Characterization of Function Spaces for the Dunkl Type Operator on the Real Line

In this work we consider the Dunkl operator on the real line, defined by $$ {\cal D}_kf(x):=f'(x)+k\dfrac{f(x)-f(-x)}{x},\,\,k\geq0. $$ We define and study Dunkl–Sobolev spaces \(L^p_{n,k}(\mathbb{R})\) , Dunkl–Sobolev spaces \({\cal L}^p_{\alpha,k}(\mathbb{R})\) of positive fractional order and generalized Dunkl–Lipschitz spaces \(\wedge^k_{\alpha,p,q}(\mathbb{R})\) . We provide characterizations of these spaces and we give some connection between them.