From the Littlewood-Paley-Stein inequality to the Burkholder-Gundy inequality
Authors: Xu Zhendong , Zhang Hao 
Published: October 2022

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:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G Superscript sans-serif upper T Baseline left-parenthesis f right-parenthesis equals left-parenthesis integral Subscript 0 Superscript normal infinity Baseline StartAbsoluteValue t StartFraction partial-differential Over partial-differential t EndFraction sans-serif upper T Subscript t Baseline left-parenthesis f right-parenthesis EndAbsoluteValue squared StartFraction normal d t Over t EndFraction right-parenthesis Superscript one half Baseline period">

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
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis sans-serif upper L Subscript p Superscript sans-serif upper T Baseline right-parenthesis Superscript negative 1 Baseline double-vertical-bar f minus normal upper F left-parenthesis f right-parenthesis double-vertical-bar Subscript p Baseline less-than-or-equal-to double-vertical-bar upper G Superscript sans-serif upper T Baseline left-parenthesis f right-parenthesis double-vertical-bar Subscript p Baseline less-than-or-equal-to sans-serif upper S Subscript p Superscript sans-serif upper T Baseline double-vertical-bar f double-vertical-bar Subscript p Baseline comma for-all f element-of upper L Subscript p Baseline left-parenthesis normal upper Omega right-parenthesis comma">

(

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.