Weak $L^2$ bound of the lacunary Carleson operator for the non-linear Fourier transform

TL;DR

This paper establishes the weak $L^2$ boundedness of a lacunary maximal function associated with the nonlinear Fourier transform valued in $SU(1,1)$, under the condition that the potential is integrable, advancing understanding in harmonic analysis and nonlinear Fourier analysis.

Contribution

It proves the weak $L^2$ boundedness of a lacunary maximal function for the nonlinear Fourier transform with $L^1$ potential, a novel result in this area.

Findings

Weak $L^2$ boundedness of the lacunary maximal function is established.

Results apply to the $SU(1,1)$-valued nonlinear Fourier transform.

Potential in $L^1$ suffices for boundedness.

Abstract

We prove the weak $L^2$ boundedness of a lacunary maximal function of the $SU(1,1)$-valued nonlinear Fourier transform if the potential is in $L^1$.