Is it true the following inequality?

Question

Is it true the following statement: Let $R>0$, there exists $C_R>0$ such that $$\left |\int_0^T z(r) dr \right |^2\leq C_R \int_0^T \left | \int_0^\xi z(r)dr \right|^2 d \xi $$ for every $z \in L^2[0,T]$ such that $\int_0^T z^2(r)dr \leq R$.

Answer 1

A counterexample:

Let $R=T=1$. For any positive integer $p$ define $z(r)=r^p$, then $z\in L^2[0,1]$ and $\int_0^1z^2(r)dr\leq1$.

Moreover, it follows \begin{align*} &I_\text{left}=\left|\int_0^1z(r)dr\right|^2=\frac1{(p+1)^2}\\ &I_\text{right}=\int_0^1\left|\int_0^\xi z(r)dr\right|^2d\xi=\int_0^1\frac{\xi^{2(p+1)}}{(p+1)^2}d\xi=\frac1{(p+1)^2(2p+3)}. \end{align*}

If the inequality $I_\text{left}\leq C\cdot I_\text{right}$ holds for some $C>0$, we must have $C\geq2p+3$. But $p$ can be arbitrarily large, so the inequality cannot hold.