Exotic multiplicity functions and heat maximal operators in certain Dunkl settings.

We consider settings of the Dunkl Laplacian, the Dunkl harmonic oscillator, and the Dunkl Ornstein-Uhlenbeck operator with the underlying group of reflections isomorphic to <inline-graphic></inline-graphic>. In each of these contexts, we admit all real-valued multiplicity functions, not necessarily bounded from below, and construct the corresponding ‘exotic’ transform or orthogonal system. This leads to new Dunkl operator-based frameworks, which generalize those yet well known, and in which harmonic analysis can reasonably be developed. To support the last claim, in all the cases we study the associated heat semigroup maximal operators and prove that they are bounded on <inline-graphic></inline-graphic>, p>1, and from <inline-graphic></inline-graphic> to weak <inline-graphic></inline-graphic>. [ABSTRACT FROM AUTHOR]