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Refined Hardy inequalities

Hajer Bahouri, Jean-Yves Chemin, Isabelle Gallagher (2006)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

The aim of this article is to present “refined” Hardy-type inequalities. Those inequalities are generalisations of the usual Hardy inequalities, their additional feature being that they are invariant under oscillations: when applied to highly oscillatory functions, both sides of the refined inequality are of the same order of magnitude. The proof relies on paradifferential calculus and Besov spaces. It is also adapted to the case of the Heisenberg group.

Research Article. Multiscale Analysis of 1-rectifiable Measures II: Characterizations

Matthew Badger, Raanan Schul (2017)

Analysis and Geometry in Metric Spaces

A measure is 1-rectifiable if there is a countable union of finite length curves whose complement has zero measure. We characterize 1-rectifiable Radon measures μ in n-dimensional Euclidean space for all n ≥ 2 in terms of positivity of the lower density and finiteness of a geometric square function, which loosely speaking, records in an L2 gauge the extent to which μ admits approximate tangent lines, or has rapidly growing density ratios, along its support. In contrast with the classical theorems...

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