From the Littlewood-Paley-Stein Inequality to the Burkholder-Gundy Inequality

Abstract

Let $\{{\mathsf{T}}_t{\}}_{t>0}$ be a symmetric diffusion semigroup on a $\sigma$-finite measure space $(\Omega ,\mathcal{A},\mu )$ and $G^{\mathsf{T}}$ the associated Littlewood-Paley $g$-function operator: $$G^{\mathsf{T}}(f)=({\int }_0^{\infty }{\left|t\frac{\partial }{\partial t}{\mathsf{T}}_t(f)\right|}^2\frac{\mathrm{d}t}{t}{)}^{\frac{1}{2}}.$$ The classical Littlewood-Paley-Stein inequality asserts that for any $1<p<\infty$ there exist two positive constants ${\mathsf{L}}_p^{\mathsf{T}}$ and ${\mathsf{S}}_p^{\mathsf{T}}$ such that $$({\mathsf{L}}_p^{\mathsf{T}}{)}^{-1}\Vert f-\mathrm{F}(f){\Vert }_p\le \Vert G^{\mathsf{T}}(f){\Vert }_p\le {\mathsf{S}}_p^{\mathsf{T}}\Vert f{\Vert }_p,\forall f\in L_p(\Omega ),$$ where $\mathrm{F}$ is the projection from $L_p(\Omega )$ onto the fixed point subspace of $\{{\mathsf{T}}_t{\}}_{t>0}$ of $L_p(\Omega )$. Recently, Xu proved that ${\mathsf{L}}_p^{\mathsf{T}}\lesssim p$ as $p\to \infty$, and raised the problem abut the optimal order of ${\mathsf{L}}_p^{\mathsf{T}}$ as $p\to \infty$. We solve Xu's open problem by showing that this upper estimate of ${\mathsf{L}}_p^{\mathsf{T}}$ is in fact optimal. Our argument is based on the construction of a special symmetric diffusion semigroup associated to any given martingale such that its square function $G^{\mathsf{T}}(f)$ for any $f\in L_p(\Omega )$ is pointwise comparable with the martingale square function of $f$. Our method also extends to the vector-valued and noncommutative setting.