Why are $L^p$ spaces for $p\not=1,2,\infty$ important? $L^p$ spaces for arbitrary $1\le p\le\infty$  are a mainstay of basic functional analysis courses, but I've only seen them "in action" when $p$ is 1, 2, or $\infty$. Can anyone give an "elementary" concrete example of an application of another $L^p$ space? 
 A: The $L^p$-spaces with $p=1,2,\infty$ get the most attention due to their special properties: $L^2$ is Hilbert, $L^\infty$ and $L^1$ are not reflexive, $L^\infty$ is not separable, $L^1$ has no predual, and so on. Viewed as plain Banach spaces, maybe the spaces $L^p$, $p\not \in\{1,2,\infty\}$ are just boring spaces with no special and interesting properties.
$L^p$-spaces have their uses in Sobolev theory as well: There, functions $u$ with first (weak) derivative are assumed in say $L^p(\Omega)$, the space capturing this is called $W^{1,p}$. In the Sobolev theory, the exponent $p$ has a crucial role. If for instance $p$ is large enough ($p$ larger than space dimension) then the functions in $W^{1,p}$ have a continuous representative. An application of these spaces is the theory of partial differential equations. Here, the value of the exponent $p$ matters to conclude smoothness of function due to Sobolev embedding theorems.
A: I would say that there are very few applications of the cases where $p \ne 1,2, \infty$. 
One example: in $\mathbb R^2$ and $\mathbb R^3$, the balls corresponding to the $l^p$ norms are so-called  super ellipses and superquadrics. These are things that are shaped somewhat like regular ellipses, but with corners that become "sharper" as $p$ increases. These are occasionally used for designing consumer products, sculptures, or the shapes of characters in fonts. But even this is a stretch; you can get roughly the same design freedom with Bézier curves and surfaces, and that's what most people would use, in practice.
I think the only reason that the cases where $p \ne 1,2, \infty$ get mentioned at all is that people can't resist the lure of abstraction; we love to have grand unifying generalizations, rather than separate enumerations of special cases, even if this means weaker results and muddied focus.
