Skip to main content
1 of 3
Tao
  • 429
  • 2
  • 9

How to use Stein-Tomas theorem to check to following inequality?

Recently, I am reading Rodnianski & Schlag Time decay for solutions of Schrödinger equations with rough and time-dependent potentials. In lemma 3.2, R&S said that by using Stein-Tomas theorem in Stein's version one has \begin{equation}\lVert \int_{\mathbb{R}^3} \frac{exp(i|x-y|)}{4 \pi |x-y|} f(y) dy\rVert_{L^4(\mathbb{R}^3)} \le C \lVert f \rVert_{L^{\frac{4}{3}}(\mathbb{R}^3)}. \end{equation} I have read some references about Stein-Tomas theorem (Fourier Restriction estimate), but I really don't know how to apply this theorem to the proof of the inequality? Can some one give me some tips?

Tao
  • 429
  • 2
  • 9