Conjugate Transforms on Dyadic Group

Abstract

In this paper we study the properties of the Lebesgue constant of the conjugate transforms. For conjugate Fejer means we will find necessary and sufficient condition on $$t$$ for which the estimation $$E\left|\widetilde{\sigma}_{n}^{\left(t\right)}f\right|\leq cE\left|f\right|$$ holds. We also prove that for dyadic irrational $$t$$ , $$L\log L$$ is the maximal Orlicz space for which the estimation $$E\left|\widetilde{\sigma}_{n}^{\left(t\right)}f\right|\leq c_{1}+c_{2}E\left(\left|f\right|\log^{+}\left|f\right|\right)$$ is valid.