This is maybe a trivial question but i need some clarification to make it clearer in my mind. Consider the fundamental solution of the equation $ \partial_{t} u - \partial^{2}_{xx}u=0$ given by the so called green function $H(t,x)=(2 \pi t)^{\frac{-d}{2}} e^{\frac{- \vert x \vert ^{2}}{2t}}$ for $t>0$ and $ x \in R $ , $H(t,x)=0 $ if $t < 0, x \in R $.

The integral: $ \int_{0}^{t} \int_{R} H(t-s,x-y) dsdy$ can be well defined ?

For me H is not L1(R^{2}) so i don't see how this integral can be defined.

Maybe it's really trivial question but it confuses me. Thank you for any element of answer.