Integral inequalities invariant under conformal transformations

Employing the Poincaré metric, we introduce conformally invariant integrals for smooth compactly supported functions defined on domains of hyperbolic type in the extended plane. For these integrals, which involve the hyperbolic radius, a smooth function, and its gradient or Laplacian, we consider conformally invariant analogues of Hardy and Rellich type inequalities with constants depending on the domain. We provide explicit estimates for the constants using numerical characteristics, namely, the maximal moduluses of the domain and a geometric constant involved in the linear hyperbolic isoperimetric inequality.
In the paper we prove several new statements. In particular, we justify a criterion for the positivity of constants for finitely–connected domains of hyperbolic type and prove several integral inequalities universal in the sense that these inequalities involve no unknown constants and are valid in each domain of hyperbolic type.
In the beginning of the paper, we briefly outline the properties of hyperbolic radius and describe several related. In particular, we mention the results by Schmidt, Osserman, Fernández, and Rodríguez on hyperbolic isoperimetric inequalities and their applications, provide the Elstrodt — Patterson — Sullivan formula for the critical exponents of convergence of the Poincaré — Dirichlet series, and present a result by Carleson and Gamelin on the maximal moduli of a domain with uniformly perfect boundary.