Decay of extremals of Morrey’s inequality

Abstract

We study the decay (at infinity) of extremals of Morrey's inequality in R n . These are functions satisfying supwhere p>n and C(p, n) is the optimal constant in Morrey's inequality. We prove that if n≥2 then any extremal has a power decay of order β for any, for all functions whose first order partial derivatives belong to L p (R n ). In a series of papers (cf.[7]-[9]), Hynd and Seuffert study this inequality and prove that there is a smallest constant C >0 such that (1.1) holds and that there are extremals of this inequality. An extremal is a function for which equality is attained in (1.1). They also prove that up to translation, rotation, dilatation and multiplication by a constant, any extremal function u satisfies 1. −Δ p u=c(δ en −δ −en ) in R n for a constant c>0,