A reverse Riesz estimate combined with a spectral gap implies a Poincaré inequality
Abstract
Working at the level of a sectorial operator A on a Banach space X and an unbounded operator ∂ defined on a subspace of X in another Banach space Y, we show that a single reverse Riesz estimate |Aαx|X ≲ |∂ x|Y for some 0 < α< 1, combined with the condition 0 ∈ ρ(A0), where A0 is the part of A on the closure of the range of A, implies the Poincaré inequality |x − P(x)|X ≲ |∂ x|Y, where P is the Abel-ergodic projection onto the kernel of A. The condition 0 ∈ ρ(A0) 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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BibTeX
@misc{arhancet2026reverse,
title = {{A reverse Riesz estimate combined with a spectral gap implies a Poincaré inequality}},
author = {Cédric Arhancet},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.08322},
eprint = {2607.08322},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.08322},
}
RIS
TY - UNPB
TI - A reverse Riesz estimate combined with a spectral gap implies a Poincaré inequality
AU - Cédric Arhancet
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2607.08322
UR - https://arxiv.org/abs/2607.08322
AB - Working at the level of a sectorial operator A on a Banach space X and an unbounded operator ∂ defined on a subspace of X in another Banach space Y, we show that a single reverse Riesz estimate |A^αx|_X ≲ |∂ x|_Y for some 0 < α< 1, combined with the condition 0 ∈ 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 ≲ |∂ x|_Y, where P is the Abel-ergodic projection onto the kernel of A. The condition 0 ∈ 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.
ER -