Bounds on the Walsh model for M^{q,*} Carleson and related operators

TL;DR

This paper extends the Walsh-analog of the Carleson-Hunt theorem by replacing the maximal operator norm with an $L^q$ norm and explores related variation-norm estimates, advancing understanding of Walsh model operators.

Contribution

It introduces bounds for Walsh model operators using $L^q$ norms, generalizing classical maximal operator results and analyzing variation-norm estimates.

Findings

Extended Walsh-analog of Carleson-Hunt theorem with $L^q$ norm

Established bounds for Walsh operators in $L^{q,*}$ spaces

Derived variation-norm estimates for related operators

Abstract

We prove an extension of the Walsh-analog of the Carleson-Hunt theorem, where the $L^\infty$ norm defining the Carleson maximal operator has been replaced by an $L^q$ maximal-multiplier-norm. Additionally, we consider certain associated variation-norm estimates.