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

Abstract

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 $$S_{q,L}(f)(x)={\left({\int }_0^{\infty }{\int }_{|y-x|<t}\Vert tL{e}^{-tL}(f)(y){\Vert }_X^q\frac{\mathrm{d}y\mathrm{d}t}{t^{d+1}}\right)}^{\frac{1}{q}},$$ $$G_{q,L}(f)={\left({\int }_0^{\infty }{\Vert tL{e}^{-tL}(f)(y)\Vert }_X^q\frac{\mathrm{d}t}{t}\right)}^{\frac{1}{q}}.$$ We show that for $\underline{\mathrm{any}}$ Banach space $X$, $1\le p<\infty$ and $1<q<\infty$ and $f\in C_c({\mathbb{R}}^d)\otimes X$, there hold \begin{align*} p^{-\frac{1}{q}}\| S_{q,{\sqrtΔ}}(f) \|_p \lesssim_{d, γ, β} \| S_{q,L}(f) \|_p \lesssim_{d, γ, β} p^{\frac{1}{q}}\| S_{q,{\sqrtΔ}}(f) \|_p, \end{align*} \begin{align*} p^{-\frac{1}{q}}\| S_{q,L}(f) \|_p \lesssim_{d, γ, β} \| G_{q,L}(f) \|_p \lesssim_{d, γ, β} 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 $p\to 1$. 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 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.

References in corpus (1)

• Conical square function estimates in UMD Banach spaces and applications to H-infinity functional calculi