infinte sums, inequalities, optimal rate.

Question

Suppose we have a sequence $\{a_k\}_{k=1}^\infty$ such that $$ \sum_{k=1}^\infty k^{2s}a_k^2<1\,\,\,\text{for some s > 1 }. $$ I need to show for all $\gamma>1$ that $$ \frac{\sum^n_{l=1}l^{\gamma}\sum_{k=1}^n k^{-\gamma}a_k^2}{\sum_{k=1}^n a_k^2}\lesssim n^2 $$ or find any sharper rate than $n^2$. So i need that $l^\gamma$ and $k^{-\gamma}$ somehow cancel each other out in some sense. Would you think this is possible? I would be glad for any suggestions or opinion!