Applications of the Funk-Hecke theorem to smoothing and trace estimates

For a wide class of Kato-smoothing estimates with radial weights, the Funk-Hecke theorem is used to generate a new expression for the optimal constant in terms of the Fourier transform of the weight, from which several applications are given. For example, we are able to easily establish a unified theorem, assuming natural power-like asymptotic estimates for the Fourier transform of the weight, from which many well-studied smoothing estimates immediately follow, as well as sharpness of the decay and smoothness exponents. Furthermore, observing that the weight has an everywhere positive Fourier transform in many well-studied cases, our approach allows sharper information regarding the optimal constant and extremisers, substantially extending earlier work of Simon. These observations are very closely related to the Mizohata-Takeuchi conjecture regarding the equivalence of weighted L-2 bounds for the Fourier extension operator on the sphere and the uniform boundedness of the X-ray transform of the weight. For radial weights, this has been independently established by Barcelo-Ruiz-Vega and Carbery-Soria; we provide a short alternative proof in three and higher dimensions of this equivalence when the Fourier transform of the weight is positive, with the optimal relationship between constants. Finally, our approach works for the closely connected trace theorems on the sphere where analogous results are given, including the optimal constant and characterisation of extremisers for the inhomogeneous H-s (R-d) -> L-2 (Sd-1) trace theorem. (C) 2015 Elsevier Inc. All rights reserved.