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

Nageswari Shanmugalingam
2000 Revista matemática iberoamericana  
This paper studies a possible de nition of Sobolev spaces in abstract metric spaces, and answers in the a rmative the question whether this de nition yields a Banach space. The paper also explores the relationship between this de nition and the Haj lasz spaces. For specialized metric spaces the Sobolev embedding theorems are proven. Di erent v ersions 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
more » ... f moduli of path families.
doi:10.4171/rmi/275 fatcat:hfrepcsgxvbgvofzbg7e6bxaj4