The aim of this paper is to study Sobolev-type embeddings and their optimality. We work in the frame of rearrangement-invariant norms and unbounded domains. We establish the equivalence of a Sobolev embedding to the boundedness of a certain Hardy operator on some cone of positive functions. This Hardy operator is then used to provide optimal domain and range rearrangement-invariant norm in the ...

This paper deals with some extensions of Hardy-Hilbert’s inequality with the best constant factors by introducing two parameters λ and α and using the Beta function. The equivalent form and some reversions are considered.

In this paper, we give a new Hardy-Hilbert’s integral inequality with some parameters and a best constant factor. It includes an overwhelming majority of results of many papers.

We establish a generalization to Riemannian manifolds of the Caffarelli-KohnNirenberg inequality. The applied method is based on the use of conformal Killing vector fields and Enzo Mitidieri’s approach to Hardy inequalities. 2000 AMS Mathematics Classification numbers: 58E35, 26D10

New inequalities concerning functions of the form f (xy) similar to Hardy-Hilbert’s integral inequality are presented. A new class of functions denoted by (p, q)−-Hö(I) is defined. Many other new inequalities are also given.