Decay of extremals of Morrey's inequality

Abstract

We study the decay (at infinity) of extremals of Morrey's inequality in ${\mathbb{R}}^n$. These are functions satisfying $$\underset{x\ne y}{\sup }\frac{|u(x)-u(y)|}{|x-y{|}^{1-\frac{n}{p}}}=C(p,n)\Vert \nabla u{\Vert }_{L^p({\mathbb{R}}^n)},$$ where $p>n$ and $C(p,n)$ is the optimal constant in Morrey's inequality. We prove that if $n\ge 2$ then any extremal has a power decay of order $\beta$ for any $$\beta <-\frac{1}{3}+\frac{2}{3(p-1)}+\sqrt{{\left(-\frac{1}{3}+\frac{2}{3(p-1)}\right)}^2+\frac{1}{3}}.$$