On Lipschitz spaces in the Dunkl setting -- semigroup approach

TL;DR

This paper establishes that Lipschitz spaces defined via the Dunkl-Poisson semigroup are equivalent to classical Lipschitz spaces, extending classical harmonic analysis concepts to the Dunkl setting.

Contribution

It introduces and characterizes Lipschitz spaces in the Dunkl setting using semigroup methods, showing their equivalence to classical Lipschitz spaces.

Findings

Lipschitz spaces in the Dunkl setting coincide with classical Lipschitz spaces.

Semigroup approach effectively characterizes Lipschitz regularity.

Provides a framework for harmonic analysis in Dunkl theory.

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 $k\geq 0$. Analogously to the classical theory, we say that a bounded measurable function $f$ defined on $\mathbb R^N$ belongs to the inhomogeneous Lipschitz space $\Lambda_k^\beta$, $\beta>0$, if $$\sup_{t>0} t^{m-\beta} \Big\|\frac{d^m}{dt^m} P_tf\Big\|_{L^\infty}<\infty,$$ where $m=[\beta]+1$. We prove that the spaces $\Lambda^\beta_k$ coincide with the classical Lipschitz spaces.