arXiv:math/0607059v2 [math.CA] 13 Jul 2006 AVERAGE DECAY ESTIMATES FOR FOURIER TRANSFORMS OF MEASURES SUPPORTED ON CURVES LUCA BRANDOLINI GIACOMO GIGANTE ALLAN GREENLEAF ALEXANDER IOSEVICH ANDREAS SEEGER GIANCARLO TRAVAGLINI Abstract. We consider Fourier transforms  µ of densities supported on curves in R d . We obtain sharp lower and close to sharp upper bounds for the decay rates of ‖ µ(R·)‖ L q (S d−1 ) , as R →∞. 1. Introduction and Statement of Results In this paper we investigate the relation between the geometry of a curve Γ in R d , d> 2, and the spherical L q average decay of the Fourier transform of a smooth density µ compactly supported on Γ. Let Γ be a smooth (C ∞ ) immersed curve in R d with parametrization t → γ (t) defined on a compact interval I and let χ ∈ C ∞ be supported in the interior of I . Let µ ≡ µ γ,χ be defined by (1.1) 〈µ, f 〉 =  f (γ (t))χ(t)dt and define by  µ(ξ )=  exp(−i〈ξ,γ (t)〉)χ(t)dt its Fourier transform. For a large parameter R we are interested in the behavior of  µ(Rω) as a function on the unit sphere, in particular in the L q norms (1.2) G q (R) ≡ G q (R; γ,χ) :=   |  µ(Rω)| q dω  1/q where dω is the rotation invariant measure on S d−1 induced by Lebesgue measure in R d . The rate of decay depends on the number of linearly inde- pendent derivatives of the parametrization of Γ. Indeed if one assumes that for every t the derivatives γ ′ (t), γ ′′ (t), ..., γ (d) (t) are linearly independent then from the standard van der Corput’s lemma (see [20, page 334]) one gets G ∞ (R) = max ω |  µ(Rω)| = O(R −1/d ). If one merely assumes that at most d − 1 derivatives are linearly independent then one cannot in general expect a decay of G ∞ (R); one simply considers curves which lie in a hyperplane. However Marshall [15] showed that one gets an optimal estimate for the L 2 average decay, namely (1.3) G 2 (R)= O(R −1/2 ) Date : Revised version, July 13, 2006. Research supported in part by NSF grants. 1