Spectral estimates of the pp-Laplace Neumann operator in conformal regular domains

In this paper we study spectral estimates of the pp-Laplace Neumann operator in conformal regular domains Ω⊂R2Ω⊂R2. This study is based on (weighted) Poincaré–Sobolev inequalities. The main technical tool is the theory of composition operators in relation with the Brennan’s conjecture. We prove that if the Brennan’s conjecture holds for any p∈(4/3,2)p∈(4/3,2) and r∈(1,p/(2−p))r∈(1,p/(2−p)) then the weighted (r,p)(r,p)-Poincare–Sobolev inequality holds with the constant depending on the conformal geometry of ΩΩ. As a consequence we obtain classical Poincare–Sobolev inequalities and spectral estimates for the first nontrivial eigenvalue of the pp-Laplace Neumann operator for conformal regular domains.