Let $f_n\colon (0,T) \to \mathbb{R}$ be functions such that $$\lVert f_n \rVert_{L^\infty(0,T)} \leq C$$ for all $n$, i.e. they are uniformly bounded. Furthermore, $$f_n(t) = \sum_{j=1}^n a_{jn}\chi_{I_{jn}}(t)$$ where $a_{jn} \in \mathbb{R}$ and for each $n$, $\{I_{jn}\}_j$ is a partition of $(0,T)$ into $n$ subintervals with measure $T/n$ (i.e., it is a uniform partition).
By taking the $L^p$ norm for any $p < \infty$,
$$\sum_{j=1}^n \frac 1n|a_{jn}|^p \leq C$$ holds uniformly in $n$. My question is, can this be improved by having
$$\sum_{j=1}^n \frac{1}{n^\alpha}|a_{jn}|^p \leq C$$ for some $\alpha <1$ and some $p \geq 1$, again with $C$ independent of $n$?