  We study the two-weighted estimate
  \[
  \bigg\|\sum_{k=0}^na_k(x)\int_0^xt^kf(t)dt|L_{q,v}(0,\infty)\bigg\|\leq
  c\|f|L_{p,u}(0,\infty)\|,\tag{$*$}
  \]
  where the functions $a_k(x)$ are not assumed to be positive. It is shown that
for $1<p\leq q\leq\infty$, provided that the weight $u$ satisfies the certain
conditions, the estimate $(*)$ holds if and only if the estimate
  \[
  \sum_{k=0}^n\bigg\|a_k(x)\int_0^xt^kf(t)dt|L_{q,v}(0,\infty)\bigg\|
  \leq c\|f|L_{p,u}(0,\infty)\|.\tag{$**$}
  \]
  is fulfilled. The necessary and sufficient conditions for $(**)$ to be valid
are well-known. The obtained result can be applied to the estimates of
differential operators with variable coefficients in some weighted Sobolev
spaces.
