We prove sharp estimates, with respect to the affine arclength measure, for the restriction of the Fourier transform to a class of curves in R d that includes curves of finite type. This measure possesses certain invariance and mitigation properties which are important in establishing uniform results.
We establish near-optimal mixed-norm estimates for the X-ray transform restricted to polynomial curves with a weight that is a power of the affine arclength. The bounds that we establish depend only on the spatial dimension and the degree of the polynomial. Some of our results are new even in the well-curved case.
Abstract. We establish endpoint Lebesgue space bounds for convolution and restricted X-ray transforms along curves satisfying fairly minimal differentiability hypotheses, with affine and Euclidean arclengths. We also explore the behavior of certain natural interpolants and extrapolants of the affine and Euclidean versions of these operators.
We announce a Fourier restriction result for general polynomial curves in R d . Measuring the Fourier restriction with respect to the affine arclength measure of the curve, we obtain a universal bound for the class of all polynomial curves of bounded degree. Our method relies on establishing a geometric inequality for general polynomial curves which is of interest in its own right. There are applications of this geometric inequality to other problems in euclidean harmonic analysis. To cite this article: S. Dendrinos, J. Wright, C. R. Acad. Sci. Paris, Ser. I 346 (2008).
RésuméRestrictions de Fourier et courbes polynomiales ; une inégalité géométrique. Le résultat que nous annonçons sur les restrictions de Fourier vaut pour des courbes polynomiales générales dans R d . Il permet de contrôler la norme L q de la transformée de Fourier relativement à la mesure d'arc affine (dont nous rappelons la définition) à la norme L p de la fonction, pour des p et q convenables. La borne est universelle pour toutes les courbes polynomiales de degré donné. Notre méthode repose sur une inégalité géométrique concernant les courbes polynomiales qui est intéressante en elle même, et s'applique à d'autres problèmes d'analyse harmonique euclidienne. Pour citer cet article : S. Dendrinos, J. Wright, C. R. Acad. Sci. Paris, Ser. I 346 (2008).
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