ON q-DEFORMATED HORMANDER MULTIPLIER THEOREM

Abstract

The main purposes of this work, we introduce the q-deformed Fourier multiplier A_g defined on the space L²_q(Rq) through the framework of the q²-Fourier transform, while also extending the functional setting of L^p_q(Rq) with 1 ≤ p < ∞. Our approach provides a natural extension of classical Fourier multiplier theory into the q-deformed setting, which is relevant in the context of quantum groups and noncommutative analysis. Furthermore, we establish several key q-analogues of classical harmonic analysis inequalities for the q²-Fourier transform, including the Paley inequality, Hausdorff-Young inequality, Hausdorff-Young-Paley inequality, and Hardy-Littlewood inequality. These results not only generalize their classical counterparts but also open new avenues for analysis on q-deformed spaces. As a significant application, we prove a q-deformed version of the Ho¨rmander multiplier theorem, which provides sufficient conditions for the boundedness of multipliers in the q-deformed setting. This work sets the stage for further developments in the field of q-deformed harmonic analysis.