Let $f \in L^p(\mathbb{R}^n)$ with rational $1 \le p \le2$. I'm primary interested in methods to generalize the Fourier transforamtion from $L^1(\mathbb{R}^n)$ to $L^p(\mathbb{R}^n)$ with $p$ as above without deal with distribution theory at all. By Plancharel-theorem there is a natural way to extend Fourier transformation to $L^2(\mathbb{R}^n)$ by exploit density of $L^1(\mathbb{R}^n) \cap L^2(\mathbb{R}^n)$ in $L^2(\mathbb{R}^n)$ and a limit argument.
For general $1 \le p \le2$ I found here an interesting approach but nowhere literature references where following approach is explaned in detail and where I can find an existence & uniqueness proof of the claim:
For $1 \le p \le2$ the functions in $L^p(\mathbb{R}^n)$ can be decomposed into a fat tail part in $L^2$ plus a fat body part in $L^1$.
Is it done in most naive way that every $f \in L^p(\mathbb{R}^n)$ has decomposition $f=g+h$ with $g \in L^2(\mathbb{R}^n), h \in L^1(\mathbb{R}^n)$? Why it should exist and up to what is such decomposition unique? Can the decompostion be constructed always explicitely of is it just an existence statement?
does anyone know books where these questions are answered?