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We prove a weighted version of the Hardy-Littlewood-Sobolev inequality for radially symmetric functions, and show that the range of admissible power weights appearing in the classical inequality due to Stein and Weiss can be improved in this particular case.2000 Mathematics Subject Classification. 26D10, 47G10, 31B10.

We consider the approximation of Poisson type problems where the source is given by a singular measure and the domain is a convex polygonal or polyhedral domain. First, we prove the well-posedness of the Poisson problem when the source belongs to the dual of a weighted Sobolev space where the weight belongs to the Muckenhoupt class. Second, we prove the stability in weighted norms for standard finite element approximations under the quasi-uniformity assumption on the family of meshes.2010 Mathematics Subject Classification. Primary: 65N30; Secondary: 65N15, 35B45.

We obtain improved fractional Poincaré and Sobolev Poincaré inequalities including powers of the distance to the boundary in John, s-John domains and Hölder-α domains, and discuss their optimality.2010 Mathematics Subject Classification. Primary: 26D10; Secondary: 46E35.