I was helping a student study for a functional analysis exam and the question came up as to when, in practice, one needs to consider the Banach space $L^p$ for some value of $p$ other than the obvious ones of $p=1$, $p=2$, and $p=\infty$. I don't know much analysis and the best thing I could think of was Littlewood's 4/3 inequality. In its most elementary form, this inequality states that if $A = (a\_{ij})$ is an $m\times n$ matrix with real entries, and we define the norm
$$\|A\| = \sup\biggl(\left|\sum\_{i=1}^m \sum\_{j=1}^n a\_{ij}s\_it\_j\right| : |s\_i| \le 1, |t\_j| \le 1\biggr)$$
then
$$\biggl(\sum\_{i,j} |a\_{ij}|^{4/3}\biggr)^{3/4} \le \sqrt{2} \|A\|.$$
Are there more convincing examples of the importance of "exotic" values of $p$? I remember wondering about this as an undergraduate but never pursued it. As I think about it now, it does seem a bit odd from a pedagogical point of view that none of the textbooks I've seen give any applications involving specific values of $p$. I didn't run into Littlewood's 4/3 inequality until later in life.


[Edit: Thanks for the many responses, which exceeded my expectations! Perhaps I should have anticipated that this question would generate a big list; at any rate, I have added the big-list tag. My choice of which answer to accept was necessarily somewhat arbitrary; all the top responses are excellent.]



Why do we care about $L^p$ spaces besides $p = 1$, $p = 2$, and $p = \infty$?
Huge chunks of the theory of nonlinear PDEs rely critically on analysis in $L^p$-spaces.


* Let's take the 3D Navier-Stokes equations for example. Leray proved in 1933 existence of *a weak* solution to the corresponding Cauchy problem with initial data from the space $L^2(\mathbb R^3)$. Unfortunately, it is still a major open problem whether the Leray weak solution is unique. But if one chooses the initial data from $L^3(\mathbb R^3)$, then Kato showed that there is *a unique strong* solution to the Navier-Stokes equations (which is known to exist locally in time). $L^3$ is the "weakest" $L^p$-space of initial data which is known to give rise to unique solutions of the 3D Navier-Stokes.
* In some cases the structure of the equations suggests the choice of $L^p$ as the most natural space to work in. For instance, many equations stemming from non-Newtonian fluid dynamics and image processing involve the $p$-Laplacian 
$\nabla\left(|\nabla u|^{p-2}\nabla u\right)$ with $1 < p < \infty.$ Here the $L^p$-space
and $L^p$-based Sobolev spaces provide a natural framework to study well-posedness and regularity issues.




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* Yet another example from harmonic analysis (which goes back to Paley and Zigmund, I think). Let
$$F(x,\omega)=\sum\limits\_{n\in\mathbb Z^d} g\_n(\omega)c\_ne^{inx},\quad x\in \mathbb T^d,$$
where $g\_n$ is a sequence of independent normalized Gaussians and $(c\_n)$ is a non-random element of $l^2(\mathbb Z^d)$. Then the function $F$ *belongs almost surely to any* $L^p(\mathbb T^d)$, $2\leq p <\infty$ and it *does not* belong almost surely to $L^{\infty}(\mathbb T^d)$.


There have been very recent applications of this resut to the existence of solutions to the nonlinear Schrodinger equations with random initial data (due to Burq, Gérard, Tzvetkov et al).


