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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

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