Best constants in the vector-valued Littlewood-Paley-Stein theory

Let $L$ be a sectorial operator of type $\alpha$ ($0 \leq \alpha < \pi/2$) on $L^2(\mathbb{R}^d)$ with the kernels of $\{e^{-tL}\}_{t>0}$ satisfying certain size and regularity conditions. Define $$S_{q,L}(f)(x) = \left(\int_0^{\infty}\int_{|y-x| < t} \|tL{e^{-tL}} (f)(y) \|_X^q \,\frac{{\rm d} y{\rm d} t}{t^{d+1}} \right)^{\frac{1}{q}},$$ $$G_{q,{L}}(f)=\left( \int_0^{\infty} \left\|t{L}{e^{-t{L}}} (f)(y) \right\|_X^q \,\frac{{\rm d} t}{t}\right)^{\frac{1}{q}}.$$ We show that for $\underline{\mathrm{any}}$ Banach space $X$, $1 \leq p < \infty$ and $1 < q < \infty$ and $f\in C_c(\mathbb R^d)\otimes X$, there hold \begin{align*} p^{-\frac{1}{q}}\| S_{q,{\sqrt{\Delta}}}(f) \|_p \lesssim_{d, \gamma, \beta} \| S_{q,L}(f) \|_p \lesssim_{d, \gamma, \beta} p^{\frac{1}{q}}\| S_{q,{\sqrt{\Delta}}}(f) \|_p, \end{align*} \begin{align*} p^{-\frac{1}{q}}\| S_{q,L}(f) \|_p \lesssim_{d, \gamma, \beta} \| G_{q,L}(f) \|_p \lesssim_{d, \gamma, \beta} p^{\frac{1}{q}}\| S_{q,L}(f) \|_p, \end{align*} where $\Delta$ is the standard Laplacian; moreover all the orders appeared above are {\it optimal} as $p\rightarrow1$. This, combined with the existing results in [29, 33], allows us to resolve partially Problem 1.8, Problem A.1 and Conjecture A.4 regarding the optimal Lusin type constant and the characterization of martingale type in a recent remarkable work due to Xu [48]. Several difficulties originate from the arbitrariness of $X$, which excludes the use of vector-valued Calder\'on-Zygmund theory. To surmount the obstacles, we introduce the novel vector-valued Hardy and BMO spaces associated with sectorial operators; in addition to Mei's duality techniques and Wilson's intrinsic square functions developed in this setting, the key new input is the vector-valued tent space theory and its unexpected amalgamation with these `old' techniques.