Optimal Sobolev inequalities in the hyperbolic space

2023/05/11 by Mihula, Zdeněk
#46E30 #46E35 (Primary) 26D10 (Secondary) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2305.06797

Abstract

We find the optimal function norm on the left-hand side of the mth order Sobolev type inequality ‖u‖_Y(ℍⁿ) ≤ C ‖∇_g^m 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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