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On the (\(1,p\))-Poincaré inequality. (English) Zbl 1287.46029

Summary: We show that \(s\)-John domains satisfy the (\(1,p\))-Poincaré inequality for all finite \(p>p_0\). We prove that the lower bound \(p_0\) is sharp. We formulate a conjecture concerning (\(q,p\))-Poincaré inequalities in \(s\)-John domains, \(1\leq q\leq p\).

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
26D10 Inequalities involving derivatives and differential and integral operators
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References:

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