Integrability green function

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.

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• the integral over all $x$ is a Gaussian integral which gives unity. – Carlo Beenakker Dec 6 at 23:06
• so we can consider the L1 norm on R^{2} of H ? – nour Dec 7 at 1:33
• you are looking for a solution to the diffusion equation, which is given by the Green function; this solution is integrable, and by particle number conservation is equal to its value at $t=0$, hence it integrates to unity. – Carlo Beenakker Dec 7 at 9:50