Lets suppose that I have the following system $$ u_t = Au_{xx}+ Bu. $$ If I solve this using Fourier transforms, I will get something like $$ \hat{u}_t = D(\xi) \hat{u}, $$ where $\xi$ is the Fourier variable. So the solution will be $\hat{u}(t,\xi) = e^{tD(\xi)}\hat{u}_0(\xi)$. I want to estimate the solution in $H^s$ with $s>1/2$ and I would like to get a estimate of the form $$||u(t)||_{H^s} \leq ke^{-ct}||u_0||_{H^s}.$$
So I have a question: My teacher told us in class that it is enough to obtain the estimate $$\sup_{\xi \in \mathbb{R}} ||e^{t D(\xi)}|| \leq k e^{-ct}.$$ But I don't see how this estimate (which involves the Fourier variable) will be enough to get the desired estimate (which involves the original variable).
Any hint will be appreciated.