We discuss a new method for obtaining a local limit theorem (LLT) from a known distributional limit theorem. The method rests on a simple analytic inequality (essentially due to Hardy, Landau, and Littlewood) which can be applied directly after quantifying the smoothness of the distribution of interest. These smoothness terms are non-trivial to handle and so we also provide new (probabilistic) tools for this purpose. We illustrate our approach by showing LLTs with rates for the magnetization in the Curie-Weiss model at high temperature and for some counts in an Erdos-Renyi random graph. This is joint work with Adrian Roellin.