Problem 1. a) Prove the semi-group property of the heat kernel:

T(t+s,-) = [(t, .) *I(s,-), t,s> 0.

b) Show that

||D₂D₂F(t,·)||L¹(Rn) ≤ Ct−³/2,

where C=C(n) > 0 is a constant.

(Hint: You may want to use the result of Part a) to prove Part b).)