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

Abstract

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: $$G^{\mathbb{P}}(f)=({\int }_0^{\infty }t|\frac{\partial }{\partial t}{\mathbb{P}}_t(f){|}^{2}dt{)}^{\frac{1}{2}}.$$ The classical Littlewood-Paley $g$-function inequality asserts that for any $1<p<\infty$ there exist two positive constants ${\mathsf{L}}_{t,p}^{\mathbb{P}}$ and ${\mathsf{L}}_{c,p}^{\mathbb{P}}$ such that $$({\mathsf{L}}_{t,p}^{\mathbb{P}}{)}^{-1}\Vert f{\Vert }_p\le \Vert G^{\mathbb{P}}(f){\Vert }_p\le {\mathsf{L}}_{c,p}^{\mathbb{P}}\Vert f{\Vert }_p,f\in L_p({\mathbb{R}}^d).$$ We determine the optimal orders of magnitude on $p$ of these constants as $p\to 1$ 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 $$S^{\Delta }(f)=(\sum _{R\in \Delta }|S_R(f){|}^2{)}^{\frac{1}{2}}.$$ For $1<p<\infty$ there exist two positive constants ${\mathsf{L}}_{c,p,d}^{\Delta }$ and ${\mathsf{L}}_{t,p,d}^{\Delta }$ such that $$({\mathsf{L}}_{t,p,d}^{\Delta }{)}^{-1}\Vert f{\Vert }_p\le \Vert S^{\Delta }(f){\Vert }_p\le {\mathsf{L}}_{c,p,d}^{\Delta }\Vert f{\Vert }_p,f\in L_p({\mathbb{R}}^d).$$ We show that $${\mathsf{L}}_{t,p,d}^{\Delta }{\approx }_d({\mathsf{L}}_{t,p,1}^{\Delta }{)}^d\text{ and }{\mathsf{L}}_{c,p,d}^{\Delta }{\approx }_d({\mathsf{L}}_{c,p,1}^{\Delta }{)}^d.$$ 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.