Isoperimetric inequalities for Bergman analytic content

Abstract

The Bergman $p$-analytic content ($1\le p<\infty$) of a planar domain $\Omega$ measures the $L^p(\Omega )$-distance between $\overline{z}$ and the Bergman space $A^p(\Omega )$ of holomorphic functions. It has a natural analogue in all dimensions which is formulated in terms of harmonic vector fields. This paper investigates isoperimetric inequalities for Bergman $p$-analytic content in terms of the St Venant functional for torsional rigidity, and addresses the cases of equality with the upper and lower bounds.

18 pages. To appear in Indiana University Mathematics Journal