On support theorems for the X-Ray transform with incomplete data - PowerPoint PPT Presentation

On support theorems for the X-Ray transform with incomplete data Aleksander Denisiuk University of Warmia and Mazury in Olsztyn, Poland denisjuk@matman.uwm.edu.pl Irvine, June 9, 2012 1 / 42

Introduction Boman-Quinto support theorems [BQ] Boman-Quinto support theorems—revised Introduction 2 / 42

Weighted X-ray Transform X ⊂ R n is open Introduction ● Boman-Quinto Y ⊂ G n is an immersed real-analytic n -dimensional support theorems ● [BQ] submanifold of the set of lines— line complex Boman-Quinto support Z = { ( x, l ) ∈ X × Y | x ∈ l } —the incidence relation ● theorems—revised µ ( x, l ) ∈ C ∞ ( Z ) is a weight function ● l ( a, ξ ) = { x = a + ξt } is a line parameterization ● � R µ f ( l ) = R µ f ( a, ξ ) = l ( a,ξ ) f ( x ) µ ( x, l ( a, ξ )) dt ● 3 / 42

Introduction Boman-Quinto support theorems [BQ] Boman-Quinto support theorems—revised Boman-Quinto support theorems [BQ] 4 / 42

Admissible complexes in R 3 Given a non-planar real analytic surface W ⊂ R 3 . Y Introduction Type I: Boman-Quinto is the set of all lines l , tangent to W , such that W has support theorems [BQ] nonzero directional curvature along l at point of tangency. Boman-Quinto support Given a nonsingular real analytic curve γ ∈ R 3 . Y is Type II: theorems—revised the set of lines intersecting this curve non-tangentially Type III: Given a closed simple nonsingular real analytic curve of directions θ ⊂ S 2 . Y is the set of lines with directions on θ . 5 / 42

Type I Introduction Theorem 1. Let Y be an open connected subset of type I Boman-Quinto complex defined by W . Assume that Y is an embedded support theorems [BQ] submanifold of the set of all lines. In case there is a plane P Boman-Quinto support tangent to W at non-discrete set of points, assume that no theorems—revised line in Y is contained in P . Let X be an open set in R 3 disjoint from W and let µ ( x, l ) be real analytic function on Z that is never zero. Let f ∈ E ′ ( X ) . If R µ f | Y = 0 and some line in Y is disjoint from supp f , then every line in Y is disjoint from supp f . 6 / 42

Type II Introduction Theorem 2. Let Y be an open connected subset of type II Boman-Quinto complex defined by γ . Assume that Y is an embedded support theorems [BQ] submanifold of the set of all lines. If γ is a plane curve, Boman-Quinto support assume that no line in Y is contained in a plane containing γ . theorems—revised Let X be an open set in R 3 disjoint from γ and let µ ( x, l ) be real analytic function on Z that is never zero. Let f ∈ E ′ ( X ) . If R µ f | Y = 0 and some line in Y is disjoint from supp f , then every line in Y is disjoint from supp f . 7 / 42

Type III Introduction Theorem 3. Let Y be an open connected subset of type II Boman-Quinto complex defined by θ . Assume that θ is not a great circle support theorems [BQ] of S 2 . Boman-Quinto support Let X be an open set in R 3 disjoint from γ and let µ ( x, l ) be theorems—revised real analytic function on Z that is never zero. Let f ∈ E ′ ( X ) . If R µ f | Y = 0 and some line in Y is disjoint from supp f , then every line in Y is disjoint from supp f . 8 / 42

Theorem of Hörmander Theorem 4. Let X be an open subset of R n , f ∈ D ′ ( x ) , and Introduction Boman-Quinto x 0 a boundary point of the support of f , and assume that support theorems [BQ] there is a C 2 function F such that F ( x 0 ) = 0 , dF ( x 0 ) � = 0 , and Boman-Quinto support F ( x ) ≤ 0 on supp f . Then ( x 0 , ± dF ( x 0 )) ∈ WF A ( f ) . theorems—revised 9 / 42

Double fibration Introduction Λ = N ∗ ( Z ) Boman-Quinto Z support theorems π X π Y p X p Y [BQ] < > < > Boman-Quinto T ∗ X \ 0 T ∗ Y \ 0 X Y support theorems—revised N ∗ ( Z ) ⊂ T ∗ X \ 0 × T ∗ Y \ 0 ● p X : Z → X has surjective differential ( Y is a regular line ● complex) 10 / 42

Admissible complexes p − 1 Introduction � � cone C x = ∪ p Y ⊂ X X ( x ) ● Boman-Quinto support theorems for non-critical x C x is two-dimensional ● [BQ] l ∈ Y is non-critical , if not all of its points are critical. ● Boman-Quinto support complex of lines is admissible, if ∀ non-critical x ∈ l C x ● theorems—revised has the same tangent plane along l 11 / 42

Proposition Introduction Proposition 5 (cf. [GU]) . Let Y be a regular real analytic Boman-Quinto admissible line complex. Let l 0 ∈ Y and assume f ∈ E ′ ( X ) support theorems [BQ] and R µ f ( l ) = 0 for all l ∈ Y in a neighborhood of l 0 . Let Boman-Quinto support x ∈ l 0 ∩ X and let ξ ∈ T ∗ x ( X ) be conormal to l 0 , but not theorems—revised conormal to the tangent plane to C x along l 0 . Then ( x, ξ ) / ∈ WF A ( f ) . 12 / 42

Proof of the proposition Introduction Let Λ 0 ⊂ Λ be a set of ( x, ξ, l, η ) such that ξ is not ● Boman-Quinto conormal to C x along l . support theorems [BQ] R µ as a Fourier integral operator with Lagrangian ● Boman-Quinto support manifold Λ theorems—revised Λ 0 is a local canonical graph ● R µ is analytic elliptic, when microlocally resticted to Λ 0 ● R µ f = 0 near l 0 ⇒ ( x, ξ ) / ∈ WF A ( f ) for ( x, ξ, l 0 , η ) ∈ Λ 0 ● 13 / 42

Characteristic paths Let x 0 ∈ RP n . Characteristic path with pivot point x 0 is the Introduction ● Boman-Quinto p − 1 � � smooth path in p Y X ( x 0 ) . support theorems [BQ] Boman-Quinto support Proposition 6. Let the hypotheses of theorem Type I theorems—revised (Type II, Type III) hold. Let f ∈ E ′ ( X ) and assume R µ f = 0 on Y . Let l ( s ) : [ a, b ] → Y be a characteristic path and assume l ( a ) does not meet supp f and the pivot point of the path is disjoint from supp f . Then l ( s ) ∩ supp f = ∅ for a ≤ s ≤ b 14 / 42

Proof of proposition 6 Introduction Reduce to the case of pivot point at infinity ● Boman-Quinto Construct a “wedge neighbourhood” of l ( s ) in X : support theorems ● [BQ] Boman-Quinto D ( s, τ ) , D ( s, 0) = l ( s ) , ( τ = ( τ 1 , τ 2 ) , � τ � ≤ ε ) ✦ support D ( a, τ ) ∩ supp f = ∅ theorems—revised ✦ no conormal ¯ ξ to ∂D (¯ s ) at ¯ x is conormal to C ¯ ✦ x along l (¯ s ) ∋ ¯ x Let ¯ s = sup { s 1 ∈ [ a, b ] | D ( s ) ∩ supp f = ∅ for a ≤ s ≤ s 1 } ● s ) , ¯ x ∈ ∂D (¯ ξ ⊥ ∂D (¯ D (¯ s ) meets supp f at some point ¯ s ) ● x, ¯ Proposition 11 implies that (¯ ξ ) / ∈ WF A ( f ) ● Hörmander’s theorem implies that f = 0 near ¯ x ● s = b . So, l ( b ) ∩ supp f = ∅ The only possibility is ¯ ● 15 / 42

Introduction Boman-Quinto support theorems [BQ] Boman-Quinto support theorems—revised Boman-Quinto support theorems—revised ❖ Bibliography 16 / 42

Introduction Boman-Quinto support theorems [BQ] Boman-Quinto support theorems—revised Proposition—revised ❖ Bibliography 17 / 42

Completeness condition Introduction Let Y be a n -dimensional complex of lines ● Boman-Quinto x ( t ) = ξ ( u ) t + β ( u ) be a local parameterization support theorems ● [BQ] y 0 ∈ Y , ω ∈ R n ∗ , ω � = 0 , ω ⊥ y 0 ● Boman-Quinto support theorems—revised Definition 7 (cf. [Pa]) . Line y 0 satisfies a weak completeness ❖ Bibliography condition for ω at x 0 = x ( t 0 ) ∈ y 0 = y ( u 0 ) , if a germ of the map Π ω : Y × R → R × R n , Π ω : ( u, t ) �→ ( � ω, ˙ x � , x ( t )) is a diffeomorphism at ( u 0 , t 0 ) . 18 / 42

ω -critical points Y is an n -dimensional line complex, y ∈ Y , ω ⊥ y Introduction ● Boman-Quinto support theorems [BQ] Definition 8. ● Point x ( t ) ∈ y is ω -critical, if the weak Boman-Quinto support completeness condition is not held at h theorems—revised Line y is ω -critical, if all its point are ω -critical ❖ Bibliography ● The set of conormals ω ⊥ y for which y is ω -critical is ● called the set of critical conormals, and is denoted by Ω y 19 / 42

ω -critical lines Introduction Lemma 9. Let y 0 = y ( u 0 ) ∈ Y , ω ⊥ y 0 . A point x = x ( u 0 , t 0 ) Boman-Quinto is ω -critical ⇐ ⇒ P ω ( t 0 ) = 0 , where polynomial support theorems [BQ] � � ∂ξ ω, � n P ω ( t ) = ∂u k P k ( t ) . Boman-Quinto k =1 support theorems—revised Proof. ❖ Bibliography � � � � � � � � ω, ∂ξ ω, ∂ξ ω, ∂ξ . . . 0 � � ∂u n ∂u 1 ∂u 2 � � � � ∂u 1 t + ∂β 1 ∂ξ 1 ∂u 2 t + ∂β 1 ∂ξ 1 ∂u n t + ∂β 1 ∂ξ 1 ξ 1 � � . . . � � ∂u 1 ∂u 2 ∂u n � � � ∂u 1 t + ∂β 2 ∂ξ 2 ∂u 2 t + ∂β 2 ∂ξ 2 ∂u n t + ∂β 2 ∂ξ 2 � P ω ( t ) = det ξ 2 . . . � � ∂u 1 ∂u 2 ∂u n � � . . . . ... � � . . . . . . . . � � � � � � ∂ξ n ∂u 1 t + ∂β n ∂u 2 t + ∂β n ∂ξ n ∂u n t + ∂β n ∂ξ n ξ n . . . � � ∂u 1 ∂u 2 ∂u n � � 20 / 42

Admissible complexes and critical normals Theorem 10. Let Y be an n -dimensional line complex in R n . Introduction Boman-Quinto The following properties are equivalent: support theorems [BQ] Boman-Quinto 1. Y is admissible support For all non-critical line y ∈ K , for all ω ∈ R n ∗ , ω ⊥ y , 2. theorems—revised ❖ Bibliography either y is ω -critical, or all its ω -critical points are critical. For all non-critical y ∈ Y dim Ω y = n − 2 . 3. 21 / 42