The strong version of nonlinear Carleson conjecture fails

TL;DR

This paper demonstrates the failure of the strong nonlinear Carleson conjecture for Dirac equations by analyzing the unboundedness of certain maximal functions related to transmission coefficients.

Contribution

It provides a proof that the strong version of the nonlinear Carleson conjecture does not hold for Dirac equations and Krein systems.

Findings

Maximal function associated with the transmission coefficient argument is unbounded.

The strong nonlinear Carleson conjecture fails for Dirac equations.

Analysis applies to Krein systems as well.

Abstract

In the context of the Dirac equation with square-summable potential, we study the Jost solutions and prove that the maximal function associated with the argument of the transmission coefficient is unbounded. We also show that the strong version of the nonlinear Carleson conjecture fails for Dirac equations and Krein systems.