I have seen how to extend the Fourier transform as an operator from $L^p(\mathbb{R})$ to $L^p(\mathbb{R})$ with $1 \leq p < \infty$ using Schwartz's class functions and tempered distributions. So I defined the Fourier transform of a tempered distribution $T$ in $\mathcal{S}'$ as the tempered distribution $\widehat{T}$ such that $\widehat{T}(f)=T(\widehat{f})$ for every $f \in \mathcal{S}$. This way, using the density of Schwartz's functions in $L^p$ spaces, we can extend the Fourier transform. Now, since Schwartz's function shouldn't be dense in $L^\infty$, what do I do?
The only thing coming to my mind would be, given $f \in L^\infty$, introducing a dumping function $e^{-\epsilon |x|}$. This way the Fourier transform of $e^{-\epsilon |x|} f$ should be well defined and there should be some kind of convergence.
This question arises because I'd need to use Young's inequality combined with Fourier transform, and it would be really nice to have something like $||\widehat{f}||_\infty \lesssim ||f||_\infty$