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Best constants in the vector-valued Littlewood-Paley-Stein theory (2401.13932v1)

Published 25 Jan 2024 in math.FA

Abstract: Let $L$ be a sectorial operator of type $\alpha$ ($0 \leq \alpha < \pi/2$) on $L2(\mathbb{R}d)$ with the kernels of ${e{-tL}}_{t>0}$ satisfying certain size and regularity conditions. Define $$ S_{q,L}(f)(x) = \left(\int_0{\infty}\int_{|y-x| < t} |tL{e{-tL}} (f)(y) |Xq \,\frac{{\rm d} y{\rm d} t}{t{d+1}} \right){\frac{1}{q}},$$ $$G{q,{L}}(f)=\left( \int_0{\infty} \left|t{L}{e{-t{L}}} (f)(y) \right|Xq \,\frac{{\rm d} t}{t}\right){\frac{1}{q}}.$$ We show that for $\underline{\mathrm{any}}$ Banach space $X$, $1 \leq p < \infty$ and $1 < q < \infty$ and $f\in C_c(\mathbb Rd)\otimes X$, 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.

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