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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.

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    $\begingroup$ This is too cheap to qualify as more than a comment, but $p={\frac{1+\sqrt{5}}{2}}$ is the only case where $\ell^p$ is dual to $\ell^{p^2}$. $\endgroup$ Dec 13, 2010 at 8:56
  • $\begingroup$ @Aaron Perhaps if there were some natural abstract connection between $\ell^p$ and $\ell^{p^2}$, you'd have a winner. $\endgroup$ Dec 13, 2010 at 9:02

5 Answers 5

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  1. 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 $p\geq 4/3$.

  2. 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 $b_n=O(a_n)$.

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  • $\begingroup$ Interesting, and close to what I want, but this doesn't so much single out the space $\ell^{4/3}$ as single out the number $4/3$. If it could be combined with a condition characterizing $\ell^p$'s for $p\leq 4/3$, that would be perfect. $\endgroup$ Dec 13, 2010 at 7:51
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    $\begingroup$ Also, if you use the higher dimensional version by Bohnenblust and Hille, you can also similarly characterize all numbers $p = \frac{2m}{m+1}$ for $m\in \mathbb{N}$. (The inequality is $$ \|\hat{a}\|_{\ell^p(\mathbb{N}^m)} \leq 2^{(m-1)/2} \|\hat{a}\| $$ where on the RHS is $\sup \hat{a}(x_1,\ldots,x_m)$ with each $x_i\in \ell^\infty(\mathbb{N})$.) $\endgroup$ Dec 13, 2010 at 12:25
  • $\begingroup$ Dear Andrey: I got confused by where you put the if and only if. Sorry. Deleted the irrelevant comments. $\endgroup$ Dec 13, 2010 at 16:54
  • $\begingroup$ Dear Willie, many thanks for your comment. $\endgroup$ Dec 13, 2010 at 17:15
  • $\begingroup$ In fact there is nothing special in Littlewood's inequality. It is a manifistation of general result of Kwapien: any operator from $\mathcal{L}_1$ space to $\mathcal{L}_p$ space is $(r,1)$-summing for $r^{-1}=1-|p^{-1}-2^{-1}|$ and $p>1$. For details see page 208 in Absolutely Summing Operators by Joe Diestel $\endgroup$
    – Norbert
    May 30, 2014 at 0:49
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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.

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  • $\begingroup$ Are there easy counterexamples in the remaining cases? $\endgroup$ Dec 21, 2010 at 3:38
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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.

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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 $\mathbb{T}^2_{t,x}=[0,2\pi]^2$.

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  • $\begingroup$ @Piero. Isn't it a special case of a Strichartz inequality ? $\endgroup$ Dec 13, 2010 at 8:30
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    $\begingroup$ @Denis: yes indeed, it is the 1D Strichartz inequality for the Scroedinger equation, and it was proved by Zygmund around 1950 $\endgroup$ Dec 13, 2010 at 11:24
  • $\begingroup$ P.S. notice that the other Strichartz inequalities are for a $L^p_tL^q_x$ norm with p different from q. $\endgroup$ Dec 13, 2010 at 11:25
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$p$-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)

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  • $\begingroup$ There are p-stable distributions for all 0<p≤2. I think this is usually attributed to Paul Levy. You can find this in many books. I particularly like the construction in Chung's A course in probability theory. $\endgroup$ Dec 13, 2010 at 10:05
  • $\begingroup$ Ah correct. I'll edit accordingly. $\endgroup$ Dec 13, 2010 at 11:46
  • $\begingroup$ A functional analytic version and amplification of this is that if $p<2$, then $\ell_r$ embeds isometrically isomorphically into $L_p$ for $p\le r\le 2$, while if $r$ is not in this range, then $\ell_r$ does not embed even isomorphically into $L_p$. $\endgroup$ Dec 14, 2010 at 0:51
  • $\begingroup$ what is an 'isomorphic' embedding ? $\endgroup$ Dec 14, 2010 at 11:04
  • $\begingroup$ An isomorphic embedding is a linear homeomorphism into. $\endgroup$ Dec 14, 2010 at 23:02

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