Conductor Sobolev-type estimates and isocapacitary inequalities Joan CerdaJoaquim MartinPilar Silvestre 46E3028A12convexitylower estimatesSobolev spacesrearrangement invariant spacesSobolev-type inequalities In this paper we present an integral inequality connecting a function space (quasi-)norm of the gradient of a function to an integral of the corresponding capacity of the conductor between two level surfaces of the function, which extends the estimates obtained by V. Maz'ya and S. Costea, and sharp capacitary inequalities due to V. Maz'ya in the case of the Sobolev norm. The inequality, obtained under appropriate convexity conditions on the function space, gives a characterization of Sobolev type inequalities involving two measures, necessary and sufficient conditions for Sobolev isocapacitary type inequalities, and self-improvements for integrability of Lipschitz functions. Indiana University Mathematics Journal 2012 text pdf 10.1512/iumj.2012.61.4709 10.1512/iumj.2012.61.4709 en Indiana Univ. Math. J. 61 (2012) 1925 - 1947 state-of-the-art mathematics http://iumj.org/access/