  We consider a process given as the solution of a one-dimensional stochastic
differential equation with irregular, path dependent and time-inhomogeneous
drift coefficient and additive noise. H\"older continuity of the Lebesgue
density of that process at any given time is achieved using a different
approach than the classical ones in the literature. Namely, the H\"older
regularity of the densities is obtained via a control problem by identifying
the stochastic differential equation with the worst global H\"older constant.
Then we generalise our findings to a larger class of diffusion coefficients.
The novelty of this method is that it is not based on a variational calculus
and it is suitable for non-Markovian processes.
