
arXiv: 2003.11843
AbstractDunkl operators may be regarded as differential‐difference operators parameterized by finite reflection groups and multiplicity functions. In this paper, the Littlewood–Paley square function for Dunkl heat flows in is introduced by employing the full “gradient” induced by the corresponding carré du champ operator and then the boundedness is studied for all . For , we successfully adapt Stein's heat flows approach to overcome the difficulty caused by the difference part of the Dunkl operator and establish the boundedness, while for , we restrict to a particular case when the corresponding Weyl group is isomorphic to and apply a probabilistic method to prove the boundedness. In the latter case, the curvature‐dimension inequality for Dunkl operators in the sense of Bakry–Emery, which may be of independent interest, plays a crucial role. The results are dimension‐free.
Integral operators, curvature-dimension condition, Dunkl process, Dunkl heat flow, Singular and oscillatory integrals (Calderón-Zygmund, etc.), Probability (math.PR), Processes with independent increments; Lévy processes, Dunkl operator, FOS: Mathematics, Potential operators, Jump processes on general state spaces, Diffusion processes, Littlewood-Paley square function, Mathematics - Probability
Integral operators, curvature-dimension condition, Dunkl process, Dunkl heat flow, Singular and oscillatory integrals (Calderón-Zygmund, etc.), Probability (math.PR), Processes with independent increments; Lévy processes, Dunkl operator, FOS: Mathematics, Potential operators, Jump processes on general state spaces, Diffusion processes, Littlewood-Paley square function, Mathematics - Probability
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