Conformal invariants of hyperbolic planar domains

Abstract

We consider planar hyperbolic domains and conformally invariant functionals defined as sharp constants for Hardy type inequalities. We study relationships between these functionals and optimal constants in hyperbolic isoperimetric inequalities. The studied Hardy type inequalities involve weight functions depending on a hyperbolic radius of a domain and are conformally invariant. We prove that the positivity of Hardy constants is connected with existence of some hyperbolic isoperimetric inequalities of a special kind. We also prove a comparison theorem for Hardy constants with different numerical parameters and we study the relationships between the linear hyperbolic isoperimetric inequality in a domain and Euclidean maximum modulus of this domain. In the proofs, an essential role is played by characteristics of domains with uniformly perfect boundary. In addition, we generalize certain results from the papers J.L. Fernández, J.M. Rodríguez, “The exponent of convergence of Riemann surfaces, bass Riemann surfaces”, Ann. Acad. Sci. Fenn. Series A. I. Mathematica. 15, 165–183 (1990); V. Alvarez, D. Pestana, J.M. Rodríguez, “Isoperimetric inequalities in Riemann surfaces of infinite type”, Revista Matemática Iberoamericana, 15:2, 353–425 (1999).