确定性与随机最大正则性的必要条件
Necessary conditions for deterministic and stochastic maximal regularity
- Delft University of Technology（代尔夫特理工大学）
- Karlsruhe Institute of Technology (KIT)（卡尔斯鲁厄理工学院）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
该论文研究巴拿赫空间几何在确定性与随机最大正则性中的作用，构造反例表明UMD假设不可省略，证明随机卷积算子的R-有界性条件$(S_p)$的必要性及相关等价性等结论。
中文摘要 AI 辅助
我们研究巴拿赫空间几何在确定性与随机最大正则性中的作用。我们首先构造一个例子，表明在Weis基于R-扇性刻画最大$L^p$-正则性的结论中，UMD假设不可省略。将该构造与基础空间的2-凹化上随机最大正则性和确定性最大正则性的等价性相结合，我们得到一个作用于2型UMD巴拿赫函数空间上的算子，该算子具有角度为0的有界$H^\text{∞}$演算，但对所有$p∈[2,∞)$均不满足随机最大$L^p$-正则性（SMR$_p$）。受此例子启发，我们更深入地研究了SMR$_p$背后的巴拿赫空间几何假设，即随机卷积算子的R-有界性条件$(S_p)$。对于2型UMD空间$X$，我们证明该条件不仅充分，而且对两类典型测试算子也是必要的：拉德马赫空间上的对角乘子，以及对$q>2$，$L^q(\boldsymbol{R}^d;X)$上的拉普拉斯算子。最后，我们证明其区间核与指数核表述等价，且在端点$p=2$处，条件$(S_2)$成立当且仅当$X$同构于希尔伯特空间。
英文摘要
We study the role of Banach space geometry in deterministic and stochastic maximal regularity. We first construct an example showing that the UMD assumption in the characterisation of maximal $L^p$-regularity in terms of $R$-sectoriality cannot be omitted. Combining this construction with an equivalence between stochastic maximal regularity and deterministic maximal regularity on the $2$-concavification of the underlying space, we obtain an operator on a UMD Banach function space of type $2$ that has a bounded $H^\infty$-calculus of angle zero, but fails stochastic maximal $L^p$-regularity (SMR$_p$) for every $p\in[2,\infty)$. Motivated by this example, we study the Banach space geometry hypothesis underlying SMR$_p$ more closely. This is an $R$-boundedness condition $(S_p)$ for stochastic convolution operators. For UMD spaces $X$ of type $2$, we show that this condition is not only sufficient, but also necessary for two canonical test operators: a diagonal multiplier on a Rademacher space and, for $q>2$, the Laplacian on $L^q(\mathbb R^d;X)$. Finally, we prove that its interval-kernel and exponential-kernel formulations are equivalent and that, at the endpoint $p=2$, condition $(S_2)$ holds if and only if $X$ is isomorphic to a Hilbert space.
