In this paper on Page 4, in the last line of the proof, the author asserts that if a radial function $u:\mathbb{R_t}\times \mathbb{R}^2\to \mathbb{R}$, smooth outside the origin $(0_t,0_x)$, admits for some $T,R>0$ a uniform bound on $||\nabla^2 u||_{L^2B(t+R/2)}$ for $0<t<T$ then the radial function can be smoothly extended to a neighborhood of $(0_t,0_x)$.
The author claims this follows from standard results, but I'm having a hard time figuring out which standard results he's referring to.