Poincaré-Sobolev Type Inequalities in a Ball and in -Sobolev Spaces

Abstract

Abstract We study the variations on a strange semi-continuity result and the function spaces between the bounded mean oscillations with the multipliers of the critical Sobolev spaces . We show the weak semi-continuity of the duality product in Sobolev spaces . We consider some new characterizations of Sobolev spaces and the degree- independent Sobolev extensions on locally uniform domains with a quantitative Sobolev inequality in bounded variation . We investigate the Sobolev inequalities with remainder terms and refined convex Sobolev inequalities . We discuss the Poincare Problem in -Sobolev spaces and the shape of extremal functions for the Poincare-Sobolev- . type inequalities in a ball