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We prove some new Strichartz estimates for a class of dispersive equationswith radial initial data. In particular, we obtain up to some endpoints thefull radial Strichartz estimates for the Schr\"odinger equation. The ideas ofproof are based on Shao's ideas \cite{Shao} and some ideas in \cite{GPW} totreat the non-homogeneous case, while at the endpoint we need to use subtletools to overcome some logarithmic divergence. We also apply the improvedStrichartz estimates to the nonlinear problems. First, we prove the small datascattering and large data LWP for the nonlinear Schr\"odinger equation withradial critical $\dot{H}^s$ initial data below $L^2$; Second, for radial datawe improve the results of the $\dot{H}^s\times \dot{H}^{s-1}$ well-posednessfor the nonlinear wave equation in \cite{SmithSogge}; Finally, we obtain thewell-posedness theory for the fractional order Schr\"odinger equation in theradial case.

Improved Strichartz estimates for a class of dispersive equations in the  radial case and their applications to nonlinear Schr\"odinger and wave  equation