On a sharp estimate for Hankel operators and Putnam's inequality

TL;DR

This paper provides a sharp estimate for Hankel operators on weighted Bergman spaces, improving Putnam's inequality for classical Bergman spaces and applying it to the de St. Venant inequality, connecting operator theory with geometric analysis.

Contribution

It introduces a sharper norm estimate for Hankel operators with anti-analytic symbols, resolving a recent conjecture and enhancing classical inequalities in operator theory.

Findings

Improved the classical Putnam inequality by a factor of 1/2 for Bergman spaces.

Provided a new proof of the de St. Venant inequality using operator estimates.

Established a sharp norm estimate for Hankel operators with anti-analytic symbols.

Abstract

We obtain a sharp norm estimate for Hankel operators with anti-analytic symbol for weighted Bergman spaces. For the classical Bergman space, the estimate improves the corresponding classical Putnam inequality for commutators of Toeplitz operators with analytic symbol by a factor of $1/2$, answering a recent conjecture by Bell, Ferguson and Lundberg. As an application, this yields a new proof of the de St. Venant inequality, which relates the torsional rigidity of a domain with its area.