Language:   Search:   Contact
Zentralblatt MATH has released its new interface!
For an improved author identification, see the new author database of ZBMATH.

Query:
Fill in the form and click »Search«...
Format:
Display: entries per page entries
Zbl 1129.46026
Heikkinen, Toni; Koskela, Pekka; Tuominen, Heli
Sobolev-type spaces from generalized Poincaré inequalities.
(English)
[J] Stud. Math. 181, No. 1, 1-16 (2007). ISSN 0039-3223; ISSN 1730-6337/e

The Poincaré inequality for the classical Sobolev spaces $W^1_p (B)$ in a ball $B$ in $\Bbb R^n$ is given by $$\frac{1}{\vert B\vert } \int_B \vert u-u_B \vert \, dx \le C r_B \left( \frac{1}{\vert B\vert } \, \int_B \vert \nabla u(x) \vert ^pc \,dx \right)^{1/p}, \quad 1\le p < \infty,$$ where $u_B$ is the mean value and $r_B$ the radius of the ball $B$. Let $(X,d,\mu)$ be a metric space with doubling measure $\mu$. A substitute of the gradient $g = \vert \nabla u\vert$ is given by a function $g: \, X \to [0,\infty]$ with $$\vert u(\gamma (0)) - u(\gamma (l))\vert \le \int_\gamma g \, ds$$ for all rectifiable curves $\gamma: [0,l] \to X$. The paper studies what happens if one replaces $\vert \nabla u\vert$ in the Poincaré inequality by such an upper gradient $g$.
[Hans Triebel (Jena)]
MSC 2000:
*46E35 Sobolev spaces and generalizations
46E30 Spaces of measurable functions
26D10 Inequalities involving derivatives, diff. and integral operators

Keywords: Sobolev spaces; Poincaré inequality; upper gradient

Highlights
Master Server