I'm trying to solve this exercice:
Let $\omega(y)=y^{-4}$ and $L^{1}(\mathbb{R},\omega)$ the space of measurable functions $g:\mathbb{R}\rightarrow\mathbb{R}$ so that $g\omega$ is Lebesgue integrable and the norm is defined as $||g||_{L^{1}(\mathbb{R},\omega)}=\int_\mathbb{R}|g(y)|\omega(y)dy$
Calculate the norm of the operator: $$Tg(x)=\int_\mathbb{R}e^{-xy^{4}}g(y)dy$$
Now, this is the first time I've encountered norms with integral signs and L-spaces with an "associated" function, so it's blurring my thinking...
There are some things I don't understand:
1/ Is: $||Tg||=\int_\mathbb{R}|(\int_\mathbb{R}e^{-yy^{4}}g(y)dy)|y^{-4}dy$ ? I don't think so because I'd have two integrals for y...
2/ Is it then: $||Tg||=\int_\mathbb{R}|e^{-xy^{4}}g(y)|y^{-4}dy$ ? Then $|||T|||\leq 1$ ,right?
These questions on the steps I'm following aside, my goal is to get it done so any other approach is more than welcome if clear enough.
Note: This is from an introduction exercice to Fourier series so if your answer uses any of that, please make small steps in your reasoning.