Hardy Inequalities and Functional Analysis on Manifolds

Hardy inequalities form a cornerstone of functional analysis on manifolds, providing estimates that compare integrals of a function and its gradient relative to the manifold’s geometry. Originating from classical inequalities in Euclidean space, these results have been generalised to curved settings, non-compact spaces and weighted contexts. On a Riemannian manifold, curvature and topology influence the best constants and the existence of remainder terms. Such inequalities underpin spectral estimates for Laplace and Schrödinger operators, furnish control on singular potentials and inform uncertainty principles. Extensions include Rellich inequalities of higher order and weighted versions on metric measure spaces, reflecting the interplay between geometry and analysis. Recent efforts have focused on identifying sharp constants, establishing integral identities that yield remainder terms, and unifying approaches across settings as diverse as hyperbolic space, Carnot groups and spaces equipped with subelliptic structures. These developments deepen understanding of the analytic structure of manifolds and bear upon quantum mechanics, the study of elliptic and parabolic partial differential equations, and geometric flows.

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Hardy Inequalities and Functional Analysis on Manifolds publication trend

The graph below shows the total number of articles in hardy inequalities and functional analysis on manifolds across all publications each year (not limited to Nature Index journals).

Technical terms

Hardy inequality: An integral estimate relating the norm of a function to the norm of its gradient, often involving a singular weight.

Riemannian manifold: A smooth manifold endowed with an inner product on each tangent space, inducing geometry and volume.

Witten–Laplace operator: A deformation of the Laplace–Beltrami operator that incorporates a potential function, central to analysis on metric measure spaces.

Rellich inequality: A higher-order analogue of the Hardy inequality, comparing norms of higher-order derivatives.

Bessel pair: A pair of weight functions used to construct sharp integral identities and remainder terms in Hardy–Rellich contexts.

References

1. Some Inequalities of Hardy Type Related to Witten–Laplace Operator on Smooth Metric Measure Spaces. Mathematics (2022).
2. Improved Hardy and Rellich inequalities on Riemannian manifolds. Transactions of the American Mathematical Society (2009).
3. Hardy-Poincaré, Rellich and uncertainty principle inequalities on Riemannian manifolds. Transactions of the American Mathematical Society (2013).
4. A unified approach to weighted Hardy type inequalities on Carnot groups. Discrete and Continuous Dynamical Systems (2017).
5. Hardy–Rellich and second order Poincaré identities on the hyperbolic space via Bessel pairs. Calculus of Variations and Partial Differential Equations (2022).