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Sobolev-type spaces from generalized Poincaré inequalities. (English) Zbl 1129.46026
The Poincaré inequality for the classical Sobolev spaces $$W^1_p (B)$$ in a ball $$B$$ in $$\mathbb R^n$$ is given by $\frac{1}{| B| } \int_B | u-u_B | \, dx \leq C r_B \left( \frac{1}{| B| } \, \int_B | \nabla u(x) | ^pc \,dx \right)^{1/p}, \quad 1\leq p < \infty,$ 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 $$g = | \nabla u|$$ is given by a function $$g: \, X \to [0,\infty]$$ with $| u(\gamma (0)) - u(\gamma (l))| \leq \int_\gamma g \, ds$ for all rectifiable curves $$\gamma: [0,l] \to X$$. The paper studies what happens if one replaces $$| \nabla u|$$ in the Poincaré inequality by such an upper gradient $$g$$.

##### MSC:
 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 26D10 Inequalities involving derivatives and differential and integral operators
##### Keywords:
Sobolev spaces; Poincaré inequality; upper gradient
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