a unique element of D0(X × Y ), called the tensor product u ⊗ v, such that 1 1 hu ⊗ v; φ ⊗ i = hu; φihv; i for φ 2 Cc (X) ; 2 Cc (Y ): 1 1 The proof relies on the fact that fφ ⊗ jφ 2 Cc (X); 2 Cc (Y )g is dense in 1 Cc (X × Y ).
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