Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

Zbl 0938.46037
Franchi, Bruno; Hajłasz, Piotr; Koskela, Pekka
Definitions of Sobolev classes on metric spaces. (English)
[J] Ann. Inst. Fourier 49, No.6, 1903-1924 (1999). ISSN 0373-0956; ISSN 1777-5310/e

Let $(S,d,\mu)$ be a set $S$ equipped with a metric $d$ and a locally finite Borel measure $\mu$ satisfying the doubling condition. There are several ways to introduce Sobolev spaces: Let $1\le p<\infty$. Then $u\in M^1_p(S,d,\mu)$ if $u\in L_p(S)$ and $$|u(x)- u(y)|\le d(x,y)(g(x)+ g(y))\quad\text{for some }0\le g\in L_p(S).$$ Or: $u\in P^1_p(S,d,\mu)$ if for some $0\le g\in L_p(S)$ and some $C>0$, $\lambda\ge 1$, $$\not\mkern-7mu\int^\infty_B|u- u_B|d\mu\le Cr\Biggl(\not\mkern-7mu\int_{\lambda B} g^p d\mu\Biggr)^{1/p},$$ where $B$ is a ball of radius $r$, $u_B$ is the average, and $\not\mkern-7mu\int$ the average value of the integral (Poincaré inequality). The authors study the relations of these two possibilities. They apply their results to Sobolev spaces defined via vector fields of first-order differential operators.
[H.Triebel (Jena)]

MSC 2000:: *46E35 Sobolev spaces and generalizations

Keywords: Sobolev spaces; metric spaces; doubling measures; Carnot-Carathéodory spaces; Hörmander's rank condition; Poincaré inequality; doubling condition

Cited in: Zbl 1026.49029

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Article Article Journal Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]