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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
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