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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 $p\geq 4/3$.

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 $b_n=O(a_n)$.

Littlewood's $4/3$-inequality singles out $\ell^{4/3}$.

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 $p\geq 4/3$.