Assume that a sequence of random variables, $(X_t)_{t\geq 0}$, converges in distribution to a random variable $X_0$, as $t\to 0$. Also assume that $X_t$ and $X_0$ have $C^{\infty}$-probability density functions, $f_{X_t}$ and $f$, respectively.
In general this does not imply that $f_{X_t}(x)\to f(x)$, as $t\to 0$, but does this allow me to uniformly bound the probability density functions, i.e. find a constant $K$ and a $t_0>0$ such that $\sup_x f_{X_t}(x)<K$ for all $t<t_0$?