Strichartz type estimates for solutions to the Schrödinger equation

In this article, we show the necessary and sufficient conditions for the inequality ‖ u ‖ L t q L x r ≲ ‖ u ‖ X s , b , \begin{equation*} \|u\|_{L_t^qL_x^r}\lesssim \|u\|_{X^{s,b}}, \end{equation*} where ‖ u ‖ X s , b ≔ ‖ u ^ ( τ , ξ ) ⟨ ξ ⟩ s ⟨ τ + | ξ | 2 ⟩ b ‖ L τ , ξ 2 \|u\|_{X^{s,b}}≔\|\hat {u}(\tau ,\xi )\langle \xi \rangle ^s\langle \tau +|\xi |^2\rangle ^b \|_{L_{\tau ,\xi }^2} . These estimates are also referred to as Strichartz estimates related to Schrödinger equation. We also give a new proof of the maximal function estimates for solutions to Schrödinger and Airy equations.