Newtonian spaces: An extension of Sobolev spaces to metric measure spaces

By Nageswari Shanmugalingam. 2000

802 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Mechanics

View Paper: https://www.semanticscholar.org/paper/93ee82306e358d39ca2a40a71313c39f9a2156f5

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In Applied Mathematics

In Computational Mechanics

Citations in the corpus

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].”
The Poincare inequality is an open ended condition (2008, methodology): “Here we consider the Sobolev space H1,p(X), p ≥ 1, introduced by Cheeger in [5], the Newtonian space N1,p(X) introduced by Shanmugalingam in [46], and the Sobolev space M1,p(X) introduced by Haj lasz in [16].”
Metric and geometric quasiconformality in Ahlfors regular Loewner spaces (2001, background): “The following concept was introduced by Shanmugalingam [62] in connection with her study of Sobolev functions on arbitrary metric spaces.”
Space of Ricci Flows I (2009, background): “10 of [75], or the discussion on page 440 of [13].”
Pointwise properties of functions of bounded variation in metric spaces (2014, neutral): “The collection of all upper gradients, together, play the role of the modulus of the weak gradient of a Sobolev function in the metric setting, see [14].”
Yau's gradient estimates on Alexandrov spaces (2010, methodology): “ we have, for almost all r ∈ (0,δ0), (2.15) ? ∂Bp(r) f(x) − f(p) dvol(x) = r2 2 · ? Σp Hp f(ξ,ξ)dξ+o(r2). 2.2. Sobolev spaces. Several diﬀerent notions of Sobolev spaces have been established, see[8, 19, 43, 20, 24]1. They coincide each other on Alexandrov spaces. Let M be an n-dimensional Alexand”
CALCULUS, HEAT FLOW AND CURVATURE-DIMENSION BOUNDS IN METRIC MEASURE SPACES (2019, background): “In more recent times, Shanmugalingam [2000] adapted this concept to the metric measure setting, with the introduction of theNewtonian spaceN (X; d; m); notice that the notion of p-Modulus immediately extends to the metric measure setting, understanding curvilinear integrals as in (2-1).”
A difference characterization of Besov and Triebel-Lizorkin spaces on RD-spaces (2009, background): “Since Hajłasz in [13] introduced Sobolev spaces on any metric measure space, a series of papers has been devoted to the construction and investigation of Sobolev spaces of various types on metric measure spaces; see, for example, [15, 30, 8, 10, 31, 42, 20, 21, 28, 5, 14, 9, 49].”
Riemannian Ricci curvature lower bounds in metric measure spaces with -finite measure (2012, methodology): “Shanmugalingam proved in [30] that functions with an upper gradient in L(2)(X, n) are absolutely continuous along a Mod2,n-a.”
Lebesgue points for Sobolev functions on metric spaces (2002, background): “However, it is easy to modify our argument to cover the spaces defined in [3], [6], [12], [13] and [21].”