This is a continuation of my question Spaces of functions similar to $L^p$ but with different local and global sizes.
I have been bothered by the fact that the $L^p$ norm on $\mathbb R^n$, which is ubiquitous in real analysis, measures the local and global sizes of a function using the same exponent. As a result, there are functions (eg. $f(x)=|x|^{-a}$ for $a>0$) that are in some $L^p$ when you restrict them to $\{|f|\leq \lambda\}$, and some $L^q$ when you restrict them to $\{|f|>\lambda\}$, but are not in any $L^r$ on all of $\mathbb R^n$ because necessarily $p>q$.
I intuitively feel like the local and global $L^p$ness of a function have nothing to do with each other, and therefore it is natural to measure them with separate exponents. In my previous post, sandwich pointed out that Wiener amalgam spaces do something like this. Nonetheless, these are not very commonly encountered spaces (as seen by the Wikipedia article, which is a stub). This motivates my question: why have usual $L^p$ spaces been so successful? Am I missing something very natural about measuring the local and global parts of a function with the same $p$? Or perhaps these amalgam warrant more attention than they've seen.