A reverse Riesz estimate combined with a spectral gap implies a Poincaré inequality

Working at the level of an Abel-ergodic sectorial operator $A$ on a Banach space $X$ and an unbounded operator $\mathrm{\partial }$ defined on a subspace $X$ in another Banach space $Y$, we show that a single reverse Riesz estimate $\mathrm{\parallel }{A}^{\alpha }x{\mathrm{\parallel }}_X\lesssim \mathrm{\parallel }\mathrm{\partial }x{\mathrm{\parallel }}_Y$ for some $0<\alpha <1$, combined with the condition $0\in \rho (A_0)$, where $A_0$ is the part of $A$ on the closure of the range of $A$, implies the Poincaré inequality $\mathrm{\parallel }x-P(x){\mathrm{\parallel }}_X\lesssim \mathrm{\parallel }\mathrm{\partial }x{\mathrm{\parallel }}_Y$, where $P$ is the Abel-ergodic projection onto the kernel of $A$. The condition $0\in \rho (A_0)$ 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, and considerably extend, a recent theorem of Jiao, Luo, Zanin and Zhou [CMP2024] on (possibly noncommutative) ${\mathrm{L}}^p$-spaces. 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 quantum groups, 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.