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

TL;DR

This paper determines the precise asymptotic behavior of the best constants in Littlewood-Paley inequalities for the Poisson semigroup and dyadic square functions, including their dependence on the dimension and the range of p.

Contribution

It provides the optimal order of magnitude for constants in Littlewood-Paley inequalities for Poisson and dyadic square functions, extending results to general test functions and the torus.

Findings

Optimal order of constants as p approaches 1 and infinity.

Dimension-dependent relations for dyadic square function constants.

Extension of results to the torus and general test functions.

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: $$G^{\mathbb{P}}(f)=\Big(\int_0^\infty t|\frac{\partial}{\partial t} \mathbb{P}_t(f)|^2dt\Big)^{\frac12}.$$ 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.