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 $(\Omega , \mathscr {A}, \mu )$ and $G^{\mathsf {T}}$ the associated Littlewood-Paley $g$-function operator: \[ G^{\mathsf {T}}(f)=\Big (\int _0^\infty \left |t\frac {\partial }{\partial t} \mathsf {T}_t(f)\right |^2\frac {\mathrm {d}t}{t}\Big )^{\frac 12}. \] The classical Littlewood-Paley-Stein inequality asserts that for any $1<p<\infty$ there exist two positive constants $\mathsf {L}^{\mathsf {T}}_{p}$ and $\mathsf {S}^{\mathsf {T}}_{p}$ such that \[ \big (\mathsf {L}^{\mathsf {T}}_{ p}\big )^{-1}\big \|f-\mathrm {F}(f)\big \|_{p}\le \big \|G^{\mathsf {T}}(f)\big \|_{p} \le \mathsf {S}^{\mathsf {T}}_{p}\big \|f\big \|_{p}\,,\quad \forall f\in L_p(\Omega ), \] 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 $\mathsf {L}^{\mathsf {T}}_{ p}\lesssim p$ as $p\rightarrow \infty$, and raised the problem about the optimal order of $\mathsf {L}^{\mathsf {T}}_{ p}$ as $p\rightarrow \infty$. We solve Xu’s open problem by showing that this upper estimate of $\mathsf {L}^{\mathsf {T}}_{ p}$ is in fact optimal. Our argument is based on the construction of a special symmetric diffusion semigroup associated with 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.