AI 中文总结
研究在巴拿赫空间中，由阿贝尔遍历扇形算子\(A\)及无界算子\(\partial\)构成的系统，通过反向里斯估计与谱隙条件，推出庞加莱不等式及伴随散度不等式，该方法统一且灵活，能恢复并扩展相关定理，适用于多种几何情形。
AI 中文摘要
在巴拿赫空间\(X\)上的阿贝尔遍历扇形算子\(A\)以及另一个巴拿赫空间\(Y\)中的子空间\(X\)上定义的无界算子\(\partial\)层面上，我们证明对于某个\(0 < \alpha < 1\)的单个反向里斯估计\(\|A^\alpha x\|_X \lesssim \|\partial x\|_Y\)，结合条件\(0 \in \rho(A_0)\)（其中\(A_0\)是\(A\)在\(A\)值域闭包上的部分），蕴含庞加莱不等式\(\|x - P(x)\|_X \lesssim \|\partial x\|_Y\)，其中\(P\)是到\(A\)核上的阿贝尔遍历投影。\(0 \in \rho(A_0)\)是谱隙的自然抽象替代，在希尔伯特情形下已很尖锐。我们还得到一个伴随散度不等式。论证简短但原理真正具有统一性：它在相同基础上涵盖交换和非交换情形且可用于任意巴拿赫空间。结果，我们恢复并大幅扩展了焦、罗、扎宁和周[CMP2024]关于（可能非交换的）\(\mathrm{L}^p\)空间的一个近期定理。然后我们展示该方法在广泛几何情形下的灵活性，从黎曼流形、李群、度量测度空间、自旋流形到真正的非交换情形如量子群、舒尔乘子半群、\(q\)-奥恩斯坦 - 乌伦贝克半群和量子环面，在其中我们有时建立新不等式，有时从单个原理恢复经典不等式。
英文摘要
Working at the level of a sectorial operator $A$ on a Banach space $X$ and an unbounded operator $\partial$ defined on a subspace of $X$ in another Banach space $Y$, we show that a single reverse Riesz estimate $\|A^αx\|_X \lesssim \|\partial x\|_Y$ for some $0 < α< 1$, combined with the condition $0 \in ρ(A_0)$, where $A_0$ is the part of $A$ on the closure of the range of $A$, implies the Poincaré inequality $\|x - P(x)\|_X \lesssim \|\partial x\|_Y$, where $P$ is the Abel-ergodic projection onto the kernel of $A$. The condition $0 \in ρ(A_0)$ is the natural abstract substitute for a spectral gap, and is sharp already in the Hilbertian case. We also obtain a companion divergence inequality. The arguments are remarkably short, yet the principle is genuinely unifying: it covers commutative and noncommutative situations on the same footing and can be used with arbitrary Banach spaces. As a consequence, we recover Poincaré inequalities associated with hypercontractive semigroups and extend their validity to semigroups satisfying only a reduced spectral gap and a reverse Riesz estimate. We then illustrate the flexibility of the method across a wide spectrum of geometries, ranging from Riemannian manifolds, Lie groups, metric measure spaces, spin manifolds to genuinely noncommutative settings such as group von Neumann algebras, semigroups of Schur multipliers, $q$-Ornstein-Uhlenbeck semigroups and quantum tori, where we sometimes establish new inequalities and otherwise recover classical ones from a single principle.
Comments60 pages, improvements, new results
