Idiosyncratic characterizations of $\ell^p$, for $p\not=1,2,\infty$ Do there exist, either in the literature or in folklore, theorems
that characterize some particular $\ell^p$ space(s) ($p\not=1,2,\infty$)?
Such a theorem should reveal the particular space(s) as somehow idiosyncratic,
in the sense that no obvious modification of the characterization works
for general $\ell^p$ spaces.  
Thus it would not be interesting here to learn, say,
that $\ell^3$ alone has a dual isomorphic to $\ell^{3/2}$; 
obviously this just specializes a general fact from the theory of all
the $\ell^p$'s. 
 A: The following theorem is due to Plotkin and Rudin and characterizes $p \neq 2,4,6,\dots.$
Theorem: (Plotkin-Rudin): Let $0< p< \infty$ and $p \neq 2,4,6,\dots$. Let $(\Omega,\mu)$ and $(\Omega',\nu)$ be two
probability measure spaces. Let finally $n$ be a positive integer and $f_1, \dots f_n
\in L_p(\mu)$, $g_1, \dots g_n \in L_p(\nu)$.
Assume that for all complex numbers $z_1, \dots z_n \in \mathbb C$, 
$$\int |1 + z_1 f_1 + \dots z_n f_n |^p d \mu = \int |1 + z_1 g_1 + \dots z_n g_n|^p d \nu.$$
Then $(f_1 ,\dots f_n)$ and $(g_1 ,\dots g_n)$ form two equimeasurable
families. This means that the ${\mathbb R}^n$-valued random
variables $(f_1 ,\dots f_n)$ and $(g_1 ,\dots g_n)$ have the same distribution.
A: Perhaps if you look at applications to other domains and you admit the Lebesgue space $L^p(\mathbb R^n)$... The space $L^n(\mathbb R^n)$ is critical for Navier-Stokes in space dimension $n$. For instance a theorem of T. Kato says that if the initial data is small in $L^3(\mathbb R^3)$, then the Navier-Stokes equation for an incompressible fluid admits a unique solution, global in time. Removing the smallness assumption is worth a million dollars. The exponent $p=3$ is the only one for which such a result holds true.
A: If you are looking for a true characterization of some $\ell^p$ space (an if and only if) I suspect no example satisfies your demands (probably it's only my ignorance :). There are of course inequalities which are known to be true for special values of $p$ and are an open problem for other values, at least for $L^p$ (and a discrete analogue seems reasonable). Best candidate for this kind of inequalities is $p=4$ since the norm is the square of a square and can be represented in a fairly reasonable way using Fourier transform and convolutions. This method allows to prove e.g. Zygmund's inequality
$$ \|\sum c_{n}e^{i(n^2t+nx)}\|^2_{L^4(\mathbb{T}^2)}\le C \sum|c_n|^2 $$
for $L^4$ on $\mathbb{T}^2_{t,x}=[0,2\pi]^2$.
A: *

*Littlewood's $4/3$-inequality singles out $\ell^{4/3}$. 
Namely, given a real valued array $\hat{a}=(\hat a_{m,n}:(m,n)\in\mathbb N^2)$, the norm $\|\hat a\|_{\ell_p}$ is finite for all $\hat a$ such that 
$$\sup \left\{\left|\sum\limits_{m\in\mathcal M,n\in\mathcal N}\hat a_{m,n}x_my_n\right|:x_m,y_n\in[-1,1],\mathcal M,\mathcal N\mbox{ are finite}\right\} < \infty$$
if and only if $p\geq 4/3$.

*The second example is somewhat tangential to the question but I find it worth mentioning. It is concerned with the peculiar asymptotics of $L^4$-norms of the Hermite functions (see, e.g., Lectures on Hermite and Laguerre expansions by Thangavelu, Lemma 1.5.2). 

Proposition. As $n\to\infty$ the Hermite functions satisfy the estimates 
  $$\|h_n\|_{p}\sim\begin{cases} n^{\frac{1}{2p}-\frac{1}{4}}, &  1\leq p< \infty, \\\ \\\ n^{-\frac{1}{8}}\log n, &  p=4, \\\  \\\ n^{-\frac{1}{6p}-\frac{1}{12}}, & 4 < p\leq \infty. \end{cases} $$
  Here $a_n\sim b_n$  means $a_n=O(b_n)$ and $b_n=O(a_n)$. 

A: $p$-stability singles out $0 < p \le 2$. Specifically, there is no probability distribution $P$ such that the linear combination $\sum^n a_i X_i$ is distributed as $\|a\|_p Y$, where $X_1 ... X_n$ and $Y$ are random variables distributed according to $P$, if $p$ is not in the range $(0, 2]$. 
For $p = 0.5, 1, 2$ these distributions have closed-form expressions. 
(note: updated to reflect Gideon Schectman's comment)
