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Table 1 The assumptions of p , q , and N imposed on Theorems 3.1 , 3.2 , 3.3 , and 3.4

From: Estimation of Sobolev embedding constant on a domain dividable into bounded convex domains

Theorem p q N
3.1 \(2< p\leq\frac{2N}{N-1}\) (N>1), 2<p<∞ (N = 1) \(q\geq\frac{p}{p-1}\) N ≥ 1
3.2 \(2< p\leq\frac{2N}{N-2}\) (N>2), 2<p<∞ (N = 2) q = 2 N ≥ 2
3.3 \(q\leq p<\frac{qN}{N-q}\) (N>q), q ≤ p<∞ (N = q) q ≥ 1 N ≥ q
3.4 p = ∞ q ≥ 1 N<q