Optimal orders of the best constants in the Littlewood-Paley inequalities

Let $\{\mathbb{P}_t\}_{t>0}$ be the classical Poisson semigroup on $\mathbb{R}^d$ and $G^{\mathbb{P}}$ the associated Littlewood-Paley $g$-function operator: $$G^{\mathbb{P}}(f)=\Big(\int_0^\infty t|\frac{\partial}{\partial t} \mathbb{P}_t(f)|^2dt\Big)^{\frac12}.$$ The classical Littlewood-Paley $g$-function inequality asserts that for any $1<p<\infty$ there exist two positive constants $\mathsf{L}^{\mathbb{P}}_{t, p}$ and $\mathsf{L}^{\mathbb{P}}_{c, p}$ such that $$ \big(\mathsf{L}^{\mathbb{P}}_{t, p}\big)^{-1}\big\|f\big\|_{p}\le \big\|G^{\mathbb{P}}(f)\big\|_{p} \le \mathsf{L}^{\mathbb{P}}_{c,p}\big\|f\big\|_{p}\,,\quad f\in L_p(\mathbb{R}^d). $$ We determine the optimal orders of magnitude on $p$ of these constants as $p\to1$ and $p\to\infty$. We also consider similar problems for more general test functions in place of the Poisson kernel. The corresponding problem on the Littlewood-Paley dyadic square function inequality is investigated too. Let $\Delta$ be the partition of $\mathbb{R}^d$ into dyadic rectangles and $S_R$ the partial sum operator associated to $R$. The dyadic Littlewood-Paley square function of $f$ is $$S^\Delta(f)=\Big(\sum_{R\in\Delta} |S_R(f)|^2\Big)^{\frac12}.$$ For $1<p<\infty$ there exist two positive constants $\mathsf{L}^{\Delta}_{c,p, d}$ and $ \mathsf{L}^{\Delta}_{t,p, d}$ such that $$ \big(\mathsf{L}^{\Delta}_{t,p, d}\big)^{-1}\big\|f\big\|_{p}\le \big\|S^\Delta(f)\big\|_{p}\le \mathsf{L}^{\Delta}_{c,p, d}\big\|f\big\|_{p},\quad f\in L_p(\mathbb{R}^d). $$ We show that $$\mathsf{L}^{\Delta}_{t,p, d}\approx_d (\mathsf{L}^{\Delta}_{t,p, 1})^d\;\text{ and }\; \mathsf{L}^{\Delta}_{c,p, d}\approx_d (\mathsf{L}^{\Delta}_{c,p, 1})^d.$$ All the previous results can be equally formulated for the $d$-torus $\mathbb{T}^d$. We prove a de Leeuw type transference principle in the vector-valued setting