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 , {equation*} \|u\|L_t^qL_x^r \|u\|_{Xs,b}, {equation*} where ‖ u ‖ X s , b ≔ ‖ u ^ ( τ , ξ ) ⟨ ξ ⟩ s ⟨ τ + | ξ | 2 ⟩ b ‖ L τ , ξ 2 \|u\|_{Xs,b}≔\|û(τ ,ξ ) ξ ^s τ +|ξ |^2 ^b \|_{Lτ ,ξ^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.