# Conditions on the inequality with a gauge norm

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 $$\mathbb{R}_+$$.