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

Nageswari Shanmugalingam — 2000

Revista Matemática Iberoamericana

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...

Singular functions on metric measure spaces.

Ilkka Holopainen;  Nageswari Shanmugalingam — 2002

Collectanea Mathematica

On relatively compact domains in metric measure spaces we construct singular functions that play the role of Green functions of the p-Laplacian. We give a characterization of metric spaces that support a global version of such singular function, in terms of capacity estimates at infinity of such metric spaces. In addition, when the measure of the space is locally Q-regular, we study quasiconformal invariance property associated with the existence of global singular functions.

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