Conjugate Transforms on Dyadic Group

TL;DR

This paper investigates the Lebesgue constants of conjugate transforms on dyadic groups, establishing conditions for boundedness and maximal Orlicz spaces for irrational parameters.

Contribution

It provides necessary and sufficient conditions for boundedness of conjugate Fejér means and identifies the maximal Orlicz space for dyadic irrational parameters.

Findings

Necessary and sufficient condition on t for bounded conjugate Fejér means.

For dyadic irrational t, L log L is the maximal Orlicz space for certain estimates.

Establishment of bounds for conjugate transforms in specific function spaces.

Abstract

In this paper we study the properties of the Lebesgue constant of the conjugate transforms. For conjugate Fej\'er means we will find necessary and sufficient condition on $t$ for which the estimation $E\left\vert \widetilde{% \sigma }_{n}^{\left( t\right) }f\right\vert \lesssim E\left\vert f\right\vert $ holds . We also prove that for dyadic irrational $t$, $L\log L $ is maximal Orlicz space for which the estimation $E\left\vert \widetilde{% \sigma }_{n}^{\left( t\right) }f\right\vert \lesssim 1+E\left( \left\vert f\right\vert \log ^{+}\left\vert f\right\vert \right) $ is valid.