Limiting behaviour of $\sum_{j=0}^\infty|\sum_{k=1}^na_{j+k}|^p$ I would like to prove or disprove the following statement.

Suppose that $\{a_j:j\ge0\}$ is an absolutely summable real sequence, i.e. $\sum_{j=0}^\infty|a_j|<\infty$. Then
  $$
\sum_{j=0}^\infty\biggl|\sum_{k=1}^na_{j+k}\biggr|^p=O(n)\quad\text{as}\quad n\to\infty,
$$
  where $1<p<2$.

For example, if $a_j=r^j$  with $|r|<1$, then the statement is true and even stronger result holds since
$$
\sum_{j=0}^\infty\biggl|\sum_{k=1}^nr^{j+k}\biggr|^p=\sum_{j=0}^\infty|r|^{pj}\biggl|\sum_{k=1}^nr^k\biggr|^p\to\sum_{j=0}^\infty|r|^{pj}\biggl|\sum_{k=1}^\infty r^k\biggr|^p\quad\text{as}\quad n\to\infty.
$$
I tried to use Hölder's inequality and the triangle inequality for the $\ell^p$ norm, but I could not obtain the result. By Hölder's inequality,
$$
\sum_{j=0}^\infty\biggl|\sum_{k=1}^na_{j+k}\biggr|^p
\le\sum_{j=0}^\infty\biggl|\Bigl(\sum_{k=1}^n|a_{k+j}|^p\Bigr)^{1/p}n^{1-1/p}\biggr|^p=n^{p-1}\sum_{k=1}^n\sum_{j=0}^\infty|a_{k+j}|^p\le n^p\sum_{j=0}^\infty|a_j|^p,
$$
but this inequality is not accurate enough. The assumption of absolute summability needs to be somehow used, but I have no idea how to do that.
Any help is much appreciated!
 A: Let $s = \sum_{j\ge 0} |a_{j}|$ and $S_{j} = a_{j+1}+\ldots+a_{j+n}$. As $s<\infty$ one has that there exists $N\ge 0$ such that $\sum_{j\ge N}|a_j|<1$. Therefore for any $j\ge N$ any $n\in {\mathbb N}$, it can bee seen that
$$
|S_j| = |a_{j+1}+\ldots+a_{j+n}|\le |a_{j+1}|+\ldots+|a_{j+n}|\le \sum_{j\ge N}|a_j| <1.
$$
Thus as $p\ge 1$, one has $|S_j|^p<|S_j|$ and consequently
$$
|S_j|^p\le |S_j|\le |a_{j+1}|+\ldots+|a_{j+n}|
$$ 
So
$$
\sum_{j\ge N} |S_j|^p
\le 
\sum_{j>N}|S_j|\le \sum_{j\ge N} |a_{j+1}|+\ldots+|a_{j+n}|= \sum_{j\ge N} |a_{j+1}|+\ldots+\sum_{j\ge N}|a_{j+n}|
$$ 
As for any $k=2,\ldots,n$ one has that
$$
\sum_{j\ge N} |a_{j+k}|\le \sum_{N\le j \le N+k-2} |a_{j+1}| + \sum_{j\ge N} |a_{j+k}|= \sum_{j\ge N} |a_{j+1}|,
$$ 
it can be seen that
$$
\sum_{j\ge N} |S_j|^p\le n \sum_{j\ge N} |a_{j+1}|
$$ 
Therefore
$$
\sum_{j\ge 0} |S_j|^p< n \sum_{j\ge N} |a_{j+1}| + \sum_{ 0\le j< N} |S_j|^p.
$$ 
So as $\sum_{j\ge N} |a_{j+1}|<1$
$$
\sum_{j\ge 0} |S_j|^p< n  + c.
$$ 
where $c = \sum_{ 0\le j< N} |S_j|^p$ is a constant.
