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

Abstract

Let { T t } t > 0 \{\mathsf {T}_t\}_{t>0} be a symmetric diffusion semigroup on a σ \sigma -finite measure space ( Ω , A , μ ) (\Omega , \mathscr {A}, \mu ) and G T G^{\mathsf {T}} the associated Littlewood-Paley g g -function operator: \[ G T ( f ) = ( ∫ 0 ∞ | t ∂ ∂ t T t ( f ) | 2 d t t ) 1 2 . G^{\mathsf {T}}(f)=\Big (\int _0^\infty \left |t\frac {\partial }{\partial t} \mathsf {T}_t(f)\right |^2\frac {\mathrm {d}t}{t}\Big )^{\frac 12}. \] The classical Littlewood-Paley-Stein inequality asserts that for any 1 > p > ∞ 1>p>\infty there exist two positive constants L p T \mathsf {L}^{\mathsf {T}}_{p} and S p T \mathsf {S}^{\mathsf {T}}_{p} such that \[ ( L p T ) − 1 ‖ f − F ( f ) ‖ p ≤ ‖ G T ( f ) ‖ p ≤ S p T ‖ f ‖ p , ∀ f ∈ L p ( Ω ) , \big (\mathsf {L}^{\mathsf {T}}_{ p}\big )^{-1}\big \|f-\mathrm {F}(f)\big \|_{p}\le \big \|G^{\mathsf {T}}(f)\big \|_{p} \le \mathsf {S}^{\mathsf {T}}_{p}\big \|f\big \|_{p}\,,\quad \forall f\in L_p(\Omega ), \] where F \mathrm {F} is the projection from L p ( Ω ) L_p(\Omega ) onto the fixed point subspace of { T t } t > 0 \{\mathsf {T}_t\}_{t>0} of L p ( Ω ) L_p(\Omega ) .

Recently, Xu proved that L p T ≲ p \mathsf {L}^{\mathsf {T}}_{ p}\lesssim p as p → ∞ p\rightarrow \infty , and raised the problem about the optimal order of L p T \mathsf {L}^{\mathsf {T}}_{ p} as p → ∞ p\rightarrow \infty . We solve Xu’s open problem by showing that this upper estimate of L p T \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 T ( f ) G^{\mathsf {T}}(f) for any f ∈ L p ( Ω ) f\in L_p(\Omega ) is pointwise comparable with the martingale square function of f f . Our method also extends to the vector-valued and noncommutative setting.