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