Conformally invariant integral inequalities

Using the Poincaré metric we define conformally invariant integrals for the gradient and the Laplacian of smooth test functions defined on hyperbolic type domains of the extended complex plane.
For these integrals we consider Hardy and Rellich type integral inequalities that contain certain domain constants. The main problem is to obtain explicit estimates of these domain constants using some numerical characteristics of hyperbolic type domains. In the talk we will describe several known results, including our results, as well as some new facts.