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We study the Hardy inequality when the singularity is placed on the boundaryof a bounded domain in $\mathbb{R}^n$ that satisfies both an interior andexterior ball condition at the singularity. We obtain the sharp Hardy constant$n^2/4$ in case the exterior ball is large enough and show the necessity of thelarge exterior ball condition. We improve Hardy inequality with the bestconstant by adding a sharp Sobolev term. We next produce criteria that lead tocharacterizing maximal potentials that improve Hardy inequality. Breaking thecriteria one produces successive improvements with sharp constants. Ourapproach goes through in less regular domains, like cones. In the case of acone, contrary to the smooth case, the Sobolev constant does depend on theopening of the cone.

Sharp Hardy and Hardy--Sobolev inequalities with point singularities on the boundary