On Lipschitz spaces in the Dunkl setting -- semigroup approach

Abstract

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 $\hat{I}{›}_k^2$, ${\hat{I}}^2>0$, if $$\underset{t>0}{\sup }{t}^{m-{\hat{I}}^2}\Vert \frac{d^m}{d{t}^m}{P}_{t}f{\Vert }_{L^{\infty }}<\infty ,$$ where $m=[{\hat{I}}^2]+1$. We prove that the spaces $\hat{I}{›}_k^2$ coincide with the classical Lipschitz spaces.

32 pages