I would like to know if the following inequality can be true : consider a double sequence $(u_{i,j})_{(i,j)\in (\mathbb{N}^\star)^2}$ of real numbers and a real $p\geq 1$, do we have $$ \sum_{j\geq 1} \left\vert \sum_{i\geq j} \frac{u_{i1}+\dots+u_{ii}}{i} \right\vert^p \leq C(p) \sum_{j\geq 1} \left( \sum_{i\geq j} |u_{ij}| \right)^p , $$ where $C(p)>0$ only depends on $p$. One checks easily that $C(1)=1$ is convenient. I do not know if an interpolation theorem could give the answer.

Thanks.