Universal Hardy–Sobolev inequalities on hypersurfaces of Euclidean space

In this paper, we study Hardy–Sobolev inequalities on hypersurfaces of [Formula: see text], all them involving a mean curvature term and having universal constants independent the hypersurface. We first consider celebrated Sobolev inequality Michael–Simon Allard, in our codimension one framework. Using their ideas, but simplifying presentations, give quick easy-to-read proof inequality. Next, establish two new Hardy hypersurfaces. One originates from an application to regularity theory stable solutions semilinear elliptic equations. The other one, which prove by exploiting “ground state” substitution, improves Carron. With same method, also obtain improved or Hardy–Poincaré