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 ${a_j:jge0}$ is an absolutely summable real sequence, i.e. $sum_{j=0}^infty|a_j|<infty$. Then
$$
sum_{j=0}^inftybiggl|sum_{k=1}^na_{j+k}biggr|^p=O(n)quadtext{as}quad ntoinfty,
$$
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
$$
sum_{j=0}^inftybiggl|sum_{k=1}^nr^{j+k}biggr|^p=sum_{j=0}^infty|r|^{pj}biggl|sum_{k=1}^nr^kbiggr|^ptosum_{j=0}^infty|r|^{pj}biggl|sum_{k=1}^infty r^kbiggr|^pquadtext{as}quad ntoinfty.
$$

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,
$$
sum_{j=0}^inftybiggl|sum_{k=1}^na_{j+k}biggr|^p
lesum_{j=0}^inftybiggl|Bigl(sum_{k=1}^n|a_{k+j}|^pBigr)^{1/p}n^{1-1/p}biggr|^p=n^{p-1}sum_{k=1}^nsum_{j=0}^infty|a_{k+j}|^ple n^psum_{j=0}^infty|a_j|^p,
$$
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!

Answer

Let $s = sum_{jge 0} |a_{j}|$ and $S_{j} = a_{j+1}+ldots+a_{j+n}$. As $s<infty$ one has that there exists $Nge 0$ such that $sum_{jge N}|a_j|<1$. Therefore for any $jge N$ any $nin {mathbb N}$, it can bee seen that
$$
|S_j| = |a_{j+1}+ldots+a_{j+n}|le |a_{j+1}|+ldots+|a_{j+n}|le sum_{jge N}|a_j| <1.
$$
Thus as $pge 1$, one has $|S_j|^p<|S_j|$ and consequently
$$
|S_j|^ple |S_j|le |a_{j+1}|+ldots+|a_{j+n}|
$$
So
$$
sum_{jge N} |S_j|^p
le
sum_{j>N}|S_j|le sum_{jge N} |a_{j+1}|+ldots+|a_{j+n}|= sum_{jge N} |a_{j+1}|+ldots+sum_{jge N}|a_{j+n}|
$$
As for any $k=2,ldots,n$ one has that
$$
sum_{jge N} |a_{j+k}|le sum_{Nle j le N+k-2} |a_{j+1}| + sum_{jge N} |a_{j+k}|= sum_{jge N} |a_{j+1}|,
$$
it can be seen that
$$
sum_{jge N} |S_j|^ple n sum_{jge N} |a_{j+1}|
$$
Therefore
$$
sum_{jge 0} |S_j|^p< n sum_{jge N} |a_{j+1}| + sum_{ 0le j< N} |S_j|^p.
$$
So as $sum_{jge N} |a_{j+1}|<1$
$$
sum_{jge 0} |S_j|^p< n + c.
$$
where $c = sum_{ 0le j< N} |S_j|^p$ is a constant.

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