Multilinear Dunkl Multiplier Operators

In this paper, we explore the boundedness of multilinear Dunkl multipliers in three different cases. Firstly, we prove that under summation Hörmander conditions, multilinear Dunkl multipliers have \(L^p(dw)\) $L^p(dw)$ boundedness in the space \(L^{p_1}_{rad,>0}(dw)\times L^{p_2}_{rad,>0}(dw)\times \cdots \times L^{p_N}_{rad,>0}(dw)\) $L_{rad,>0}^{p_1}(dw)\times L_{rad,>0}^{p_2}(dw)\times \cdots \times L_{rad,>0}^{p_N}(dw)$ , where \(1\le p_i, p\le \infty \) $1\le p_i,p\le \infty$ , and \(\frac{1}{p_1}+\frac{1}{p_2}+\cdots +\frac{1}{p_N}=\frac{1}{p}\) $\frac{1}{p_1}+\frac{1}{p_2}+\cdots +\frac{1}{p_N}=\frac{1}{p}$ . Secondly, by proving the norm of a specific Carleson measure, we demonstrate that under restricted general Hörmander conditions, the \(L^1(dw)\) $L^1(dw)$ boundedness of multilinear Dunkl multipliers can be achieved in the space \(L^2_{rad,>0}(dw)\times L^2_{rad,>0}(dw)\times \cdots \times L^{\infty }_{rad, >0}\) $L_{rad,>0}^2(dw)\times L_{rad,>0}^2(dw)\times \cdots \times L_{rad,>0}^{\infty }$ . Finally, by proving the Littlewood-Paley inequality under Dunkl transforms, we further establish that under general Hörmander conditions, the \(L^p(dw)\) $L^p(dw)$ boundedness of multilinear Dunkl multipliers can be attained in the space \(L^{p_1}_{rad}(dw)\times L^{p_2}_{rad}(dw)\times \cdots \times L^{p_N}_{rad}(dw)\) $L_{\mathit{rad}}^{p_1}(dw)\times L_{\mathit{rad}}^{p_2}(dw)\times \cdots \times L_{\mathit{rad}}^{p_N}(dw)$ , where \(2<p_1,..., p_N<\infty \) $2<p_1,...,p_N<\infty$ , \(1<p<\infty \) $1<p<\infty$ , and \(\frac{1}{p_1}+\frac{1}{p_2}+\cdots +\frac{1}{p_N}=\frac{1}{p}\) $\frac{1}{p_1}+\frac{1}{p_2}+\cdots +\frac{1}{p_N}=\frac{1}{p}$ .