A reverse Riesz estimate combined with a spectral gap implies a Poincar\'e inequality

Working at the level of an Abel-ergodic sectorial operator $A$ on a Banach space $X$ and an unbounded operator $\partial$ defined on a subspace $X$ in another Banach space $Y$, we show that a single reverse Riesz estimate $\|A^\alpha x\|_X \lesssim \|\partial x\|_Y$ 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 $\|x - P(x)\|_X \lesssim \|\partial x\|_Y$, 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.