  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:
  $$G^{\mathsf{T}}(f)=\Big(\int_0^\infty \left|t\frac{\partial}{\partial t}
\mathsf{T}_t(f)\right|^2\frac{\mathrm{d}t}{t}\Big)^{\frac12}.$$
  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 abut 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 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.
