A variant of a general inequality of the Hardy-Knopp type.
Let be a weight on . Assume that is continuous on . Let the operator be given at measurable non-negative function on by We characterize weights on for which there exists a positive constant such that the inequality holds for every . Such inequalities have been used in the study of optimal Sobolev embeddings and boundedness of certain operators on classical Lorenz spaces.
A closed subset of the real line which is right porous but is not -left-porous is constructed.
A non-homogeneous Hardy-like inequality has recently been found to be closely related to the knowledge of the lowest eigenvalue of a large class of Dirac operators in the gap of their continuous spectrum.