  The endpoint Strichartz estimate $\|e^{it\Delta} f\|_{L_t^2 L_x^\infty}
\lesssim \|f\|_{L^2}$ is known to be false in two space dimensions. Taking
averages spherically on the polar coordinates $x=\rho\omega$, $\rho>0$,
$\omega\in\mathbb{S}^1$, Tao showed a substitute of the form $\|e^{it\Delta}
f\|_{L_t^2L_\rho^\infty L_\omega^2} \lesssim \|f\|_{L^2}$. Here we address a
weighted version of such spherically averaged estimates. As an application, the
existence of solutions for the inhomogeneous nonlinear Schr\"odinger equation
is shown for $L^2$ data.
