Let $\Phi(x)=\int_0^x \phi(y)\,dy$, $x \in \mathbb{R}_+$, be an N-function, and let $u$ be locally inferable on $\mathbb{R}_+$. Consider the gauge norm $$ \rho_{\Phi,u}(f)=\inf\{\lambda>0: \int_{\mathbb{R}_+}\Phi\left(\frac{|f(x)|}{\lambda}\right)u(x)\,dx\leq 1\}, $$where $f \in M_+(R_+)$.

Suppose that $\Phi(2t)\approx \Phi(t)$ for $t\gg1$ and $u(\infty)=\infty$.

Question: Under which conditions would the following inequality hold: $$ \rho_{\Phi, u}\left(t^{-1}\int_0^t f^*\right)\leq C \rho_{\Phi,u}(f^*), $$ where $f^*$ is a nonincreasing rearrangement of $f$ on $\mathbb{R}_+$.


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