Consequences of the Funk–Hecke theorem for smoothing and trace estimates
Neal Bez
Saitama University, Saitama, Japan
email: [email protected]
Hiroki Saito
Kogakuin University, Tokyo, Japan
email: [email protected]
Mitsuru Sugimoto
Nagoya University, Nagoya, Japan
email: [email protected]
Abstract We use the Funk–Hecke theorem to establish a new expression for the optimal constant
of a broad class of Kato-smoothing estimates. This expression is in terms of the Fourier transform
of the weight; when this is a positive function we are able to obtain a number of rather strong
conclusions. For example, assuming power-like asymptotics for the Fourier transform of the
weight and the smoothing function, we may quickly deduce a full characterisation of which
exponents permit a smoothing estimate to hold. In a different direction, we substantially extend
\
a result of Simon [3] to prove that whenever w ∈ L1 (0, ∞) with w(|
· |) > 0, and d ≥ 3, the
estimate
Z Z
|D1/2 eit∆ f (x)|2 w(|x|) dxdt ≤ kwkL1 (0,∞) kf k2L2 (Rd )
R
Rd
holds, where the constant is optimal and, moreover, we can show that there are no extremisers.
This result has very close connections with the Mizohata–Takeuchi conjecture which is concerned
with weighted L2 bounds for the Fourier extension operator on spheres, and the L∞ norm of the
X-ray transform of the weight. In the case of radial weights, this was verified independently by
Barceló–Ruiz–Vega [1] and Carbery–Soria [2]. Using our approach, we will give a concise and
alternative proof for d ≥ 3 when the Fourier transform of the weight is positive. The approach
based on the Funk–Hecke theorem applies equally well to trace theorems on the sphere, and this
will also be discussed during the talk.
Bibliography
[1] Barceló, J. A., Ruiz, A. and Vega, L., Weighted estimates for the Helmholtz equation and some applications,
J. Funct. Anal., 150, 356–382 (1997).
[2] Carbery, A. and Soria, F., Pointwise Fourier inversion and localisation in Rn , J. Fourier Anal. Appl., 3,
847–858 (special issue) (1997).
[3] Simon, B. Best constants in some operator smoothness estimates, J. Funct. Anal., 107, 66–71 (1992).