Vector-valued Littlewood-Paley-Stein theory
报告题目(Title)：Vector-valued Littlewood-Paley-Stein theory
报告人(Speaker)：许全华教授 (哈尔滨工业大学 & 法国贝桑松大学)
地点(Place)：腾讯会议号 202358667
时间(Time)：2020年7月31日16:00-17:00
邀请人(Inviter)：杨大春，袁文，周渊
报告摘要
Let $(T_t)_{t>0}$ be a symmetric diffusion Markov semigroup on a $\sigma$-finite measure space $(\O, \mathcal F, \mu)$ and $\mathsf F$ its fixed point space. A celebrated classical theorem of Stein asserts that for every $1
$$ \|f-\mathsf F(f)\|_{L_p(\Omega)}\approx \left\|\left(\int_0^\infty\Big |t\frac{\partial}{\partial t} T_tf\Big|^2\,\frac{dt}t \right)^{1/2} \right \|_{L_p(\Omega)} \,, \quad\forall\, f\in L_p(\Omega),$$
where the equivalence constants depend only on $p$. This result considerably extends the classical Littlewood-Paley inequality that concerns the classical Poisson semigroup on $R^d$.
In view of vector-valued harmonic analysis and martingale theory, it is natural to consider Stein's theorem for functions with values in a Banach space $X$. It is then quite easy to show that the equivalence
$$ \|f-\mathsf F(f)\|_{L_p(\Omega;X)}\approx \left\|\left(\int_0^\infty\big \|t\frac{\partial}{\partial t} T_t f\big\|_X^2\,\frac{dt}t\right)^{1/2}\right \|_{L_p(\Omega;X)}\,,\quad\forall\, f\in L_p(\Omega;X)$$
holds for some $1
We have investigated this problem in a series of articles (the first going back to 1998). In the latest one still in finalization, we largely weaken the assumption on $(T_t)_{t>0}$ that is only supposed to be a semigroup of positive contractions on $L_p(\Omega)$ for one fixed $1
