We are given some transfer function $G(s)$
And the 2-norm and $\infty$ norms are given by
$$ {||G||}_2 = (\frac{1}{2\pi}\int_{-\infty}^{\infty} |G(j\omega)|^{2} \text{d}\omega)^{1/2} $$
$${||G||}_\infty = \text{sup}|G(j\omega)| $$
I want to see if either or both of these norms are sensitive to some delay.
We can define the new system $$H(s) = G(s) \text{e}^{-s\tau}$$
And compare ${||G||}$ to ${||H||}$
I am trying to do this using the definition of norms that I provided, but I am not getting anywhere.
Is there an easier way to approach this? Perhaps in the time domain?
Note that deriving the $H_2$ or $H_\infty$ norm of a delayed transfer function is not an straightforward task and there is very little chance you could achieve it by using the formal definition you mentioned.
On the other hand, you can find some papers on this matter that gave the problem a try for some special cases. For example, this paper is the most famous one which has derived a relationship for the $H_2$ norm based on the corresponding delayed Lyapunov equation.
For the $H_\infty$ case, there is a paper by R. H. Korogui et al. which more or less might be something you are looking for. There is also this paper on computing the $H_\infty$ norm of delayed systems. But it doesn't mention any transfer function and the analysis is done on the state-space representation with state delays.
Overall, the problem does not have a definitive answer in general and you should consider minimizing its scope.