Optimal Monotonicity of $L^p$ Integral of Conformal Invariant Green Function

TL;DR

This paper establishes optimal monotonicity principles for the $L^p$ integral of Green functions in planar domains, deriving new inequalities and connecting them to classical geometric and analytic inequalities on Riemannian surfaces.

Contribution

It introduces a sharp one-dimensional power integral estimate and new isoperimetric inequalities, linking Green function integrals to geometric curvature conditions and classical inequalities.

Findings

New analytic and geometric isoperimetric inequalities discovered.

Optimal monotonicity principles established for Green function integrals.

Connections made between these principles and classical inequalities like Nash-Sobolev and Moser-Trudinger.

Abstract

Both analytic and geometric forms of an optimal monotone principle for $L^p$-integral of the Green function of a simply-connected planar domain $\Omega$ with rectifiable simple curve as boundary are established through a sharp one-dimensional power integral estimate of Riemann-Stieltjes type and the Huber analytic and geometric isoperimetric inequalities under finiteness of the positive part of total Gauss curvature of a conformal metric on $\Omega$. Consequently, new analytic and geometric isoperimetric-type inequalities are discovered. Furthermore, when applying the geometric principle to two-dimensional Riemannian manifolds, we find fortunately that $\{0,1\}$-form of the induced principle is midway between Moser-Trudinger's inequality and Nash-Sobolev's inequality on complete noncompact boundary-free surfaces, and yet equivalent to Nash-Sobolev's/Faber-Krahn's eigenvalue/Heat-kernel-upper-bound/Log-Sobolev's inequality on the surfaces with finite total Gauss curvature and quadratic area growth.