Heisenberg Uniqueness Pairs for the Parabola
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Heisenberg Uniqueness Pairs for the Parabola Per Sjölin
Received: 31 January 2012 / Revised: 25 September 2012 / Published online: 25 January 2013 © Springer Science+Business Media New York 2013
Abstract Let Γ denote the parabola y = x 2 in the plane. For some simple sets Λ in the plane we study the question whether (Γ, Λ) is a Heisenberg uniqueness pair. For example we shall consider the cases where Λ is a straight line or a union of two straight lines. Keywords Fourier transforms · Heisenberg uniqueness pairs Mathematics Subject Classification 42B10 1 Introduction Let μ denote a finite complex-valued Borel measure in R2 . The Fourier transform of μ is defined by μ(x, ˆ y) = e−i(xξ +yη) dμ(ξ, η) for (x, y) ∈ R2 . R2
Let Γ denote the parabola y = x 2 in R2 . We assume that supp μ ⊂ Γ and that μ is absolutely continuous with respect to the arc length measure on Γ . Also let Λ be a subset of R2 . Following Hedenmalm and Montes-Rodríguez [3] we say that (Γ, Λ) is a Heisenberg uniqueness pair (or only uniqueness pair) if μ(x, ˆ y) = 0 for (x, y) ∈ Λ implies that μ is the zero measure. The case where Γ is a hyperbola was discussed in [3], and Sjölin [5] and Lev [4] studied the case where Γ is a circle. For further results see also Canto-Martin, Hedenmalm, and Montes-Rodríguez [1]. Communicated by Hans G. Feichtinger. P. Sjölin () Department of Mathematics, Royal Institute of Technology, 100 44 Stockholm, Sweden e-mail: [email protected]
J Fourier Anal Appl (2013) 19:410–416
411
We shall here let Γ denote the parabola y = x 2 . If μ has the above properties it is clear that there exists a measurable function f on R such that f (t) 1 + 4t 2 dt < ∞
and R2 hdμ = Thus we have
R
R h(t, t
√ 1 + 4t 2 dt if h is continuous and bounded in R2 .
2 )f (t)
μ(x, ˆ y) =
R
2 e−i(xt+yt ) f (t) 1 + 4t 2 dt,
(x, y) ∈ R2 .
We shall prove the following theorem, where Γ denotes the parabola y = x 2 . Theorem 1 (i) Let Λ = L where L is a straight line. Then (Γ, Λ) is a uniqueness pair if and only if L is parallel to the x-axis. (ii) Let Λ = L1 ∪ L2 , where L1 and L2 are different straight lines. Then (Γ, Λ) is a uniqueness pair. (iii) Assume that L1 and L2 are different straight lines, which are not parallel to the x-axis. Also assume that E1 ⊂ L1 , and E2 ⊂ L2 and that E1 and E2 have positive one-dimensional Lebesgue measure. Set Λ = E1 ∪ E2 . Then (Γ, Λ) is a uniqueness pair. Remark When we talk about the one-dimensional Lebesgue measure of a subset E of a straight line L in the plane, we identify L with R. We also remark that Heisenberg uniqueness pairs are somewhat related to the notion of annihilating pairs (see Havin and Jöricke [2]). To give the definition of this concept we let S and Σ be subsets of R. Following [2] we say that the pair (S, Σ) is mutually annihilating if ψ ∈ L2 (R), supp ψ ⊂ S, supp ψˆ ⊂ Σ implies that ψ = 0. Here ψˆ denotes the Fourier transform of ψ . We refer to [2] for results on annihilating pairs.
2 Lemmas and Proofs We let the function f be defined as in the Introduction and s
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