  In this article, we provide two-weight inequalities for the heat flow on the
whole space by applying the sparse domination. For power weights, such
inequalities were given by several authors. Owing to the sparse domination, we
can treat general weights in Muckenhoupt classes. As a application, we present
the local and global existence results for the Hardy-H\'enon parabolic
equation, which has a potential belonging to a Muckenhoupt class.
