Given a $k$-dimensional subspace in $\ell^n_p$, is there a way to bound the value of $$ \sum_{i=1}^k \|a_i\|_{\ell^p}^2 $$ for $a_i$ an orthonormal (for the "standard" underlying $\ell^n_2$) basis.

More precisely, are there estimates for the average (for the usual probability measure on $O(k)$) value or the minimal one.

EDIT: Some results using martingales or volume ratios seem to produce estimates for similar quantities, which made me believe there are known estimates for this.

Also, it seems the case where this methods apply is when $p \leq 2$, so this should be an added to the hypothesis.

EDIT: Of course, when $k=1$, one may use Hölder's inequality to get that it is $\leq n^{\frac{2}{p}-1}$ (bound attained when $X$ is spanned by a vector with all coefficients equal).