Boundedness of the pseudo-differential operators generated by 1D-Dunkl operator

This article is devoted to the study of pseudo-differential operators generated by Dunkl operators, focusing primarily on their boundedness properties. We establish that, under a set of suitable assumptions on the symbols and the underlying function spaces, these operators are bounded on specific Banach spaces. In addition, we define the composition of pseudo-differential operators generated by Dunkl operators and rigorously prove that this composition also inherits boundedness properties under appropriate conditions. The analysis is carried out using techniques based on the Dunkl transform, which provides a powerful tool for handling operators associated with reflection groups and allows for the derivation of precise estimates. Beyond the theoretical development, we illustrate an application of the obtained results, demonstrating how these boundedness properties can be employed to address complex problems in mathematical physics and harmonic analysis. Overall, the work contributes to a deeper understanding of Dunkl analysis and the structure of pseudo-differential operators in this context. The results presented not only consolidate existing knowledge but also open new perspectives for further investigations in the field, providing a solid foundation for future research on Dunkl operators and their applications in both theoretical and applied analysis.