Heisenberg Uniqueness pairs

Transcript

1 Heisenberg Uniqueness pairs Philippe Jaming Bordeaux Fourier Workshop 2013, Renyi Institute Joint work with K. Kellay

2 Heisenberg Uniqueness Pairs µ : finite measure on R 2 µ(x, y) = R 2 e i(sx+ty) dµ(s, t). Γ : finite union of disjoint curves M(Γ) : measures supported on Γ AC(Γ) : µ M(Γ) absolutely continuous w.r.t arc length. Λ R 2 : set of lines Definition (Γ, Λ) Heisenberg Uniqueness Pair (HUP) if µ AC(Γ), µ = 0 on Λ µ = 0.

3 Heisenberg Uniqueness Pairs µ : finite measure on R 2 µ(x, y) = R 2 e i(sx+ty) dµ(s, t). Γ : finite union of disjoint curves M(Γ) : measures supported on Γ AC(Γ) : µ M(Γ) absolutely continuous w.r.t arc length. Λ R 2 : set of lines Definition (Γ, Λ) Heisenberg Uniqueness Pair (HUP) if µ AC(Γ), µ = 0 on Λ µ = 0.

4 Heisenberg Uniqueness Pairs µ : finite measure on R 2 µ(x, y) = R 2 e i(sx+ty) dµ(s, t). Γ : finite union of disjoint curves M(Γ) : measures supported on Γ AC(Γ) : µ M(Γ) absolutely continuous w.r.t arc length. Λ R 2 : set of lines Definition (Γ, Λ) Heisenberg Uniqueness Pair (HUP) if µ AC(Γ), µ = 0 on Λ µ = 0.

5 Heisenberg Uniqueness Pairs µ : finite measure on R 2 µ(x, y) = R 2 e i(sx+ty) dµ(s, t). Γ : finite union of disjoint curves M(Γ) : measures supported on Γ AC(Γ) : µ M(Γ) absolutely continuous w.r.t arc length. Λ R 2 : set of lines Definition (Γ, Λ) Heisenberg Uniqueness Pair (HUP) if µ AC(Γ), µ = 0 on Λ µ = 0.

6 Heisenberg Uniqueness Pairs µ : finite measure on R 2 µ(x, y) = R 2 e i(sx+ty) dµ(s, t). Γ : finite union of disjoint curves M(Γ) : measures supported on Γ AC(Γ) : µ M(Γ) absolutely continuous w.r.t arc length. Λ R 2 : set of lines Definition (Γ, Λ) Heisenberg Uniqueness Pair (HUP) if µ AC(Γ), µ = 0 on Λ µ = 0.

7 Heisenberg Uniqueness Pairs µ : finite measure on R 2 µ(x, y) = R 2 e i(sx+ty) dµ(s, t). Γ : finite union of disjoint curves M(Γ) : measures supported on Γ AC(Γ) : µ M(Γ) absolutely continuous w.r.t arc length. Λ R 2 : set of lines Definition (Γ, Λ) Heisenberg Uniqueness Pair (HUP) if µ AC(Γ), µ = 0 on Λ µ = 0.

8 Heisenberg Uniqueness Pairs µ : finite measure on R 2 µ(x, y) = R 2 e i(sx+ty) dµ(s, t). Γ : finite union of disjoint curves M(Γ) : measures supported on Γ AC(Γ) : µ M(Γ) absolutely continuous w.r.t arc length. Λ R 2 : set of lines Definition (Γ, Λ) Heisenberg Uniqueness Pair (HUP) if µ AC(Γ), µ = 0 on Λ µ = 0.

9 References H. HEDENMALM & A. MONTES-RODRÍGUEZ Heisenberg uniqueness pairs and the Klein-Gordon equation. Ann. of Math. (2), 173 (2011), N. LEV Uniqueness theorems for Fourier transforms. Bull. Sci. Math., 135 (2011), P. SJÖLIN Heisenberg uniqueness pairs and a theorem of Beurling and Malliavin. Bull. Sci. Math., 135 (2011), P. SJÖLIN Heisenberg uniqueness pairs for the parabolla. Jour. Four. Anal. Appl., to appear.

10 References H. HEDENMALM & A. MONTES-RODRÍGUEZ Heisenberg uniqueness pairs and the Klein-Gordon equation. Ann. of Math. (2), 173 (2011), N. LEV Uniqueness theorems for Fourier transforms. Bull. Sci. Math., 135 (2011), P. SJÖLIN Heisenberg uniqueness pairs and a theorem of Beurling and Malliavin. Bull. Sci. Math., 135 (2011), P. SJÖLIN Heisenberg uniqueness pairs for the parabolla. Jour. Four. Anal. Appl., to appear.

11 References H. HEDENMALM & A. MONTES-RODRÍGUEZ Heisenberg uniqueness pairs and the Klein-Gordon equation. Ann. of Math. (2), 173 (2011), N. LEV Uniqueness theorems for Fourier transforms. Bull. Sci. Math., 135 (2011), P. SJÖLIN Heisenberg uniqueness pairs and a theorem of Beurling and Malliavin. Bull. Sci. Math., 135 (2011), P. SJÖLIN Heisenberg uniqueness pairs for the parabolla. Jour. Four. Anal. Appl., to appear.

12 Aim Aim Get rid of explicit computation get a hint on the geometric nature of the problem Unify proofs.

13 Aim Aim Get rid of explicit computation get a hint on the geometric nature of the problem Unify proofs.

14 Aim Aim Get rid of explicit computation get a hint on the geometric nature of the problem Unify proofs.

15 HUP 2 Inv 1 Fix (s 0, t 0 ), (x 0, y 0 ) R 2. Then ( Γ, Λ ) is a HUP if and only if ( Γ (s0, t 0 ), Λ (x 0, y 0 ) ) is a HUP. Inv 2 Fix T a linear invertible transformation R 2 R 2 and denote by T its adjoint. Then ( Γ, Λ ) is a HUP if and only if ( T 1 (Γ), T (Λ) ) is a HUP. If Γ = {s, γ(s), s I} (Γ, λ) HUP f L 1 (I) s.t. (x, y) Λ f (s)e i(sx+γ(s)y) ds = 0 f = 0 I

16 HUP 2 Inv 1 Fix (s 0, t 0 ), (x 0, y 0 ) R 2. Then ( Γ, Λ ) is a HUP if and only if ( Γ (s0, t 0 ), Λ (x 0, y 0 ) ) is a HUP. Inv 2 Fix T a linear invertible transformation R 2 R 2 and denote by T its adjoint. Then ( Γ, Λ ) is a HUP if and only if ( T 1 (Γ), T (Λ) ) is a HUP. If Γ = {s, γ(s), s I} (Γ, λ) HUP f L 1 (I) s.t. (x, y) Λ f (s)e i(sx+γ(s)y) ds = 0 f = 0 I

17 HUP 2 Inv 1 Fix (s 0, t 0 ), (x 0, y 0 ) R 2. Then ( Γ, Λ ) is a HUP if and only if ( Γ (s0, t 0 ), Λ (x 0, y 0 ) ) is a HUP. Inv 2 Fix T a linear invertible transformation R 2 R 2 and denote by T its adjoint. Then ( Γ, Λ ) is a HUP if and only if ( T 1 (Γ), T (Λ) ) is a HUP. If Γ = {s, γ(s), s I} (Γ, λ) HUP f L 1 (I) s.t. (x, y) Λ f (s)e i(sx+γ(s)y) ds = 0 f = 0 I

18 Basic lemma γ(s)

19 Basic lemma γ(s) t 3 t 1 t 4 t 2 s 1 s 2 s 3 s 4 s

20 Basic lemma γ(s) t 3 t 1 t t 4 t 2 γ 1 0 s 1 s 2 s 3 s 4 (t) γ 1 1 (t) γ 1 2 (t) γ 1 3 (t) γ 1 s 4 (t)

21 Basic lemma R f (s)e iγ(s)x ds =

22 Basic lemma R f (s)e iγ(s)x ds = m sk+1 k=0 s k f (s)e iγ(s)x ds

23 Basic lemma R f (s)e iγ(s)x ds = = m sk+1 k=0 s k tk+1 m k=0 t k f (s)e iγ(s)x ds ( f γ 1 k (t) ) γ ( γ 1 k (t) )eitx dt

24 Basic lemma R f (s)e iγ(s)x ds = = = m sk+1 k=0 s k tk+1 m k=0 t k max tk min t k m k=0 f (s)e iγ(s)x ds ( f γ 1 k (t) ) γ ( γ 1 k (t) )eitx dt 1 [tk,t k+1 ](t) f ( γ 1 k (t) ) γ ( γ 1 k (t) )eitx dt.

25 Basic lemma R f (s)e iγ(s)x ds = = = m sk+1 k=0 s k tk+1 m k=0 t k max tk min t k m k=0 f (s)e iγ(s)x ds ( f γ 1 k (t) ) γ ( γ 1 k (t) )eitx dt 1 [tk,t k+1 ](t) f ( γ 1 k (t) ) γ ( γ 1 k (t) )eitx dt. Thus (Γ, {x = 0}) HUP

26 Basic lemma R f (s)e iγ(s)x ds = = = Thus (Γ, {x = 0}) HUP m sk+1 k=0 s k tk+1 m k=0 t k max tk min t k m k=0 m k=0 f (s)e iγ(s)x ds ( f γ 1 k (t) ) γ ( γ 1 k (t) )eitx dt 1 [tk,t k+1 ](t) f ( γ 1 k (t) ) γ ( γ 1 k (t) )eitx dt. 1 [tk,t k+1 ](t) f ( γ 1 k (t) ) γ ( γ 1 k (t) ) = 0.

27 Example 1 γ is one-to-one, then (Γ, {x = 0}) HUP

28 Example 2 γ has one local extrema.

29 Example 2 γ has one local extrema. t γ 1 (t) γ 1 + (t)

30 Example 2 γ has one local extrema. t γ 1 (t) γ 1 + (t) f ( γ+ 1 (t)) γ ( γ+ 1 (t)) = f ( γ 1 (t)) γ ( γ 1 (t)).

31 Lemma f ( γ+ 1 (t)) γ ( γ+ 1 (t)) = f ( γ 1 (t)) γ ( γ 1 (t)). ( γ(α+ ) ), β = γ 1 ( γ(β+ ) ), and for α +, β + > s 1 and α = γ 1 we have β β+ f (s) ds = f (s) ds α + α

32 Theorem : two lines γ smooth, a > b, ψ(t) = γ(t) + at and χ(t) = γ(t) + bt ψ (resp. χ) have a unique local minimum in s 1 (resp s 2 ) ψ(t), χ(t) + when t ±. f L 1 (R) s.t. f (s)e iψ(s)x ds = f (s)e iχ(s)x ds = 0 then f = 0. R R

33 Theorem : two lines γ smooth, a > b, ψ(t) = γ(t) + at and χ(t) = γ(t) + bt ψ (resp. χ) have a unique local minimum in s 1 (resp s 2 ) ψ(t), χ(t) + when t ±. f L 1 (R) s.t. f (s)e iψ(s)x ds = f (s)e iχ(s)x ds = 0 then f = 0. R R

34 Theorem : two lines γ smooth, a > b, ψ(t) = γ(t) + at and χ(t) = γ(t) + bt ψ (resp. χ) have a unique local minimum in s 1 (resp s 2 ) ψ(t), χ(t) + when t ±. f L 1 (R) s.t. f (s)e iψ(s)x ds = f (s)e iχ(s)x ds = 0 then f = 0. R R

35 Theorem : two lines γ smooth, a > b, ψ(t) = γ(t) + at and χ(t) = γ(t) + bt ψ (resp. χ) have a unique local minimum in s 1 (resp s 2 ) ψ(t), χ(t) + when t ±. f L 1 (R) s.t. f (s)e iψ(s)x ds = f (s)e iχ(s)x ds = 0 then f = 0. R R

36 Theorem : two lines γ smooth, a > b, ψ(t) = γ(t) + at and χ(t) = γ(t) + bt ψ (resp. χ) have a unique local minimum in s 1 (resp s 2 ) ψ(t), χ(t) + when t ±. f L 1 (R) s.t. f (s)e iψ(s)x ds = f (s)e iχ(s)x ds = 0 then f = 0. R R

37 Proof 1

38 Proof 1 σ 0 σ 1

39 Proof 1 ψ 1 ( ψ(σ1 ) ) ψ(σ 1 ) σ 0 σ 1

40 Proof 1 ( χ ψ 1 ( ψ(σ1 ) )) ψ(σ 1 ) ψ 1 ( ψ(σ1 ) ) σ 0 σ 1 σ 2 σ k +.

41 Proof 2 If f 0, σ k+1 σ k f (s) ds 0 σk σ k+1 f (s) ds = f (s) ds = k 0 σk+1 σ k σk+1 σ k f (s) ds. f (s) ds = +.

42 Proof 2 If f 0, σ k+1 σ k f (s) ds 0 σk σ k+1 f (s) ds = f (s) ds = k 0 σk+1 σ k σk+1 σ k f (s) ds. f (s) ds = +.

43 Proof 2 If f 0, σ k+1 σ k f (s) ds 0 σk σ k+1 f (s) ds = f (s) ds = k 0 σk+1 σ k σk+1 σ k f (s) ds. f (s) ds = +.

44 A better picture t = γ(s) σ 2 σ 0 σ 1 t = as

45 Hyperbola, case 1 t = γ(s) σ 2 σ 1 σ 0 σ 1 σ 0 t = as σ 2 t = γ(s)

46 Hyperbola, case 2 t = γ(s) σ 1 σ 0 σ 1

47 Hyperbola, case 3 t = γ(s) J 2 I 2 I 2 I 1 J 1 J 1 J 2 I 1

48 Closed curve

49 Closed curve

50 Closed curve

51 Closed curve

52 Closed curve

53 Closed curve

54 One local max, one local min

55 One local max, one local min

56 One local max, one local min

57 One local max, one local min

58 That s all! Thank you for your attention!