A weak 2-weight problem for the Poisson-Hermite semigroup

This survey is a slightly extended version of the lecture given by the author at the \emph{VI International Course of Mathematical Analysis in Andaluc\'\i a} (CIDAMA), in September 2014. Most results are contained (in a slightly less general setting) in the earlier paper [3] (Garrig\'os, Hartzstein, Signes, Torrea and Viviani, Pointwise convergence to initial data of heat and Laplace equations, Trans. Amer. Math. Soc. 368 (9) (2016), 6575-6600).