Modulus and the Poincaré inequality on metric measure spaces

By S. Keith. 2003

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DIFFERENTIABLE STRUCTURES ON METRIC MEASURE SPACES: A PRIMER (2011, methodology): “As mentioned in the Overview, this exposition is based on the work of Cheeger [Che99] and Keith [Kei04].”
Weak curvature conditions and functional inequalities (2005, background): “It is known that the property of admitting a local Poincaré inequality is preserved under measured Gromov–Hausdorff limits [7,8].”
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.”
Moser iteration for (quasi)minimizers on metric spaces (2006, background): “By, e.g., Theorem 4 in Keith [ 23 ] (which also holds for α = 0), X supports a weak (1, 1)-Poincare inequality.”
Geometric Implications of the Poincaré Inequality (2007, methodology): “Or we may replace the upper gradient by the local Lipschitz constant, see [14].”
Geometry and Analysis of Dirichlet forms (2012, background): “1, we recall the notion of an upper gradient; see [14] and also [20, 42].”
Cheeger-harmonic functions in metric measure spaces revisited (2013, methodology): “According to [17] the Poincaré inequality here coincide s with the one for all measurable functions and their upper gradien ts, as introduced in [14].”