Let L be a sectorial operator of type \(\alpha \) ( \(0 \le \alpha < \pi /2\) ) on \(L^2({\mathbb {R}}^d)\) with the kernels of \(\{e^{-tL}\}_{t>0}\) satisfying certain size and regularity conditions. Define \(\begin{aligned} S_{q,L}(f)(x)= & \left( \int _0^{\infty }\int _{|y-x| < t} \left\| tL{e^{-tL}} (f)(y) \right\| _X^q \,\frac{\textrm{d}y\textrm{d}t}{t^{d+1}} \right) ^{\frac{1}{q}}, \\ G_{q,{L}}(f)(x)= & \left( \int _0^{\infty } \left\| t{L}{e^{-t{L}}} (f)(x) \right\| _X^q \,\frac{\textrm{d}t}{t} \right) ^{\frac{1}{q}}. \end{aligned}\) We show that for \(\underline{\textrm{any}}\) Banach space X, \(1 \le p < \infty \) and \(1< q < \infty \) and \(f\in C_c(\mathbb {R}^d)\otimes X\) , there hold \(\begin{aligned}&p^{-\frac{1}{q}}\Vert S_{q,{\sqrt{\Delta }}}(f) \Vert _p \lesssim _{d, \gamma , \beta } \left\| S_{q,L}(f) \right\| _p \lesssim _{d, \gamma , \beta } p^{\frac{1}{q}}\Vert S_{q,{\sqrt{\Delta }}}(f) \Vert _p,\\&p^{-\frac{1}{q}}\Vert S_{q,L}(f) \Vert _p \lesssim _{d, \gamma , \beta } \Vert G_{q,L}(f) \Vert _p \lesssim _{d, \gamma , \beta } p^{\frac{1}{q}}\Vert S_{q,L}(f) \Vert _p, \end{aligned}\) where \(\Delta \) is the standard Laplacian; moreover all the orders appeared above are optimal as \(p\rightarrow 1\) . This, combined with the existing results in Martínez et al. (Adv Math 203(2):430–475, 2006) and Ouyang and Xu (Can J Math 62(4):827–844, 2010), 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 (Holomorphic functional calculus and vector-valued Littlewood–Paley–Stein theory for semigroups. J Euro Math Soc, 2024. arXiv:2105.12175). Several difficulties originate from the arbitrariness of X, which excludes the use of vector-valued Calderón–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.
