Optimal Sobolev inequalities in the hyperbolic space
Abstract
We find the optimal function norm on the left-hand side of the mth order Sobolev type inequality \|u\|Y(Hn) ≤ C \|∇gm u\|X(Hn) in the n-dimensional hyperbolic space Hn, 1≤ m < n. The optimal function norm in the inequality among all rearrangement-invariant function norms is completely characterized. A variety of concrete examples of optimal function norms is provided. The examples include delicate limiting cases, and especially when m≥3, seem to provide new, improved inequalities in these limiting cases.
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