"Reverse" Triangle Inequality in weak $L^p$

Question

Consider the $L^p$-weak quasi-norm, for $p>1$, that is $$\|f\|_{L^p_w(U)}=\sup_{t>0} t \mu\big(\left\{x \in U : |f(x)| > t \right\}\big)^{1/p}$$ where $\mu$ is the Lebesgue measure on $\mathbb{R}^n$. Is it true that there exist two positive constant $C_1, C_2 > 0$ such that $$\|f-g\|_{L^p_w(U)} \ge C_1 \|f\|_{L^p_w(U)} - C_2 \|g\|_{L^p_w(U)}$$ for every $f, g \in L^p_w(U)$ ?