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

Abstract

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 of $X$ in another Banach space $Y$, we show that a single reverse Riesz estimate $\Vert A^{\alpha }x{\Vert }_X\lesssim \Vert \partial x{\Vert }_Y$ 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 $\Vert x-P(x){\Vert }_X\lesssim \Vert \partial x{\Vert }_Y$, 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 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.

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