We consider inequalities where integrals are defined in the sense of Choquet with respect to Hausdorff content. We study cases where continuously differentiable functions are defined on open, connected sets with so much regularity that there exists a pointwise estimate between the values of a function and its gradient under the maximal operator or the Riesz potential, at every point of the set. We show that certain Hardy inequalities and Poincare–Sobolev inequalities are valid in this context.