The paper introduces boundary operators for Poincaré-Einstein manifolds and proves higher order trace inequalities.
problem Proving higher order trace inequalities on Poincaré-Einstein manifolds.
method Introducing conformally covariant boundary operators and using them to set up higher order Dirichlet problems.
result Sharp higher order Sobolev trace inequalities on the ball are obtained.
In this paper, we prove several Poincaré inequalities of fractional type on conformally flat manifolds with finite total Q-curvature. This shows a new aspect of the Q-curvature on noncompact complete manifolds.
The paper studies inequalities for fractional GJMS operators on conformal infinity.
problem Deriving comparison inequalities for fractional Yamabe constants.
method Using Poincaré-Einstein manifolds and fractional GJMS operators.
result Two comparison inequalities for fractional Yamabe constants are derived.
Explicit formulas for fractional GJMS operators on hyperbolic spaces and inequalities proved.
problem Proving explicit formulas and inequalities for fractional operators on hyperbolic spaces.
method Scattering theory on hyperbolic space, Helgason-Fourier analysis, and special function analysis.
result Sharp constants in fractional Poincaré-Sobolev and Hardy-Sobolev-Maz'ya inequalities coincide with Euclidean space constants.
Under a spectral assumption on the Laplacian of a Poincaré--Einstein manifold, we establish an energy inequality relating the energy of a fractional GJMS operator of order 2γ∈(0,2) or 2γ∈(2,4) and the energy of the weighted conformal Laplacian or weighted Paneitz operator, respectively. This spectral assumption…
New RDP guarantees for heavy-tailed SDEs and SGD.
problem Characterizing differential privacy for heavy-tailed noise in learning algorithms.
method Rényi flow computations and fractional Poincaré inequalities.
result First RDP guarantees for heavy-tailed SDEs with weaker dependence on dimension.
Growth functions of Coxeter groups and the Poincare series of Kleinian and Fuchsian singularities are 2-tangle q-fractions.
Uniform Poincaré inequalities established for various metric spaces.
problem Establishing uniform Poincaré inequalities on different metric spaces.
method Proper geodesic metric spaces equipped with a Borel measure. Local Poincaré inequality and volume conditions are used to derive uniform Poincaré inequalities.
result Uniform Poincaré inequalities are established for various metric spaces including hyperbolic spaces and covers of compact spaces.
Study connects weighted isoperimetric problems to nonlocal elliptic operator extensions.
problem Sharp inequalities for weighted Poisson integrals and their extremizers.
method Formulates variational problem on conformal metric measure space.
result Sharp inequalities are linked to variational problem on CCE manifolds.
We describe a set of conformally covariant boundary operators associated to the sixth-order GJMS operator on a conformally invariant class of manifolds which includes compactifications of Poincaré--Einstein manifolds. This yields a conformally covariant energy functional for the sixth-order GJMS operator on such manifo…
New sampling method for heavy-tailed distributions using Langevin Algorithm.
problem Sampling from heavy-tailed distributions efficiently.
method Transformed Unadjusted Langevin Algorithm on specific transformations.
result Polynomial-order oracle complexities for certain heavy-tailed densities.
The study improves Poincaré and log-Sobolev inequalities on hyperbolic spaces.
problem Improving Poincaré and log-Sobolev inequalities on hyperbolic spaces.
method Establishing scale-dependent Poincaré-Hardy type identities and choosing suitable parameters, potentials, and vector fields.
result Derives new versions and substantially improves existing inequalities.
New proof shows inequality without restrictions.
problem Sharp inequality relating Yamabe invariants on Poincare-Einstein manifolds.
method New proof without restrictions.
result Sharp inequality holds without restrictions.
We prove a Poincare type inequality for differential forms on compact manifolds by means of a constructive 'globalization' of a local Poincare inequality on convex sets.
Sharp inequalities on Siegel domains and complex hyperbolic spaces established.
problem Establishing inequalities on complex hyperbolic spaces and Siegel domains.
method Helgason-Fourier analysis, Kunze-Stein phenomenon, factorization theorem.
result Sharp Hardy-Adams and Adams type inequalities on Sobolev spaces of any positive fractional order on complex hyperbolic spaces.
Study Poincaré inequality in metric spaces via separating sets.
problem Geometric characterization of Poincaré inequality in metric spaces.
method Properties of separating sets and various notions of energy.
result Equivalence of conditions for 1-Poincaré inequality.
We investigate the possibility of improving the p-Poincaré inequality ∥∇HNu∥p≥Λp∥u∥p on the hyperbolic space, where p>2 and Λp:=[(N−1)/p]p is the best constant for which such inequality holds. We prove several different, and independent, improved inequalities, one of which is …
Using an inverse system of metric graphs as in: J. Cheeger and B. Kleiner, "Inverse limit spaces satisfying a Poincaré inequality", we provide a simple example of a metric space X that admits Poincaré inequalities for a continuum of mutually singular measures.
New Poincaré inequality for differential forms on manifolds.
problem Developing inequalities for differential forms on manifolds.
method Proving a new Poincaré-type inequality and deriving new inequalities involving mean and scalar curvatures.
result Characterized the limiting case of a new inequality involving mean and scalar curvatures of the boundary.
We describe a set of conformally covariant boundary operators associated to the Paneitz operator, in the sense that they give rise to a conformally covariant energy functional for the Paneitz operator on a compact Riemannian manifold with boundary. These operators naturally give rise to a first- and third-order conform…
Proves inequality linking function deviation to gradient norm on compact manifolds.
problem Analyzing coupled elliptic systems on compact manifolds.
method Develops a new Poincaré-Sobolev inequality with a density-free reference average.
result Poincaré constant depends on the density's gradient norm.
The paper proves new inequalities for convex hypersurfaces using centro-affine geometry.
problem Proving inequalities for convex hypersurfaces.
method Introducing a flat logarithmic centro-affine geometry and using Bochner formulas.
result Established new Poincaré and Brunn-Minkowski inequalities.
The paper derives new inequalities on manifolds and applies them to convex hypersurfaces.
problem Deriving new inequalities on manifolds and convex hypersurfaces.
method Using Fourier theory and geometric implications of Poincare-type inequalities.
result Sharp Minkowski-type inequalities, including stability and Alexandrov-Fenchel inequalities.
We prove that complete Riemannian manifolds with polynomial growth and Ricci curvature bounded from below, admit uniform Poincaré inequalities. A global, uniform Poincaré inequality for horospheres in the universal cover of a closed, n-dimensional Riemannian manifold with pinched negative sectional curvature follows …
Sharp inequality linking interior and boundary Yamabe invariants on specific manifolds.
problem Relating Yamabe invariants on asymptotically Poincare-Einstein manifolds.
method Established a sharp inequality using lower Ricci curvature bounds.
result Sharp inequality relating type II Yamabe invariant of the interior to the Yamabe invariant of the conformal infinity.
We study the validity of the Lp inequality for the Riesz transform when p>2 and of its reverse inequality when p<2 on complete Riemannian manifolds under the doubling property and some Poincaré inequalities.
The article proves a Poincaré inequality for hypersurfaces and applies it to rigidity results.
problem Proving rigidity results for hypersurfaces under curvature constraints.
method Using a divergence formula for symmetric endomorphisms, the article deduces a Poincaré type inequality and applies it to higher-order mean curvature of hypersurfaces.
result The article proves several rigidity results for complete r-minimal hypersurfaces.
The paper proves inequalities on Finsler manifolds under Ricci curvature bounds.
problem Proving (p,q)-Sobolev and Nash inequalities on Finsler metric measure manifolds. method Global p-Poincaré inequality, (p,q)-Sobolev inequality, Nash inequality derivation. result Established global optimal (p,q)-Sobolev inequality with a sharp constant. Study complete manifolds with weighted Poincaré inequality and Ricci curvature bounds.
problem Understanding the structure of complete manifolds with specific curvature and inequality conditions.
method Analyzing manifolds with weighted Poincaré inequality and Ricci curvature bounds.
result Obtained splitting results for manifolds with non-zero weight function limit at infinity.
Quantifies fractional isoperimetric inequality with strong control over boundary oscillation.
problem Fractional isoperimetric inequality and its quantitative aspects.
method Regularization process with a new spirit.
result Stability estimates for fractional Cheeger inequality.
We study in this paper the fractional Yamabe problem first considered by Gonzalez-Qing on the conformal infinity (Mn,[h]) of a Poincaré-Einstein manifold (Xn+1,g+) with either n=2 or n≥3 and (Mn,[h]) is locally flat - namely (M,h) is locally conformally flat. However, as for the classic…
The paper extends Heintze-Karcher inequalities to fractional Q-curvature.
problem Extending Heintze-Karcher inequalities to fractional Q-curvature.
method Generalization of Heintze-Karcher inequalities to fractional Q-curvature on conformally compact Einstein manifolds.
result Rigidity theorems for specific values of γ.
The development of global sensitivity analysis of numerical model outputs has recently raised new issues on 1-dimensional Poincaré inequalities. Typically two kind of sensitivity indices are linked by a Poincaré type inequality, which provide upper bounds of the most interpretable index by using the other one, cheaper …
A carpet is a metric space homeomorphic to the Sierpinski carpet. We characterize, within a certain class of examples, non-self-similar carpets supporting curve families of nontrivial modulus and supporting Poincaré inequalities. Our results yield new examples of compact doubling metric measure spaces supporting Poinca…
Short note proves Poincaré inequality for 4-manifold forms.
problem Quantifying Poincaré inequality for one forms on 4-manifolds.
method Hodge theory on orbifolds, comparison of fundamental groups, spectral convergence, degeneration to orbifolds.
result First non-trivial global Poincaré inequality without higher curvature assumptions.
Sharp fractional Sobolev inequalities on closed manifolds identified.
problem Critical fractional Sobolev embedding on closed Riemannian manifolds.
method Intrinsic heat-kernel based framework, determining optimal coefficients, proving sharp inequalities.
result Sharp p-power inequality and almost sharp inequality established. Study on functional inequalities on simple edge spaces.
problem Whether classical functional inequalities hold in simple edge spaces.
method Analyzing Sobolev and Poincaré inequalities, proving optimality of Sobolev constant.
result Optimality result concerning the B-constant of the Sobolev inequality.
We prove structure theorems for complete manifolds satisfying both the Ricci curvature lower bound and the weighted Poincaré inequality. In the process, a sharp decay estimate for the minimal positive Green's function is obtained. This estimate only depends on the weight function of the Poincaré inequality, and yields …
New proof of surface group theorem for 2D Poincaré duality groups.
problem Characterizing groups with specific algebraic properties.
method Analyzing amenability and homological isoperimetric inequalities.
result Groups satisfying certain conditions are either amenable or have linear homological isoperimetric inequalities.
Sharp inequality for compactifying Poincaré-Einstein manifolds.
problem Proving a sharp relative comparison inequality for compactifying Poincaré-Einstein manifolds.
method Proved a sharp relative comparison inequality for type-I Escobar-Yamabe compactification.
result Confirms a conjecture by proving the sharp relative comparison inequality.
We show that any d-Ahlfors regular subset of Rn supporting a weak (1,d)-Poincaré inequality with respect to surface measure is uniformly rectifiable.
This paper extends the convergence analysis of Langevin Monte Carlo beyond Poincaré inequalities.
problem Analyzing convergence of Langevin Monte Carlo under various functional inequalities.
method Establishing upper and lower bounds for Langevin diffusions and LMC under weak Poincaré inequalities.
result Explicitly quantifies the effect of the initializer on the performance of LMC algorithm.
Sharp lower bound found for geodesic ball eigenvalues.
problem Finding the minimum eigenvalue for geodesic balls.
method Applied Li-Schoen's uniform Poincare inequality for non-negative Ricci curvature manifolds.
result Sharp lower bound of the first Dirichlet eigenvalue for geodesic balls.
The study proves topological rigidity for certain geometric shapes using Poincaré inequalities.
problem Proving topological rigidity for translators and self-expanders in mean curvature flow.
method Abstract structure theorem for weighted manifolds, Poincaré inequality, and topological control.
result Full topological control on translators and self-expanders under stability or curvature assumptions.
The paper derives inequalities and formulas for generalized Ricci flow.
problem Understanding and characterizing generalized Ricci flow.
method Using Bochner formula and adapted Malliavin gradient, the paper derives inequalities and characterizes generalized Ricci flow.
result Characterizations of generalized Ricci flow via inequalities for the associated Malliavin gradient.
Graphs with nonnegative Bakry-Émery curvature have volume doubling and Poincaré inequalities.
problem Proving properties of graphs with specific curvature conditions.
method Graph-theoretic modified nonlinear heat-flow method, including point-mass consequences and diffusive exit-time control.
result Volume doubling and Poincaré inequalities for graphs with nonnegative Bakry-Émery curvature.
The study improves Bochner inequality on Finsler manifolds to derive important inequalities.
problem Improving Bochner inequality on Finsler manifolds to derive new inequalities.
method Using improved Bochner inequality and its integrated form, the study derives a sharp Poincaré-Lichnerowicz inequality, a new proof for logarithmic Sobolev inequality, and an estimate of geodesic ball volumes.
result Derivation of new inequalities and estimates on Finsler manifolds.
In this paper we prove L∞ type gap theorems in Yang-Mills theory for complete four-dimensional manifolds with a weighted Poincaré inequality. We apply the theorems to a broad class of complete manifolds satisfying weighted Poincaré inequalities. In particular, we obtain a gap theorem on the Euclidean space wi…