TITLE: Why do we care about $L^p$ spaces besides $p = 1$, $p = 2$, and $p = \infty$?
QUESTION [139 upvotes]: 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.]

REPLY [2 votes]: One answer that I don't see explicitly mentioned here is that $L^\infty = \lim_{p\to\infty} L^p$, so to speak.  So if you in fact care about $L^\infty$ but there are easier theorems regarding $p<\infty$, then do what we do in analysis, and get a sequential approximation.