Recovering Product BMO from Schatten Hankel operators

TL;DR

This paper proves that small Schatten class Hankel operators on product Hardy spaces have symbols in product BMO, extending Nehari's theorem to Schatten class operators.

Contribution

It establishes that Schatten class membership implies the existence of a product BMO symbol, generalizing classical results to a Schatten class setting.

Findings

Schatten class Hankel operators imply product BMO symbols.

Extension of Nehari's theorem to Schatten class operators.

Provides conditions under which Hankel operators have BMO symbols.

Abstract

We prove that if a small Hankel operator on the product Hardy space belongs to some Schatten class $S^p$, $p < \infty$, then it has a symbol in product BMO. In other words, the conclusion of Nehari's theorem holds under the hypothesis that the operator belongs to a Schatten class.