Assume $x=(x_k)_{k\in\mathbb N}\in\ell^p$ for some $p\in(0,\infty)$, i.e., $\sum\limits_{k\in\mathbb N}|x_k|^p<\infty $. Then, is there always some $q\in(0,p)$ (possibly depending on $x$), such that $x\in\ell^q$?
I doubt it, but couldn't find a counter example. Any suggestions (or proofs that such a $q$ always exists) are welcome.
I am especially interested in the case $p=1$, so any insight on that side would also help.
Consider $x_k := \frac{1}{k \log(k)^2}$. By the Cauchy condensation test $\sum_{k = 2}^\infty \lvert x_k \rvert^p$ for $p >0$ converges iff $$ \sum_{k = 2}^\infty 2^k\lvert x_{2^k} \rvert^p = \sum_{k = 2}^\infty \frac{1}{2^{k(p-1)}k^{2p}\log(2)^{2p}} $$ converges. Note that for $p=1$ this series clearly converges. For $p \in (0, 1)$ we have $k(p-1)<0$ and therefore $$ \frac{1}{2^{k(p-1)}k^{2p}\log(2)^{2p}} \rightarrow \infty $$ as $k \rightarrow \infty$. Therefore no convergence in this case by the divergence test.