Let $(\O, \mu)$ be a measure space and $1<p<\8$. Let  $\{T_t\}_{t>0}$ be a strongly continuous semigroup of positive contractions on $L_p(\O, \mu)$  and $\{P_t\}_{t>0}$ its subordinated Poisson semigroup. It is known that there exist two positive constants $\a_p, \b_p$ (depending on $\{T_t\}_{t>0}$ too) such that

$$\a_p^{-1}\|f-\mathsf F(f)\|_{p}\le\left\|\left(\int_0^\8\Big |t\frac{\partial}{\partial t} P_t (f)\Big|^2\,\frac{dt}t\right)^{\frac12}\right\|_{p}\le \b_p\|f-\mathsf F(f)\|_{p}\,,\quad f\in L_p(\O),

$$

where $\mathsf F$ is the projection from $L_p(\O, \mu)$ onto the fixed point subspace of  $\{T_t\}_{t>0}$.

We determine the optimal orders of magnitude on $p$ of $\a_p$ and $\b_p$ as $p\to1$ and $p\to\infty$. We also consider the particular case where  $\{T_t\}_{t>0}$ is the heat semigroup on $\real^d$. Then the previous inequality is the classical Littlewood-Paley $g$-function inequality. If time permits, we will present a brief discussion on the vector-valued extension.