On Lipschitz spaces in the Dunkl setting -- semigroup approach

Let $\{P_t{\}}_{t>0}$ be the Dunkl-Poisson semigroup associated with a root system $R\subset {\mathbb{R}}^N$ and a multiplicity function $k\ge 0$. Analogously to the classical theory, we say that a bounded measurable function $f$ defined on ${\mathbb{R}}^N$ belongs to the inhomogeneous Lipschitz space ${\mathrm{\Lambda }}_k^{\beta }$, $\beta >0$, if $$\underset{t>0}{\sup }{t}^{m-\beta }\parallel \frac{d^m}{d{t}^m}{P}_{t}f{\parallel }_{L^{\mathrm{\infty }}}<\mathrm{\infty },$$ where $m=[\beta ]+1$. We prove that the spaces ${\mathrm{\Lambda }}_k^{\beta }$ coincide with the classical Lipschitz spaces.