The Poincare inequality is an open ended condition

By S. Keith, Xiao Zhong. 2008

302 citations recorded by the source corpus.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/eda7bb7619b53195e5f923dc3a1d34774a514d10

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Yuan Zhou
2. Dachun Yang
3. Pekka Koskela
4. Heli Tuominen
5. Renjin Jiang
6. T. Coulhon
7. Anders Bjorn
8. Amiran Gogatishvili

In Mathematical Physics

1. Dachun Yang
2. Yuan Zhou
3. Pekka Koskela
4. Renjin Jiang
5. Heli Tuominen
6. Anders Bjorn
7. T. Coulhon
8. Adam S. Sikora

Citations in the corpus

• Boundary regularity for p-harmonic functions and solutions of the obstacle problem on metric spaces (2006, background): “By Keith–Zhong [17] it follows that X supports a weak (1, q)-Poincaré inequality for some q ∈ [1, p), which was earlier a standard assumption.”
• A weak Kellogg property for quasiminimizers (2006, background): “By Keith–Zhong [17] it follows that X supports a weak (1, q)-Poincaré inequality for some q ∈ [1, p), which was earlier a standard assumption.”
• A difference characterization of Besov and Triebel-Lizorkin spaces on RD-spaces (2009, background): “Of particular interest is the study of functional inequalities, like Sobolev and Poincaré inequalities, on metric measure spaces; see, for example, [29, 15, 20, 12, 11, 27].”
• Pointwise Characterizations of Besov and Triebel-Lizorkin Spaces and Quasiconformal Mappings (2010, unknown): “13 of [16], [17] and the Lebesgue-Radon-Nikodym theorem.”
• Characterizations of Besov and Triebel–Lizorkin spaces on metric measure spaces (2011, background): “9] and [15]), where Ṅ1, p(X ) is the Newtonian Sobolev space introduced in [23].”
• Sobolev Mappings between Manifolds and Metric Spaces (2009, result): “Indeed, prior to the work of Keith and Zhong it was known that the spaces are equal provided that the space supports the q-Poincaré inequality for some 1 6 q p (see, for example, [29, Theorem 11.3]).”
• Moser iteration for (quasi)minimizers on metric spaces (2006, background): “It is noteworthy that by a result of Keith and Zhong [24] a complete metric space equipped with a doubling measure that supports a weak (1, p)-Poincaré inequality, admits a weak (1, q)-Poincaré inequality.”
• Measure density and extendability of Sobolev functions (2008, background): “Since X supports the (1, p)Poincaré inequality, N(X) = M(X) by [15].”
• Gradient estimates for heat kernels and harmonic functions (2017, methodology): “(i) By the self-improving property of (P2) from [77] (see Appendix A.”
• A characterization of Hajłasz–Sobolev and Triebel–Lizorkin spaces via grand Littlewood–Paley functions☆ (2009, background): “These different approaches result in the same function class if the underlying space supports a suitable Poincaré inequality [25].”
• Local Hölder continuity for doubly nonlinear parabolic equations (2010, background): “Moreover, by a recent result in [16], the weak (1, p)-Poincaré inequality and the doubling condition also imply the (1, q)-Poincaré inequality for some q < p, that is”