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.
Categories: math.FA