Lebesgue integration of oscillating and subanalytic functions Tamara - PowerPoint PPT Presentation

Lebesgue integration of oscillating and subanalytic functions Tamara Servi (University of Pisa) (joint work with R. Cluckers, G. Comte, D. Miller, J.-P. Rolin) 19th July 2015

Motivation and background

Motivation and background Oscillatory integrals. λ ∈ R , x = ( x 1 , . . . , x n ) ∈ R n ˆ R n e i λϕ ( x ) ψ ( x ) d x , where: I ( λ ) = • the phase ϕ is analytic, 0 ∈ R n is an isolated singular point of ϕ ; • the amplitude ψ is C ∞ with support a compact nbd of 0.

Motivation and background Oscillatory integrals. λ ∈ R , x = ( x 1 , . . . , x n ) ∈ R n ˆ R n e i λϕ ( x ) ψ ( x ) d x , where: I ( λ ) = • the phase ϕ is analytic, 0 ∈ R n is an isolated singular point of ϕ ; • the amplitude ψ is C ∞ with support a compact nbd of 0. These objects are classically studied in optical physics (Fresnel, Airy,...).

Motivation and background Oscillatory integrals. λ ∈ R , x = ( x 1 , . . . , x n ) ∈ R n ˆ R n e i λϕ ( x ) ψ ( x ) d x , where: I ( λ ) = • the phase ϕ is analytic, 0 ∈ R n is an isolated singular point of ϕ ; • the amplitude ψ is C ∞ with support a compact nbd of 0. These objects are classically studied in optical physics (Fresnel, Airy,...). Aim. To study the behaviour of I ( λ ) when λ → ∞ .

Motivation and background Oscillatory integrals. λ ∈ R , x = ( x 1 , . . . , x n ) ∈ R n ˆ R n e i λϕ ( x ) ψ ( x ) d x , where: I ( λ ) = • the phase ϕ is analytic, 0 ∈ R n is an isolated singular point of ϕ ; • the amplitude ψ is C ∞ with support a compact nbd of 0. These objects are classically studied in optical physics (Fresnel, Airy,...). Aim. To study the behaviour of I ( λ ) when λ → ∞ . j I ( λ ) ∼ e i λϕ ( 0 ) � − n = 1 a j ( ψ ) λ N ( ϕ ) a j ( ψ ) ∈ R , N ( ϕ ) ∈ N fixed. j ∈ N

Motivation and background Oscillatory integrals. λ ∈ R , x = ( x 1 , . . . , x n ) ∈ R n ˆ R n e i λϕ ( x ) ψ ( x ) d x , where: I ( λ ) = • the phase ϕ is analytic, 0 ∈ R n is an isolated singular point of ϕ ; • the amplitude ψ is C ∞ with support a compact nbd of 0. These objects are classically studied in optical physics (Fresnel, Airy,...). Aim. To study the behaviour of I ( λ ) when λ → ∞ . j I ( λ ) ∼ e i λϕ ( 0 ) � − n = 1 a j ( ψ ) λ N ( ϕ ) a j ( ψ ) ∈ R , N ( ϕ ) ∈ N fixed. j ∈ N n > 1 reduce to the case n = 1 by monomializing the phase (res. of sing.).

Motivation and background Oscillatory integrals. λ ∈ R , x = ( x 1 , . . . , x n ) ∈ R n ˆ R n e i λϕ ( x ) ψ ( x ) d x , where: I ( λ ) = • the phase ϕ is analytic, 0 ∈ R n is an isolated singular point of ϕ ; • the amplitude ψ is C ∞ with support a compact nbd of 0. These objects are classically studied in optical physics (Fresnel, Airy,...). Aim. To study the behaviour of I ( λ ) when λ → ∞ . j I ( λ ) ∼ e i λϕ ( 0 ) � − n = 1 a j ( ψ ) λ N ( ϕ ) a j ( ψ ) ∈ R , N ( ϕ ) ∈ N fixed. j ∈ N n > 1 reduce to the case n = 1 by monomializing the phase (res. of sing.). Suitable blow-ups act as changes of variables in R n , outside a set of measure 0.

Motivation and background Oscillatory integrals. λ ∈ R , x = ( x 1 , . . . , x n ) ∈ R n ˆ R n e i λϕ ( x ) ψ ( x ) d x , where: I ( λ ) = • the phase ϕ is analytic, 0 ∈ R n is an isolated singular point of ϕ ; • the amplitude ψ is C ∞ with support a compact nbd of 0. These objects are classically studied in optical physics (Fresnel, Airy,...). Aim. To study the behaviour of I ( λ ) when λ → ∞ . j I ( λ ) ∼ e i λϕ ( 0 ) � − n = 1 a j ( ψ ) λ N ( ϕ ) a j ( ψ ) ∈ R , N ( ϕ ) ∈ N fixed. j ∈ N n > 1 reduce to the case n = 1 by monomializing the phase (res. of sing.). Suitable blow-ups act as changes of variables in R n , outside a set of measure 0. Using Fubini and the case n = 1, one proves: n − 1 a q , k ( ψ ) λ q ( log λ ) k . I ( λ ) ∼ e i λϕ ( 0 ) � � q ∈ Q k = 0

Oscillatory integrals in several variables A more general situation. λ = ( λ 1 , . . . , λ m ) ∈ R m , x = ( x 1 , . . . , x n ) ∈ R n

Oscillatory integrals in several variables A more general situation. λ = ( λ 1 , . . . , λ m ) ∈ R m , x = ( x 1 , . . . , x n ) ∈ R n ˆ R n e i ϕ ( λ, x ) ψ ( x ) d x I ( λ ) = (the parameters λ and the variables x are “intertwined” in the expression for ϕ ).

Oscillatory integrals in several variables A more general situation. λ = ( λ 1 , . . . , λ m ) ∈ R m , x = ( x 1 , . . . , x n ) ∈ R n ˆ R n e i ϕ ( λ, x ) ψ ( x ) d x I ( λ ) = (the parameters λ and the variables x are “intertwined” in the expression for ϕ ). ˆ Example. Fourier transforms ˆ R n e − 2 π i λ · x ψ ( x ) d x . ψ ( λ ) =

Oscillatory integrals in several variables A more general situation. λ = ( λ 1 , . . . , λ m ) ∈ R m , x = ( x 1 , . . . , x n ) ∈ R n ˆ R n e i ϕ ( λ, x ) ψ ( x ) d x I ( λ ) = (the parameters λ and the variables x are “intertwined” in the expression for ϕ ). ˆ Example. Fourier transforms ˆ R n e − 2 π i λ · x ψ ( x ) d x . ψ ( λ ) = Aim. Understand the nature of I ( λ ) (depending on the nature of ϕ and ψ ).

Oscillatory integrals in several variables A more general situation. λ = ( λ 1 , . . . , λ m ) ∈ R m , x = ( x 1 , . . . , x n ) ∈ R n ˆ R n e i ϕ ( λ, x ) ψ ( x ) d x I ( λ ) = (the parameters λ and the variables x are “intertwined” in the expression for ϕ ). ˆ Example. Fourier transforms ˆ R n e − 2 π i λ · x ψ ( x ) d x . ψ ( λ ) = Aim. Understand the nature of I ( λ ) (depending on the nature of ϕ and ψ ). Tool needed. Monomialize the phase while keeping track of the different nature of the variables λ and x .

Oscillatory integrals in several variables A more general situation. λ = ( λ 1 , . . . , λ m ) ∈ R m , x = ( x 1 , . . . , x n ) ∈ R n ˆ R n e i ϕ ( λ, x ) ψ ( x ) d x I ( λ ) = (the parameters λ and the variables x are “intertwined” in the expression for ϕ ). ˆ Example. Fourier transforms ˆ R n e − 2 π i λ · x ψ ( x ) d x . ψ ( λ ) = Aim. Understand the nature of I ( λ ) (depending on the nature of ϕ and ψ ). Tool needed. Monomialize the phase while keeping track of the different nature of the variables λ and x . Natural framework and natural tool:

Oscillatory integrals in several variables A more general situation. λ = ( λ 1 , . . . , λ m ) ∈ R m , x = ( x 1 , . . . , x n ) ∈ R n ˆ R n e i ϕ ( λ, x ) ψ ( x ) d x I ( λ ) = (the parameters λ and the variables x are “intertwined” in the expression for ϕ ). ˆ Example. Fourier transforms ˆ R n e − 2 π i λ · x ψ ( x ) d x . ψ ( λ ) = Aim. Understand the nature of I ( λ ) (depending on the nature of ϕ and ψ ). Tool needed. Monomialize the phase while keeping track of the different nature of the variables λ and x . Natural framework and natural tool: Framework: ϕ, ψ globally subanalytic (i.e. definable in R an ).

Oscillatory integrals in several variables A more general situation. λ = ( λ 1 , . . . , λ m ) ∈ R m , x = ( x 1 , . . . , x n ) ∈ R n ˆ R n e i ϕ ( λ, x ) ψ ( x ) d x I ( λ ) = (the parameters λ and the variables x are “intertwined” in the expression for ϕ ). ˆ Example. Fourier transforms ˆ R n e − 2 π i λ · x ψ ( x ) d x . ψ ( λ ) = Aim. Understand the nature of I ( λ ) (depending on the nature of ϕ and ψ ). Tool needed. Monomialize the phase while keeping track of the different nature of the variables λ and x . Natural framework and natural tool: Framework: ϕ, ψ globally subanalytic (i.e. definable in R an ). Tool: the Lion-Rolin Preparation Theorem.

Oscillatory integrals in several variables A more general situation. λ = ( λ 1 , . . . , λ m ) ∈ R m , x = ( x 1 , . . . , x n ) ∈ R n ˆ R n e i ϕ ( λ, x ) ψ ( x ) d x I ( λ ) = (the parameters λ and the variables x are “intertwined” in the expression for ϕ ). ˆ Example. Fourier transforms ˆ R n e − 2 π i λ · x ψ ( x ) d x . ψ ( λ ) = Aim. Understand the nature of I ( λ ) (depending on the nature of ϕ and ψ ). Tool needed. Monomialize the phase while keeping track of the different nature of the variables λ and x . Natural framework and natural tool: Framework: ϕ, ψ globally subanalytic (i.e. definable in R an ). Tool: the Lion-Rolin Preparation Theorem. Proviso. For the rest of the talk, subanalytic means “globally subanalytic”.

Our framework: parametric integrals and subanalytic functions Def. For X ⊆ R m and f : X × R n → R , define, ∀ x ∈ X s.t. f ( x , · ) ∈ L 1 ( R n ) , ´ the parametric integral I f ( x ) = R n f ( x , y ) d y .

Our framework: parametric integrals and subanalytic functions Def. For X ⊆ R m and f : X × R n → R , define, ∀ x ∈ X s.t. f ( x , · ) ∈ L 1 ( R n ) , ´ the parametric integral I f ( x ) = R n f ( x , y ) d y . Def. For X ⊆ R m subanalytic, let � S ( X ) := { f : X → R subanalytic } and S = S ( X ) X sub.

Our framework: parametric integrals and subanalytic functions Def. For X ⊆ R m and f : X × R n → R , define, ∀ x ∈ X s.t. f ( x , · ) ∈ L 1 ( R n ) , ´ the parametric integral I f ( x ) = R n f ( x , y ) d y . Def. For X ⊆ R m subanalytic, let � S ( X ) := { f : X → R subanalytic } and S = S ( X ) X sub. (Comte - Lion - Rolin). f ∈ S ( X × R n ) ⇒ I f ∈ C ( X ) ,