Some basic inequalities in the theory of symmetric normed space

Question

I am learning about symmetric normed space and have trouble figuring out the following inequality.

Let $c_c(\mathbb{N})$ be the space of compactly supported sequence. Let $||a||_{p,\omega}:=\sup_{n}(n^{-1+\frac{1}{p}}\sum_{i=1}^n a_j^*)$ be the Calderón symmetric norm. Here $\{a_j^*\}_{j\in \mathbb{N}}$ denotes the non-increasing rearrangement of $\{|a_j|\}_{j\in \mathbb{N}}$. Let $||a||'_{p,\omega}:= \sup_{i}(i^{1/p}a_i^*)$. Then inequality to show is: \begin{equation} ||a||_{p,\omega} \leq \frac{p}{p-1}||a||'_{p,\omega} \end{equation} where $p\neq 1$.

This should be elementary but I can't figure it out. Thank you for the help.

Answer 1

With $C=\lVert a\rVert_{p,\omega}'$ there holds $a_j^*\le j^{-1/p}C$ for every $j$. Hence, $$\lVert a\rVert_{p,\omega}\le \sup_nn^{-1+1/p}\sum_{j=1}^nj^{-1/p}C\le \sup_nn^{-1+1/p}\Bigl(1+\int_1^nx^{-1/p}dx\Bigr)C=\sup_nn^{-1+1/p}\frac1{1-1/p}n^{1-1/p}C=\frac p{p-1}\lVert a\rVert_{p,\omega}'.$$