Littlewood- Paley operators Applied to the Dunkl operator on the Real line.

Abstract

We show a -Theorem for non-doubling measures and for the Cauchy integral. We study the elementary theorem of Paley- Wiener for the Dunkl transforms and the characterization of Besov spaces for the Dunkl operator on the real line. We discus the boundedness of Multilinear and bilinear littlewood- Paley operators and give sharp estimate for the bilinear operator and weighted inequalities for multilinear integral operators. We study the littlewood- Paley theory with non- doubling measures and the operators associated with the Dunkl operator on the real line. And also study the littlewood- Paley - function in the Dunkl analysis and give some remarks on the bilinear and weighted normed inequalities for littlewood- Paley theory and operators. Finally we characterized the weighted end point estimates for multilinear littlewood- Paley operators.