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

Nageswari Shanmugalingam

Revista Matemática Iberoamericana (2000)

  • Volume: 16, Issue: 2, page 243-279
  • ISSN: 0213-2230

Abstract

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This paper studies a possible definition of Sobolev spaces in abstract metric spaces, and answers in the affirmative the question whether this definition yields a Banach space. The paper also explores the relationship between this definition and the Hajlasz spaces. For specialized metric spaces the Sobolev embedding theorems are proven. Different versions of capacities are also explored, and these various definitions are compared. The main tool used in this paper is the concept of moduli of path families.

How to cite

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Shanmugalingam, Nageswari. "Newtonian spaces: An extension of Sobolev spaces to metric measure spaces.." Revista Matemática Iberoamericana 16.2 (2000): 243-279. <http://eudml.org/doc/39609>.

@article{Shanmugalingam2000,
abstract = {This paper studies a possible definition of Sobolev spaces in abstract metric spaces, and answers in the affirmative the question whether this definition yields a Banach space. The paper also explores the relationship between this definition and the Hajlasz spaces. For specialized metric spaces the Sobolev embedding theorems are proven. Different versions of capacities are also explored, and these various definitions are compared. The main tool used in this paper is the concept of moduli of path families.},
author = {Shanmugalingam, Nageswari},
journal = {Revista Matemática Iberoamericana},
keywords = {Espacios de Sobolev; Espacios de Banach; Sobolev space; metric measure space; doubling space; gradient; Poincaré inequality; embedding theorem; capacity; path family},
language = {eng},
number = {2},
pages = {243-279},
title = {Newtonian spaces: An extension of Sobolev spaces to metric measure spaces.},
url = {http://eudml.org/doc/39609},
volume = {16},
year = {2000},
}

TY - JOUR
AU - Shanmugalingam, Nageswari
TI - Newtonian spaces: An extension of Sobolev spaces to metric measure spaces.
JO - Revista Matemática Iberoamericana
PY - 2000
VL - 16
IS - 2
SP - 243
EP - 279
AB - This paper studies a possible definition of Sobolev spaces in abstract metric spaces, and answers in the affirmative the question whether this definition yields a Banach space. The paper also explores the relationship between this definition and the Hajlasz spaces. For specialized metric spaces the Sobolev embedding theorems are proven. Different versions of capacities are also explored, and these various definitions are compared. The main tool used in this paper is the concept of moduli of path families.
LA - eng
KW - Espacios de Sobolev; Espacios de Banach; Sobolev space; metric measure space; doubling space; gradient; Poincaré inequality; embedding theorem; capacity; path family
UR - http://eudml.org/doc/39609
ER -

Citations in EuDML Documents

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  1. Marcello Lucia, Michael J. Puls, The p-Royden and p-Harmonic Boundaries for Metric Measure Spaces
  2. Nicola Gigli, An Overview of the Proof of the Splitting Theorem in Spaces with Non-Negative Ricci Curvature
  3. Luigi Ambrosio, Nicola Gigli, Giuseppe Savarè, Heat Flow and Calculus on Metric Measure Spaces with Ricci Curvature Bounded Below - the Compact Case
  4. Koskela, Pekka, Metric Sobolev spaces
  5. Guillaume Lupo-Krebs, Hervé Pajot, Dimensions conformes, espaces Gromov-hyperboliques et ensembles autosimilaires
  6. Zoltán M. Balogh, Jeremy T. Tyson, Kevin Wildrick, Dimension Distortion by Sobolev Mappings in Foliated Metric Spaces
  7. Anders Björn, Jana Björn, Approximations by regular sets and Wiener solutions in metric spaces
  8. Zahra Sinaei, Riemannian Polyhedra and Liouville-Type Theorems for Harmonic Maps
  9. H. Hakkarainen, R. Korte, P. Lahti, N. Shanmugalingam, Stability and Continuity of Functions of Least Gradient
  10. Nijjwal Karak, Generalized Lebesgue points for Sobolev functions

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