For any sequence $\omega\in[1, \infty)^{\mathbb{N}}$, define the weighted $\ell^p\_\omega$-norm of the sequence $v$ by
$$\Vert v\Vert\_{\ell^p\_\omega} := \left(\sum\_{k=1}^\infty \omega\_k^p |v|\_k^p\right)^{1/p} .$$


**My question is:** If $v\in\ell^q$ and $p>q$, under which conditions on $\omega$ will $v\in\ell^p\_\omega$?




---


Obviously, since $p>q$, we can use Hölder‘s inequality to bound
$$
\Vert v\Vert\_{\ell^p\_\omega}^p
= \Vert \omega^p v^{p-q}v^q\Vert\_{\ell^1}
\le \Vert \omega^p v^{p-q}\Vert\_{\ell^\infty}\Vert v^q\Vert\_{\ell^1}
= \Vert \omega^{p/(p-q)} v\Vert\_{\ell^\infty}^{p-q} \Vert v\Vert\_{\ell^q}^q .
$$
This gives the condition $\Vert\omega^{p/(p-q)}v\Vert\_{\ell^\infty} <\infty$.
(Here, all operations on the sequences have to be understood element-wise.)


* But is this condition necessary?
* Moreover, can we find an unbounded increasing sequence $\omega$ (preferably algebraically increasing like $\omega\_k=k^s$ for $s=\frac1q-\frac1p$) such that $\ell^q\subseteq\ell^p\_\omega$?



Can a weighted $\ell^p$ norm be bounded by an unweighted $\ell^q$ norm?
The answer to both questions is no.


* Taking $q=1$, $p=2$, $v\_k = k^{-(1+\varepsilon)}$ and $\omega\_k = k^{(1+3/2\varepsilon)/2}$ for some $\varepsilon>0$ provides a counter-example to the first assertion.
* [This answer](https://math.stackexchange.com/a/2296826/303111) provides a counter-example for the second assertion when $\omega$ increases algebraically.
It can be generalized to any explicitly given sequence $\omega$.


