turkmath.org

Probability Theory presents tools to study singular integral operators and analytically difficult problems by means of stochastic processes. One such problem is to determine a general class of multipliers and so the bounded operators on function spaces. In this talk we will use a discontinuous process, namely a symmetric stable process, to show boundedness results of extended versions of classical singular integral operators. We will define Littlewood-Paley operators arising from this process and discuss the corresponding multipliers which are studied in [2]. We will introduce versions of intermediate operators appearing in the Littlewood-Paley Theory and show our recent boundedness results in [1]. Finally we will discuss the relation between these new operators and fractional derivative in its integral form and discuss multipliers defined in terms of fractional derivatives.