Full capacity–volumetry of sharp exp-integrability law

Abstract

This paper uses law of trichotomy to show a full range of capacity–volumetry of the sharp $\exp$-integrability law which covers the sharp Adams–Moser–Trudinger $\exp$-integrability law for higher order derivatives, thereby finding a new approach to a relatively complete family of the essential capacity–volumetric estimates with the optimal constants including the sharp Ahlfors–Beurling–Pólya–Szegö and Morrey–Sobolev capacity–volumetric inequalities.