Integral operators in weighted zygmund classes

This thesis consists of two chapters. The first chapter devoted to the fundamentalconcepts and main theorems. In the second chapter, the classical theorems on the maximaloperator in the Zygmund class first of are given. It is shown that these theorems also hold forthe Hardy-Littlewood-Wiener maximal function. In order to make comparation with theCalderon-Zygmund decomposition of a dyadic cube a Calderon-Zygmund decomposition of aarbitrary cube in R n are given.In the second chapter, we characterized the weights for which the weighted inequalityof the Zygmund type for the Hilbert transform hold. For the Hilbert transformation the (1.1)weak type inequality with one weight holds if and only if ρ ∈ A1 . It follows from the ?if part?of this theorem that the A1 condition is sufficient for the weighted analogue of Zygmund?sinequality, on the other hand, it turns out that A1 is not necessary. The necessary and sufficientconditions of A is established. For the strong maximal function the sufficiency conditions aregiven by Bagby and Kurtz [2]. Considering Gogatishvili [8], in finally section, for the strongmaximal function a necessary and sufficient conditions are given.Keywords: Weighted Zygmund Classes, Integral Operators, Maximal Functions.