Is it natural how $L^p$ spaces measure local and global sizes the same? 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.
 A: A bunch of things:


*

*The perspective that $L^p$ spaces have been very successful is probably skewed heavily by your, excuse me for being blunt, limited exposure to research literature. The reason that $L^p$ spaces appear frequently in textbooks is because that they are simple to define and thus serve a great purpose pedagogically. 

*That said, $L^p$ spaces (and $L^p$ based Sobolev spaces) remain in use a lot in the literature because their properties (embeddings, interpolations, behaviour under Fourier transform, duality, to name a few basic ones) are relatively well understood. If you can solve a problem using $L^p$ spaces then why not? 

*More over, concrete Banach spaces occur mostly in either analysis concerning differential equations, or occasionally in the construction of explicit counter-examples to various conjectures in Banach space theory. That is to say, concrete spaces like Lebesgue, Sobolev, Holder spaces are most frequently used as tools in studying differential equations. And tools are selected by their utility. 
In view of this, it shouldn't be surprising the "global" size is not too important for a lot of its applications. After all, a lot of PDE theory (especially in the elliptic and parabolic realms) have been developed on compact domains. For these questions the global size is pretty much irrelevant. (Recall that on a bounded domain $L^p$ injects into $L^q$ when $q \leq p$.) 
For hyperbolic equations, on the other hand, once can spatially localize solutions by the finite-speed of propagation and always work with functions of compact support. Again, in this situation the global size is of no consequence. In this realm, only when studying the global (in time) behavior of the solutions do global size come in. And in this case people are fully aware of the difference between local and global sizes. 
To give one example, consider the wave equation on spatial dimensions bigger than 1. The conservation of energy is stated in the $L^2$-based energy spaces. But physical experience also tells us that waves tend to spread-out and decay over time (think about a radio wave; the spherical wave front grows as time progresses and since the energy is conserved, the amplitude must decrease). It turns out that this is most conveniently measured when requiring the initial data to be in some $L^1$-based space. (This can be understood in terms of strong Huygens' principle and principle of superposition, as requiring that the initial data to have "global size" in $L^1$.) 

*When one wants different local and global behaviour, however, in the PDE context, one usually use $L^p\cap L^q$ instead of $L^p + L^q$; the reason is that differential equations are studied largely through a priori estimates. And if you can only prove your estimate for "a part" of the solution, then it is not really that useful. 

*However, in times, there are strong reasons to use certain $L^p$ spaces. For example, arguments requiring Hilbert-space structures can only be done in $L^2$-based spaces; for technical reasons hyperbolic equations are most naturally studied in $L^2$ based spaces. In these situations one can sometimes get by by approximating "global size" in $L^p$ using "global size" is a weighted $L^q$ space. 

*Of course, these weighted $L^q$ spaces are not translation invariant, which does make the theory not as elegant. And there are other people who feel the same way (in the linked paper they use a sophisticated modification of the Wiener Amalgam space $W(L^2, L^1)$.

*Furthermore, you probably overlooked one place where local versus global sizes come up a lot; for this, however, you need to take the Fourier transform. Basically, the distinction between local and global can also be considered on phase space too! When one does this one is lead to the well-known Besov and Triebel-Lizorkin spaces. For PDE theory these spaces are enormously useful. 

*But finally: to answer your question about why it is natural to measure locally and globally the same way. It is not so much natural as convenient. If you think about the classical symmetries of $\mathbb{R}^d$, we have the $d$ translations and the $d(d-1)/2$ rotations. $L^p$ spaces are invariant under them both. (And so would the amalgam spaces and interpolation/intersection spaces.) But $\mathbb{R}^d$ is always equipped with scaling symmetry. A really nice and convenient thing about $L^p$ spaces is that, if we write the operator 
$$ \sigma_\lambda f(x) = f(\lambda x) $$
then we have
$$ \| \sigma_\lambda f\|_p = \lambda^{-n/p} \|f\|_p.$$
This relation is very convenient when it comes to doing actual computations. If you look at something like $L^p + L^q$ or $L^p\cap L^q$, the formula becomes a lot messier (and are no longer actual equalities). In applications, if you are studying a problem with a nice scaling property, you would expect to use a space that also have nice scaling properties. 
