I was reading the so-called subordination identity that allows one to derive estimates on the Poisson operator $e^{-t\sqrt{-\Delta}}$ from estimates on the heat diffusion operator $e^{t\Delta}$. I am a bit curious as to how (if at all) one can go the other way, that is, from some estimate on the Poisson operator (which can sometimes be derived through independent means, for example, through properties of the corresponding elliptic equation) one can derive estimates on the operator $e^{t\Delta}$. I realize that my question is a bit vague, but it so happens that I am just curious to learn more on these stuff, I am not exactly stuck on any particular problem. Thanks a lot in advance!
